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Journal of Algebra 539 (2019) 377–396
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Journal of Algebra www.elsevier.com/locate/jalgebra
K-automorphisms of a weak-crossed product F -algebra over a Galois extension K/F Jayampathy Ratnayake Department of Mathematics, University of Colombo, Colombo 3, Sri Lanka
a r t i c l e
i n f o
Article history: Received 4 October 2018 Available online 22 August 2019 Communicated by Louis Rowen Keywords: Weak crossed products Lower subtractive partial order Weak Bruhat order
a b s t r a c t We compute the K-automorphism group of AK = K ⊗F Af of a weak crossed product algebra Af for a weak 2-cocycle f over a Galois extension K/F with Galois group G. The K-automorphism group of AK decomposes into its unipotent ˆ Automorphisms in H ˆ are part and the reductive part H. computed via their restriction to semi-simple part of AK . ˆ and the lowerThere is a strong relationship between H subtractive relation (≤) induced by f on G. We introduce ˆ namely Λ, which contains interesting a subgroup of H, combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lowersubtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Throughout this paper, F ⊆ K is a Galois extension of fields with Galois group G. We will write 1G for the identity element of G. A weak 2-cocycle is a function f : G ×G → K, E-mail address: [email protected]. https://doi.org/10.1016/j.jalgebra.2019.08.011 0021-8693/© 2019 Elsevier Inc. All rights reserved.
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which satisfies the 2-cocycle condition f (g, h)f (gh, k) = f g (h, k)f (g, hk) for all g, h, k ∈ G, written without the inverses. The important distinction between a weak 2-cocycle and a classical 2-cocycle is that a weak 2-cocycle can take the value 0 ∈ K. Two weak 2-cocycles f1 and f2 are cohomologous, written f1 ∼ f2 , if there is a function α : G → K × α(g)α(h)g f2 (g, h). We will denote the set of equivalence classes of such that f1 (g, h) = α(gh) weak 2-cocycles, with respect to this equivalence relation, by M 2 (G, K). The pointwise product of two weak 2-cocycles is again a weak 2-cocycle, which one can verify easily. This product induces a product in M 2 (G, K), which gives it a structure of a monoid. We call M 2 (G, K) the Brauer Monoid of K/F . The Brauer monoid is a disjoint union of abelian groups, indexed by the set of idempotents, where the identity element in each component corresponds to an idempotent in the monoid. Weak 2-cocycles that corresponds to idempotents of the Brauer monoid are the idempotent cocycles, i.e. the cocycles that take only the values 0 or 1 in K. Given a weak idempotent 2-cocycle e, the abelian subgroup determined by e is denoted by Me2 (G, K). Associated to a weak 2-cocycle f there is a relation on G, determined by g ≤f h iff f (g, g −1 h) = 0 (see [1]). This relation satisfies the following left lower subtractive property: if g ≤f h then g ≤f x ≤f h ⇐⇒ g −1 x ≤f g −1 h (right lower subtractive property can be defined similarly). The Inertial subgroup of f , Hf , is the subgroup of G defined by Hf = {g ∈ G : g ≤f 1G } = {g ∈ G : f (g, g −1 ) = 0}. The relation ≤f induces a lower subtractive partial order on the cosets G/Hf in the obvious way: g1 H ≤f g2 H ⇐⇒ g1 ≤ g2 . It was shown in [2] that the partial order on G/H is well-defined. For two weak 2-cocycles f and g, ≤f =≤g if and only if f and g are in the same component Me2 (G, K) for some idempotent e. Moreover, for every lower subtractive relation ≤ on G with the property that 1G ≤ g for all g in G, there is a corresponding idempotent weak 2-cocycle e, given by e(g1 , g2 ) = 1 ⇔ g1 ≤ g1 g2 , such that ≤e =≤. The lower subtractive property guarantees that it is a 2-cocycle. Thus, idempotent weak 2-cocycles are naturally in a one-to-one correspondence with the set of lower subtractive relations on G/H, where H varies over all subgroups of G, and with the idempotents of M 2 (G, K) (see [2] for more details on this). Weak Bruhat orders (see [3]) are important examples of lower subtractive partial orders. A more general example, which includes weak Bruhat orders, is when a group G is given with a generating set S. In this case there is a natural lower subtractive relation, which is also a partial order, on G given by the minimal words in S as follows. For x, y ∈ G, we define x ≤ y if there is a minimal word for y starting with a word (necessarily minimal) for x. This is called the standard graph of G with respect to the generating set S and we invite the reader to refer [4] for more details. Every weak 2-cocycle is cohomologous to a normalized weak 2-cocycle f satisfying the condition f (g, 1G ) = f (1G , g) = 1 for all g ∈ G. As in the classical case, such a weak 2-cocycle gives rise to a crossed product algebra Af = g∈G Kxg , where the
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multiplication is given by axg1 bxg2 = ag1 (b)f (g1 , g2 )xg1 g2 for all a, b ∈ K and g1 , g2 ∈ G. We will denote this algebra, associated to a normalized weak 2-cocycle f , by Af . Weak 2-cocycles and weak crossed products were first introduced and studied in [2]. We recall some basic properties of Af bellow and invite readers to refer [2,4–6] for further details on weak 2-cocycles, weak crossed products and weakly Azumaya algebras. The center of Af is F and hence it is an F -algebra, but in general it is not simple. The nilpotent radical of a weak crossed product algebra is zero if and only if the associated weak 2-cocycle is a classical 2-cocycle. Modulo the radical, weak crossed product algebras are central simple algebras over an intermediate extension of K/F . The equivalence relation on weak 2-cocycles given by 1-coboundaries is such that f ∼ f if and only if there is an F -algebra isomorphism α : Af → Af which is identity on K. Specifically, Af = Bf Jf where Bf = h∈Hf Kxh is the semisimple part of Af and Jf = h∈H / f Kxh is the Jacobson radical of Af consisting of nilpotent elements. Thus Af = B ⊕ J is a Wedderburn-Malcev decomposition. Moreover, Bf is a classical crossed product algebra with respect to the cocycle f |Hf ×Hf , f restricted to Hf × Hf and the center of Bf is L = K Hf . The idempotent cocycles are useful for two main reasons. First, they have a role in weakly Azumaya algebras which is similar to the role matrix algebras have in central simple algebras. It is a well-known fact that every central simple F algebra is a form of a matrix algebra. Even though it is not true that every weakly Azumaya algebra is a form of an algebra associated to a weak idempotent cocycle, one would like to know up to what extent a similar result holds for weakly Azumaya algebras. The second reason is that they present a way to study lower subtractive relations on G, such as the weak Bruhat order of a Coxeter Group (W, S), which are interesting combinatorial objects in their own right. In this paper, we compute the K-automorphism group of K ⊗F Af . We were mainly motivated towards this study because it lays the foundation to study the K-forms of a weak crossed product algebra Af , via Descent Theory, by finding H 1 (G, AutK (AK )), which we wish to present in a sequel to this paper. In Section 2 of this paper we will compute the K-automorphism group of K ⊗ Af , where Af is a weak crossed product ˜ ((13) of Definition 3), two interesting algebra. We introduce Γ (Definition 2) and Λ subgroups that carry combinatorial information about the lower subtractive relation. In Section 3 we investigate the automorphism group of an idempotent cocycle, with the help of the associated lower subtractive partial order on G/Hf . We exhibit a duality on the lower subtractive partial orders and exhibit how it is relevant to the study of automorphisms. Finally, in Section 4 we give some examples including the automorphism group of the algebra associated to the Weak-Bruhat order of a Coxeter system (Example 6 and 7) and show it is related to the automorphisms of Coxeter diagrams.
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2. K-automorphisms of K
F
Af
Let Af = g∈G Kxg be the weak crossed product algebra with respect to the normalized weak 2-cocycle f : G × G → K and Hf the inertial subgroup of f . Set n = |G|. All tensor products, unless otherwise stated, are over the field F . Consider K as an F -algebra. Then K ⊗ K is a K-algebra where the K-vector space structure comes from the first factor. Thus, for k ∈ K and x ⊗ y ∈ K ⊗ K, k.(x ⊗ y) = kx ⊗ y. The map α : K ⊗ K → K n where α(a ⊗ b) = (aσ(b))σ∈G is an isomorphism of K-algebras, where K n = {(aσ )σ∈G : aσ ∈ K} is the set of n-tuples indexed by the elements of G with the diagonal K-action. Lemma 1. AK = K⊗Af ∼ =K g∈G K n xg , where the K-algebra structure on g∈G K n xg is given by xg (aσ )σ∈G = (aσg )σ∈G xg and xg1 xg2 = (f σ (g1 , g2 ))σ∈G xg1 g2 In particular xg eτ = eτ g−1 xg , where eτ = (δτ,σ )σ∈G is the idempotent in K n with 1 in the τ -place and zeros in the other co-ordinates. n Proof. The map λ : K ⊗ Af → (K ⊗ K)xg = K xg given by λ(p ⊗ qxg ) = (pσ(q))σ∈G xg is the required K-algebra isomorphism, which we leave for the reader to verify. 2 n xg . Note that BK = Throughout this paper we will identify AK with g∈G K n n ∼ ∼ K x K ⊗B is the semisimple part of A and J = = h K K K h∈Hf g ∈H / f K xg =K K ⊗J is the Jacobson radical of AK . The decomposition AK = BK ⊕ JK is a WedderburnMalcev decomposition of AK . The following lemma is a consequence of AK being a weakly Azumaya algebra with respect to BK (see Theorem 2.1 of [7]), which can also be directly verified.
Lemma 2. The centralizer of BK in AK is the center of BK . The following lemma follows from the above. Lemma 3. If conjugation by 1 + j ∈ 1 + JK preserves BK then j = 0. Proof. If (1 + j)BK (1 + j)−1 = BK , then for any b ∈ BK there is some b ∈ BK such that b + jb = (1 + j)b = b (1 + j) = b + b j. Thus, b = b and jb = b j. Therefore, j commutes with all the elements in BK . From the previous lemma, j ∈ BK . Therefore, j = 0. 2 The following definitions were motivated by [8].
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Definition 1. ˆ = {ϕ ∈ AutK (AK ) : ϕ(BK ) ⊆ BK }, be the subgroup of automorphisms in 1. Let H AutK (AK ) which preserve BK . ˆ consisting of automorphisms ˆ : ϕ|B = idB }, be the subgroup of H 2. Let H = {ϕ ∈ H K K which restrict to the identity on BK . ˆ → AutK (BK ) Thus, H = AutBK (AK ), and it is the kernel of the homomorphism r : H ˆ to ϕ|B , the restriction of ϕ to BK . Hence H is a normal subgroup which sends ϕ ∈ H K ˆ of H and there is a short exact sequence of groups as follows. r
ˆ− 1→H→H → Im(r) → 1
(1)
For any K-automorphism ϕ of AK , ϕ(BK ) gives another Wedderburn-Malcev decomposition of AK , AK = ϕ(BK ) + JK . Because BK is uniquely determined up to a conjugation by an element in 1 + J(AK ) ([9] Theorem 11.6), there is an element 1 + j ∈ 1 + JK such that BK = (1 + j)−1 ϕ(BK )(1 + j). Furthermore, if 1 + j ∈ 1 + JK is another element such that BK = (1 + j )−1 ϕ(BK )(1 + j ), then conjugation by (1 + j)(1 + j )−1 ∈ 1 + JK takes BK to itself. Then it follows from Lemma 3 that 1 + j = 1 + j . Let Inn∗ (AK ) be the subgroup of inner automorphisms of AK determined by elements in 1 + JK . By the observations above we see that Inn∗ (AK ) ∼ = 1 + JK . Furthermore, up to a unique inner automorphism given by an element in 1 + JK , every K-automorphism of AK preserves BK . Thus, we have the following proposition. Proposition 1. Inn∗ (AK ) ∼ = 1 +JK is a normal subgroup of AutK (AK ) and AutK (AK ) = ˆ In particular Inn∗ (AK ) H. ˆ→1 1 → Inn∗ (AK ) → AutK (AK ) → H
(2)
is a split exact sequence of groups. The rest of this section (as well as of this paper) is devoted to understanding the ˆ nature of H. When the inertial subgroup is the entire Galois group G, f is a classical cocycle, ˆ∼ AK ∼ = P GLn (K). The simplest non-classical situation is = Mn (K) and AutK (AK ) = H when f has a trivial inertial group, in which case BK = K n . Even in the general case, knowing the structure of K-automorphisms of AK taking K n to itself turns out to be useful. Every K-automorphism of K n is determined by the permutation it induces on the set of minimal idempotents {eτ = (δτ,σ )σ∈G : τ ∈ G} of K n , where, δτ,σ = 1 if τ = σ and 0 otherwise. Since each minimal idempotent is identified by an element τ in G, we have AutK (K n ) ∼ = P erm(G) = Sn . We use this identification throughout.
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Let ϕ ∈ AutK (AK ) be a K-automorphism which takes K n to K n . Then, ϕ|K n : K n → K n is a K-automorphism of K n . We will denote the permutation of G obtained by restricting ϕ to K n by the same letter, ϕ. With this terminology we may write ϕ(eτ ) = eϕ(τ ) , where on the right hand side ϕ refers to the permutation induced by ϕ on G. The following proposition identifies the structure of K-automorphisms of AK that preserve K n . Proposition 2. Every ϕ ∈ AutK (AK ) that preserve K n is of the form ϕ (eτ xg ) = aτ,g eϕ(τ ) xϕ(τ )−1 ϕ(τ g)
(3)
where aτ,g ∈ K × and ϕ ∈ SG = P erm(G) satisfy the following conditions for all τ, g, g ∈ G. g ≤f g ⇐⇒ ϕ(τ )−1 ϕ(τ g) ≤f ϕ(τ )−1 ϕ(τ g )
aτ,g aτ g,g f ϕ(τ ) (ϕ(τ )−1 ϕ(τ g), ϕ(τ g)−1 ϕ(τ gg )) = aτ,gg f τ (g, g )
(4) (5)
Note that ϕ in the right hand side of (3) is r(ϕ) = ϕ|K n considered as a permutation of idempotents. Conversely, if ϕ ∈ SG satisfies condition (4) and we choose elements aτ,g ∈ K × satisfying (5) for all τ, g ∈ G, then the map ϕ obtained by extending formula (3) linearly is a K-automorphism of AK preserving K n . g g n Proof. Suppose ϕ(xg ) = h∈G ah xh where, ah ∈ K . Then, one can easily verify τ,g τ,g that ϕ(eτ xg ) = is the ϕ(τ )-coordinate of ahg . Because h∈G ah eϕ(τ ) xh , where ah ϕ is a K-algebra homomorphism ϕ(eτ xg eτ xg ) = ϕ(eτ xg )ϕ(eτ xg ). On the other τ,g σ −1 hand, (g, g ))σ∈G xgg and thus we have h∈G ah eϕ(τ ) xh × eτ xg eτ xg = eτ eτ g (f τ ,g σ h ∈G ah eϕ(τ ) xh = ϕ eτ eτ g −1 (f (g, g ))σ∈G xgg . τ,g In particular, when g = 1G (i.e. xg = 1) and simplifying we have ah eϕ(τ ) × τ,g eϕ(τ )h−1 xh = ah eϕ(τ ) eϕ(τ g−1 ) xh . The unique h ∈ G for which aτ,g = 0 is given by h −1 h = ϕ(τ ) ϕ(τ g). This obtains (3). Using (3) one can easily get that for all τ , g, g in G,
aτ,g aτ g,g f ϕ(τ ) (ϕ(τ )−1 ϕ(τ g), ϕ(τ g)−1 ϕ(τ gg )) = aτ,gg f τ (g, g ). In particular, we have g ≤f gg ⇐⇒ ϕ(τ )−1 ϕ(τ g) ≤f ϕ(τ )−1 ϕ(τ gg ) and both sides of the equality are nonzero precisely when g ≤f gg . The proof of the converse is similar and we leave the verification to the reader. 2 Consider the case where the inertial subgroup of f is trivial, i.e., Hf = {1G }. In this case, B = Kx1G ∼ J, where J = g=1G Kxg . Furthermore, = K. Therefore, Af = K BK ∼ =K K n is a commutative algebra and JK = g=1G K n xg . The lower subtractive
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partial order on G defined by f as defined in Section 1 is given by g ≤ gh ⇐⇒ f (g, h) = 0. As an application of Proposition 2, we may record the following corollary. ˆ is of the form Corollary 1. If f has trivial inertial group, then every automorphism in H given by (3) satisfying (4) and (5). Returning to the general case, let H = Hf be the inertial group of f , L = K H the fixed field of H, m = |H| = deg(K/L) and l = n/m = |G/H|. Moreover, let T = {σ1 , ..., σl }, with σ1 = 1G , be a complete set of coset representatives of H. Then Li = σi (L), for i = 1, .., l, are the distinct embeddings of L into K. Let Ki be the field K with L vector space structure twisted by σi . That is, for l ∈ L and x ∈ K, l.x = σi (l)x. The map l l l α : K ⊗F L → i=1 Kσi which sends x ⊗ l to (l.x)i=1 = (xσi (l))i=1 and extended linearly is an isomorphism of K-algebras. We will first compute the K-automorphism group of BK (Proposition 3) and then use it to investigate K-automorphisms of AK . The K- automorphisms of AK that stabilize BK (for a general inertial group) are described in Theorem 1, and is analogous to that of a trivial inertial group (Corollary 1). Recall that B is a classical crossed product algebra with respect to the Galois group H and the cocycle f1 = f |H×H . Thus, by using the crossed product notation we can write B = (K/L, H, f1 ). The Galois group of K/Li is Hi = σi Hσi−1 and there is an associated cocycle fi and a crossed product algebra Bi given by the G action on the second cohomology of G. Explicitly, fi : Hi × Hi → K is the cocycle given by fi (σi h1 σi−1 , σi h2 σi−1 ) = f σi (h1 , h2 ) = σi (f (h1 , h2 )), for h1 and h2 in H, and Bi = (K/Li , Hi , fi ). The algebra Bi is a central simple Li algebra and there is a K-isomorphism of algebras Ki ⊗L B → K ⊗Li Bi which sends w⊗vxh to w⊗σi (v)xσi hσ−1 . i Thus, all the algebras Ki ⊗L B are split and K-isomorphic to the algebra of m × m matrices, Mm (K). Therefore, we have the following isomorphisms of K-algebras. BK
∼ =K =K (K ⊗F L) ⊗L B ∼
l
i=1
Ki ⊗L B ∼ =K
l
(K ⊗Li
Bi ) ∼ =K
i=1
l
Mm (K)
i=1
Lemma 4. AutK (BK ) ∼ = Inn(BK ) Sl , where Inn(BK ) ∼ = P GLm (K)l . Hence there is a split short exact sequence of groups as follows. 1 → P GLm (K)l → AutK (BK ) → Sl → 1
(6)
l Proof. We need to compute the K-automorphism group of M := i=1 Mm (K). Any permutation of the coordinates is a K-automorphism of M and any automorphism of M up to a unique permutation preserves coordinates. Any K-automorphism which preserves the coordinates is a product of K- automorphisms of Mm (K). Because AutK (Mm (K)) = P GLm (K), the group of K- automorphisms of M that preserve the coordinates × × l l l l × × is P GL (K) = (M (K) /K ) = M (K) / K) = m m m i=1 i=1 i=1 i=1
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M × / (Z(M )× ) = Inn(M ) ∼ = Inn(BK ). Also note that P GLm (K)l is normal subgroup. In fact, for (ϕi ) ∈ P GLm (K)l and π ∈ Sl , π(ϕi )π −1 = (ϕπ(i) ). 2 We use a set T = {σ1 , ..., σl } of distinct coset representatives to compute AutK (BK ) explicitly. The set T is naturally in bijection with G/H where the latter is a G set. This gives the set T a G structure, given by g.σi = σj ⇐⇒ g.σi H = σj H. In this section we will identify the permutation group on l letters as the group of permutations of the set T . So, Sl = P erm (T ) and there is a G action on Sl given by g ϕ(σi ) = g.ϕ(g −1 .σi ), for g ∈ G. Because B is a central simple L-algebra, the center of BK is K l ∼ = K ⊗F L, and l has K-dimension l = dim(L/F ). Let e˜σi = k∈H eσi k and e˜h = i=1 eσi h . We have eσi h = e˜σi e˜h . Note that for i = 1, .., l, e˜σi commutes with every xh for h ∈ H, and hence is in the center of BK . By considering the dimension, it follows that the center of BK is the K-span of {˜ eσi |i = 1, ..., l}. Every permutation ϕ ∈ Sl = P erm(T ) lifts to a permutation on G, say ϕ, ˜ given by ϕ(σ ˜ i h) = ϕ(σi )h. For ϕ ∈ Sl , ϕ˜ ∈ P erm(G) satisfies the property ϕ(gh) ˜ = ϕ(g)h ˜ for any g ∈ G and h ∈ H. This permutation on G corresponds to the K-automorphism of K |G| = K n , which sends e˜σi to e˜ϕ(σi ) for 1 ≤ i ≤ l and fixes e˜h ’s element-wise for h ∈ H. We will extend ϕ˜ to a K-linear map on BK , which we will also denote by ϕ. ˜ Consider the K-basis {eτ xk |τ ∈ G, k ∈ H} of BK and express every τ ∈ G in the unique form τ = σi h for σi ∈ T and h ∈ H. We define ϕ˜ : BK → BK to be the K-linear extension of the following map: ϕ˜ : eσi h xk →
f σi (h, k) eϕ(σi )h xk f ϕ(σi ) (h, k)
(7)
where h, k ∈ H. We claim ϕ˜ is a K-algebra automorphism of BK . To verify this, first observe that Proposition 2 is applicable here. For k ∈ H, f σi (h,k) ϕ(τ ˜ )−1 ϕ(τ ˜ k) = k, and (7) has the form given in Proposition 2 with aσi h,k = f ϕ(σ . i ) (h,k) Condition (4) in Proposition 2 is trivially satisfied (for this case g and g in (4) are in H). Therefore, what we need to verify is (5), which reads as f σi (h, kl) σi h f σi (h, k) f σi (hk, l) ϕ(σi )h f f (k, l), (k, l) = f ϕ(σi ) (h, k) f ϕ(σi ) (hk, l) f ϕ(σi ) (h, kl) and follows from the cocycle condition on f . Moreover, for ϕ1 and ϕ2 in Sl , one can verify the equality ϕ˜2 ◦ ϕ˜1 = ϕ 2 ◦ ϕ1 by checking it on the K basis eσi h xk . It follows from this that S˜l is a subgroup of AutK (BK ) that is isomorphic to Sl . The intersection of S˜l with Inn(BK ) is trivial, because any inner automorphism of BK is identity on its center. Then from Lemma 4 it follows that AutK (BK )/InnK (BK ) ∼ = S˜l and |AutK (BK )/InnK (BK )| = l!. We summarize these observations, and a few other easy to verify facts, in the following proposition:
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Proposition 3. 1. The K- linear endomorphism, ϕ˜ : BK → BK given by (7) is a K-automorphism of BK , which takes K n back to itself. 2. The restriction of ϕ˜ to the center is the identity if and only if ϕ is the identity permutation. In fact, ϕ(˜ ˜ eσi ) = e˜ϕ(σi ) . 3. S˜l := {ϕ˜ ∈ AutK (BK )|ϕ ∈ Sl } ∼ = Sl is a subgroup of K-automorphisms of BK whose intersection with Inn(BK ) is trivial. 4. AutK (BK ) = Inn(BK ) S˜l . Equivalently, there is a split exact sequence as follows. 1 → Inn(BK ) → AutK (BK ) → S˜l → 1 × 5. Given a K-automorphism φ of BK , there is bφ ∈ BK , unique up to a central unit, −1 ˜ such that x → bφ φ(x)bφ ∈ Sl .
Thus, we have realized a subgroup of automorphisms of BK which corresponds to Sl in ˆ in the general setting, let us first recall Proposition 2 Lemma 4. To prove a result on H and note that every K-automorphism of AK which preserves BK and restricts to an automorphism of BK in S˜l also preserves K n . We will denote the conjugation by an invertible element u by Cu . ˆ consider its restriction to BK , r(φ). There exist b ∈ B × (unique up to a Given φ ∈ H, K central unit in BK ) and ϕ˜ ∈ S˜l such that r(φ)= Cb ◦ ϕ. ˜ Let φ = Cb−1 ◦φ. Then r(φ ) = ϕ˜ and hence, φ is of the form in (7). Because ϕ˜ preserves K n , so does φ and its restriction to K n is given by the permutation ϕ ˜ ∈ P erm(G), which permutes the idempotents eτ ’s. Thus, from Proposition 2, φ (eτ xg ) = aτ,g eϕ(τ ˜ ) xϕ(τ ˜ )−1 ϕ(τ ˜ g) , which satisfies conditions f σi (h,k) (4), (5) and aσi h,k = f ϕ(σ for all τ = σ h ∈ G and h, k ∈ H. i i ) (h,k)
× Note that the choice of b ∈ BK is not unique and can be replaced by vb for any × v ∈ Z(BK ) . To understand the effect of choosing vb instead of b, we need to understand the nature of the conjugation automorphism, Cv , by an element v ∈ Z(BK )× . Because −1 Z(BK ) = i=1,..,l K e˜σi ∼ ui e˜σi where ui ∈ K × and v −1 = ui e˜σi . = K G/H , v = For g ∈ G, let ug¯ = ui where gH = σi H. Then one can easily verify that Cv (eτ xg ) = uτ¯ uτ−1 ¯g eτ xg . Thus we have the following theorem.
ˆ there is a unit b ∈ B × and a unique ϕ ∈ Sl such that Theorem 1. Given φ ∈ H, K b φ (eτ xg ) b−1 = aτ,g eϕ(τ ˜ ) xϕ(τ ˜ )−1 ϕ(τ ˜ g)
(8)
where aτ,g (b) = aτ,g ∈ K ∗ and ϕ˜ ∈ SG = P erm(G) satisfies the following conditions for all τ, g, g ∈ G, h, k ∈ H and σi ∈ T : g ≤f g ⇐⇒ ϕ(τ ˜ )−1 ϕ(τ ˜ g) ≤f ϕ(τ ˜ )−1 ϕ(τ ˜ g )
(9)
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˜ ) aτ,g aτ g,g f ϕ(τ (ϕ(τ ˜ )−1 ϕ(τ ˜ g), ϕ(τ ˜ g)−1 ϕ(τ ˜ gg )) = aτ,gg f τ (g, g )
aσi h,k =
(10)
σi
f (h, k) f ϕ(σi ) (h, k)
(11)
Furthermore, b can be replaced by vb for any v ∈ Z(BK )× . If v = ui e˜σi ∈ Z(BK )× , −1 τ,g τ,g changing b to vb will change the coefficients a to c = uϕ(τ aτ,g . ˜ ) uϕ(τ ˜ g) ˆ that have the form given Definition 2. Let Γ denote the subgroup of automorphisms in H by the right hand side of Equation (8) in Theorem 1. ˆ | φ (eτ xg ) = aτ,g eϕ(τ Γ = {φ ∈ H ˜ ) xϕ(τ ˜ )−1 ϕ(τ ˜ g) , aτ,g ∈ K × , ϕ ∈ Sl and satisfies conditions (9) , (10) and (11)} Clearly Γ = r−1 (S˜l ) and hence is a subgroup. Also, let us denote the subgroup of × ˆ given by the units of BK by B ¯∼ inner automorphisms in H /K × . Because φ ◦ Cb = = BK ¯ is a normal subgroup of H. ˆ Furthermore, it follows from the last part of Cφ(b) ◦ φ, B ¯ Proposition 3 that φ ∈ B ∩ Γ if and only if it is a conjugation by a central element. Thus ¯ ∩ Γ = Z(BK )× /K × ∼ B = (K l )× /K × . Thus, we have the following corollary. ˆ = BΓ, ¯ B ¯ H ˆ and B ¯ ∩Γ∼ Corollary 2. H = (K l )× /K × . ˜ ≤ S˜l and Due to condition (9), it is natural to consider the following subgroups, Λ Λ ≤ Sl . Definition 3. Λ := {ϕ ∈ Sl |∀g1 , g2 , τ ∈ G, g ≤ g ⇐⇒ ϕ(τ ˜ )−1 ϕ(τ ˜ g) ≤ ϕ(τ ˜ )−1 ϕ(τ ˜ g )}
(12)
˜ := {ϕ˜ ∈ S˜l |ϕ ∈ Λ} Λ
(13)
We wish to remind the reader that ϕ ˜ stands for the lift of ϕ ∈ Sl to an element either in P erm(G) or in S˜l , which could be understood without ambiguity from the context. ˜ Moreover, the isomorphism Sl ∼ = S˜l , given by (7), restricts to an isomorphism Λ ∼ = Λ. ˜ ˜ Before we discuss more about Λ, it is worth noting that r(Γ) < Λ. 2.1. H : BK -automorphisms of AK ˆ (or of AutK (AK )) consisting of all automorphisms Recall that H is the subgroup of H which restrict to the identity on BK . Thus every φ ∈ H (by assumption) acts trivially on the elements xh for h ∈ H. Each φ ∈ H is of the form φ(eτ xg ) = aτ,g eτ xg , where the aτ,g τ,g satisfy conditions (10) and (11). Thus, φ(xg ) = φ( τ eτ xg ) = a eτ xg = (aτ,g )τ ∈G xg and the corresponding permutation ϕ˜ on G is the identity. Hence the condition (10)
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can be rewritten as aτ,g aτ g,g f τ (g, g ) = aτ,gg f τ (g, g ) whenever g ≤ gg . However, f (g, g ) = 0 ⇐⇒ g ≤ gg . Therefore, the condition (10) amounts to
aτ,g aτ g,g = aτ,gg whenever g ≤ gg .
(14)
Given a G-module M , written multiplicatively, we will write the action of g ∈ G on m ∈ M by g m. Definition 4 (Weak 1-cocycle (see [1])). Given a G-module M , a function a : G → M is a weak 1-cocycle, with respect to a given lower subtractive relation ≤, if a satisfies the condition a(gg ) = a(g) g a(g ) whenever g ≤ gg
(15)
a(h) = 1 for h ≤ 1G (i.e. h ∈ H)
(16)
Weak 1-cocycles with pointwise multiplication form a group, which we denote by 1 Z≤ (G, M ). Thus, 1 Z≤ (G, M ) = {a : G → M | a satisfies conditions (15) and (16)}
For each a = (aτ )τ ∈G ∈ K n and g ∈ G, define g a by g
(aτ )τ ∈G = (aτ g )τ ∈G
(17)
This defines an action of G on K n . In fact this is the action of xg ’s on K n in Proposition 1. Furthermore, conditions (14) and (11) imply that the function a : G → (K n )× , where a(g) = ag = (aτ,g )τ ∈G is a weak-1-cocycle 1 Proposition 4. The group H is isomorphic to the group of weak 1-cocycles Z≤ (G, (K n )× ), n 1 where G acts on K according to (17). Given a weak 1-cocycle a ∈ Z≤ (G, (K n )× ), the associated automorphism is given by
φ(xg ) = (aτ,g )τ ∈G xg and φ(eτ ) = eτ
(18)
Proof. We saw that for each ϕ ∈ H, there is an associated weak 1-cocycle. Moreover, since the action of G on K n is precisely the action of xg ’s on the idempotents of K n (in Proposition 1), every such weak 1-cocycle determines a K-automorphism of AK as given. 2 Remark 1. The partial order on G/H induces a partial order on T (choosing a set of coset representatives). Let ht(1) = {σ2 , σ3 , ..., σr } be the height one elements of this partial order. Then, G is generated by ht(1) ∪ H. In fact, every element of G is of the form σi1 σi2 ...σir h, where σi1 , ..., σir ∈ ht(1) (see [2]) for details). One can easily verify
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that the value of any weak 1-cocycle a at g is uniquely determined by the values of a on σi1 , σi2 , ..., σir (see (24) in Example 1). Hence, every weak 1-cocycle is determined by its values on the ht(1) elements and therefore H < |ht(1)| (K n )× . Because P GLm (K)l ∼ = Inn(BK ) ⊆ Img(r) and Img(r) ∩ S˜l = r(Γ), we can now rewrite the short exact sequence (1) as follows. Corollary 3. The exact sequence (1) can be rewritten as follows. 1 ˆ− 1 → Z≤ (G, (K n )× ) → H → Inn(BK ) r(Γ) → 1 r
(19)
3. The automorphism group of an idempotent cocycle In this section we will consider the special case where f is an idempotent cocycle. Recall that f : G × G → K is an idempotent cocycle if it takes only the values 0 and 1 in K. Such cocycles are in one-to-one correspondence with lower subtractive relations ≤ on G with the property that 1G ≤ g for all g ∈ G. ˜ We know that Inn(BK ) ≤ r(H) ˆ is a subgroup of Inn(BK ) Λ. ˆ Recall that r(H) ˜ and r(Γ) ≤ Λ. However it is not clear, in the general case, whether every permutation ˜ lifts to an automorphism of AK . Nonetheless, when the cocycle is an idempotent in Λ ˜ the map cocycle, it is clear that we can choose aτ,g = 1 in (8). Thus, given ϕ˜ ∈ Λ, ˜ ϕ˜ : eτ xg → eϕ(τ ˜ ) xϕ(τ ˜ )−1 ϕ(τ ˜ g) is a K-automorphism of AK . Therefore, r(Γ) = Λ and r ˜ ˆ onto Inn(BK ) Λ. The short exact sequence (19) for this case is maps H 1 ˜ →1 ˆ → InnK (BK ) Λ 1 → Z≤ (G, K n ) → H
(20)
Remark 2. Recall the G action on AutK (AK ) given by g ϕ = (g ⊗1) ◦ϕ ◦(g −1 ⊗1) for ϕ ∈ ˜ to an element in H, ˆ gives AutK (K ⊗ A). The lifting described above, of an element in Λ ˜ ˆ rise to a group homomorphism from Λ to H. However, it should be noted that this group homomorphism is not a G map, with respect to the G action on automorphisms. In fact ˜ is not even G stable in general. For example in the case of the Waterhouse idempotent Λ ˜ = S˜l and S˜l is not G invariant. Moreover, even though every element (see Example 3), Λ ˆ there is no canonical way of doing it. in Inn(BK ) can be lifted to an element in H, 3.1. Lower subtractive partial orders and some observations about Λ Recall that T is a set of distinct coset representatives of H, S˜l ∼ = Sl = P erm(T ) and Λ is defined by (12). The definition of Λ depends only on the lower subtractive relation defined on G. Hence it is determined by the corresponding idempotent cocycle. The group G acts from the left on G/H and hence for each γ ∈ G, there is an associated permutation of cosets in G/H which induces a permutation on T . We denote the permutation induced by γ ∈ G on T by γ¯ and it is given by γ¯ (σi ) = σj ⇐⇒ γσi H =
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σj H ⇐⇒ σj−1 γσi ∈ H. In this way, we get a group homomorphism from G to Sl , γ → γ¯ . ¯ and image of H by H. ¯ Also, Let us denote the image of G in Sl under this map by G let us write γ.¯ g for γ¯ (σi ) where g ∈ σi H. Lemma 5. Λ = {ϕ ∈ Sl | for all g1 , g2 , τ ∈ G, g1 ≤ g2 ⇐⇒
ϕ(τ ˜ )−1 . ϕ(τ.g¯1 ) ≤ ϕ(τ ˜ )−1 . ϕ(τ.g¯2 )},
¯ < Λ. and G Proof. We will leave it for the reader to verify that the condition that defines Λ can be ¯ < Λ. expressed as given here. We will outline the proof of G Recall that h ≤ g for all g ∈ G and h ∈ H. For any g, g ∈ G and h ∈ H, hg ≤ hg ⇔ (h ≤ hg ≤ hg and h ≤ hg ) ⇔ h−1 hg ≤ h−1 hg ⇔ g ≤ g , where the second if and only if is due to the lower subtractive property. Thus, left multiplication by elements of the inertial subgroup preserves the partial order. Let γ ∈ G. Note that for any τ ∈ G, γ˜¯ (τ ) = γτ h for some h ∈ H. Thus, for any g1 , g2 , τ ∈ G, there are h, h1 , h2 ∈ H, such that −1 γ˜¯ (τ )−1 γ˜¯ (τ gi ) = (γτ h) (γτ gi hi ) = h−1 gi hi
Thus; γ˜¯ (τ )−1 γ˜¯ (τ g1 ) ≤ γ˜¯ (τ )−1 γ˜¯ (τ g2 ) ⇐⇒ h−1 g1 h1 ≤ h−1 g2 h2 ⇐⇒ g1 h1 ≤ g2 h2 ⇐⇒ g1 ≤ g2 The last equivalence follows from g1 h1 ≤ g2 h2 ⇐⇒ g1 h1 ≤ g2 h2 ⇐⇒ g¯1 ≤ g¯2 ⇐⇒ g 1 ≤ g2 . 2 The ordering on G restricts to a partial order on T , which is isomorphic as partial orders to G/H. Let us denote the subgroup of permutations in Sl that preserve the ordering by Aut(≤). One can easily show that ϕ ∈ Sl is in Aut(≤) if and only if ϕ˜ preserve the ordering on G. Let us also define Λ˙ to be the order preserving permutations, that is, Λ˙ = Aut(≤) ∩ Λ. Suppose ϕ ∈ Sl is in Λ. Let σ1 ∈ T be the element such that σ1 H = H. For any γ ∈ Hϕ(σ1 )−1 , γ¯ ϕ is a permutation in Λ, which takes σ1 back to itself. Now, by taking
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˙ τ = 1 in the definition of Λ, we see that γ¯ϕ ∈ Λ˙ also preserves the order. Thus, γ¯ϕ ∈ Λ. ˙ Hence, we may conclude that ¯ and γ¯ ϕ ∈ Λ. Therefore, ϕ = γ −1 (¯ γ ϕ), where γ −1 ∈ G ˙ ˙ ¯ ¯ Λ = GΛ. Furthermore, if γ¯ ∈ G ∩ Λ, then it preserves the root σ1 , and hence γ ∈ H ˙ Therefore, we have the following ¯ Conversely any element in H ¯ is in Λ. and γ¯ ∈ H. proposition. ¯ Λ˙ and G ¯ ∩ Λ˙ = H¯f Proposition 5. Λ = G Associated to ≤, there are two right lower-subtractive relations with the same inertial group, the right-graph ≤∗ and the inverse-graph ≤− . The right graph, ≤∗ , was first introduced in [6] to study the ideal structure of Af . Definition 5. Relations ≤∗ and ≤− are defined by the following rules. 1. g ≤∗ h ⇐⇒ hg −1 ≤ h (or equivalently x ≤ y ⇐⇒ x−1 y ≤∗ y). − −1 −1 2. g ≤ h ⇐⇒ g ≤ h . The following lemma follows directly from the definition. We leave the verification to the reader. Lemma 6. ≤∗ and ≤− are right lower subtractive relations on G with the same inertial group as ≤. Dually, given a right lower-subtractive relation , there are two associated left lower subtractive relations, ∗ and − . The following properties can be easily verified. Proposition 6. If ≤ is a lower subtractive relation on G, then the following holds. 1. (≤∗ )∗ = ≤ i.e., x(≤∗ )∗ y ⇐⇒ x ≤ y − 2. (≤ )− = ≤ 3. (≤∗ )− = (≤− )∗ . Definition 6. The “dual” of a left lower subtractive relation ≤, denoted by ≤† , is the left lower subtractive relation (≤− )∗ =(≤∗ )− . Corollary 4. 1. x ≤† y ⇐⇒ y −1 x ≤ y −1 . Equivalently, x ≤ y ⇐⇒ y −1 x ≤† y −1 † 2. ≤† = ≤. The relations ≤† and ≤ share several important properties. For example the graph of ≤ on G/H is a tree (acyclic) if and only if the graph of ≤† on G/H is a tree. Moreover, for a standard graph with respect to the generating set S, the dual graph ≤† is also
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standard with generating set S −1 , the set of inverses of the elements in S. In particular if the set of height one elements is closed under taking inverses (for example all the generators considered in a Coxeter group have order 2) the dual relation is the same as the original. Let us denote the corresponding Λ for ≤† by Λ† . The statement of the following proposition also relates ≤ and ≤† and is particularly useful in the setting of this paper. Proposition 7. Λ = Λ† . Proof. Let ϕ˜ ∈ Λ. To show that ϕ˜ ∈ Λ† , we need to show g ≤† g if and only if ϕ(τ ˜ )−1 ϕ(τ ˜ g) ≤† ϕ(τ ˜ )−1 ϕ(τ ˜ g ), ∀τ ∈ G. From the definition of ≤† , the condition above amounts to showing g −1 g ≤ g −1 ⇐⇒ ϕ(τ ˜ g )−1 ϕ(τ ˜ g) ≤ ϕ(τ ˜ g )−1 ϕ(τ ˜ ), ∀τ ∈ G.
(21)
Because ϕ˜ is in Λ, we have g −1 g ≤ g −1 ⇐⇒ ϕ(τ ˜ )−1 ϕ(τ ˜ g −1 g) ≤ ϕ(τ ˜ )−1 ϕ(τ ˜ g −1 )∀τ ∈ G
(22)
Because τ is arbitrary, the equation (21) is obtained by replacing τ in (22) by τ g . Thus, we have that ϕ˜ ∈ Λ† . Thus Λ ⊆ Λ† . Applying this to Λ† , we also have Λ† ⊆ (Λ† )† = Λ. Therefore, Λ = Λ† . 2 Proposition 7 is particularly useful in computing Λ (see Example 4). It implies that, if e is the idempotent cocycle associated with ≤ and e† the idempotent cocycle associated to ≤† , then the K-automorphism groups of K ⊗F Ae and K ⊗F Ae† share Λ as a common subgroup. The group Λ will also be useful in computing the K-forms of Ae and Ae† . 3.2. Idempotent cocycles with trivial inertial group In this subsection we consider idempotent cocycles with trivial inertial subgroups. In this case BK is commutative and hence there are no nontrivial inner automorphisms. Let Λ be the subgroup of P erm(G) = Sn which satisfies the condition given in (4). Thus: Λ = {ϕ ∈ P erm(G)|∀g, g , τ ∈ G, g ≤ g ⇐⇒ ϕ(τ )−1 ϕ(τ g) ≤ ϕ(τ )−1 ϕ(τ g )} As observed just before Remark 2, ϕ ∈ Λ lifts (canonically) to a K- automorphism of ˆ by choosing aτ,g = 1 in (3) of Proposition 2. As a consequence, the image of AK , ϕ˜ in H, ˜ (1) is a split short exact sequence and H ˆ∼ r is Λ, = H Λ. We record this in the following lemma.
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Lemma 7. Lef f be an idempotent cocycle with trivial inertial subgroup. Then the sequence ˆ→Λ→1 1→H→H
(23)
ˆ∼ is split. Therefore, H = H Λ. Consider the left regular representation of G in SG . Every element γ ∈ G defines a permutation of G by left multiplication. We will denote this permutation by lγ . The map l : γ → lγ , embeds G in SG . In this way we identify G as a subgroup of SG . We have Λ = G.Λ˙ and G ∩ Λ˙ = {id}. Also, if ψ ∈ Aut(G) then for every τ ∈ G, ψ(τ )−1 ψ(tg) = ψ(g) and hence ψ ∈ Λ if and only if ψ ∈ Aut(≤). Thus, every order preserving automorphism of G is in Λ. I.e., Aut(G) Aut(≤) is a subgroup of Λ. Remark 3. Short exact sequence (23) is in fact a short exact sequence of G groups with respect to the G action on AutK (AK ) mentioned in Remark 2. The copy of G in SG is also G invariant with respect to this action, which reduces to the conjugation action of G on itself. 4. Examples Example 1. When the inertial group of the weak 2-cocycle f is the trivial subgroup {1G }. In this case B = K and hence BK is a commutative algebra. The relation ≤ is a lower subtractive partial order on G and AutK (BK ) ∼ = P erm(G). Furthermore, Λ = {ϕ ∈ SG = P erm(G)| ∀g, g , τ ∈ G, g ≤ g ⇐⇒ ϕ(τ )−1 ϕ(τ g) ≤ ϕ(τ )−1 ϕ(τ g )} ˆ < Λ and H ˆ = Γ. Note that the condition (11) is trivial. In is a subgroup of SG , r(H) ˆ fact, H, as explained in Corollary 1, is of the form (3) subject to conditions (4) and (5) in Proposition 2. Thus, AutK (AK ) ∼ = (1 + JK ) Γ. Consider the subgroup H in this case. Let ht(1) = {g1 , ..., gr } be the set of height 1 elements (i.e., the elements immediately above 1G ). Recall that G is generated by ht(1). Consider an element g ∈ G and a path 1G < gk1 < gk1 gk2 < gk1 gk2 gk3 < ... <
t
gki = g
i=1
starting from 1G and ending at g. Then one can use the cocycle condition to find the value of ag in terms of the values of agi ’s as follows. ag = a ti=1 gk = agk1 gk1 a ti=2 gk = agk1 gk1 ak2 ...( i
i
t−1 i=1
gk i )
akt
(24)
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Therefore, the value of a at g is determined by the values of a on the height 1 elements. Hence H is a subgroup of the group of functions from ht(1) to K n , M aps (ht(1), (K n )× ) = |ht(1)| (K n )× , where a is in H if and only if for every g ∈ G and for all distinct paths to g, computation in (24) evaluates to the same value. Here, G act on |ht(1)| (K n )× diagonally where the action on each K n is as in (17). Example 2. When the graph of the partial order on G/H induced by a weak 2-cocycle f is a tree. Let a : G → K n a weak 1-cocycle. Given any g ∈ G, because there is a unique path from 1G H to gH, it follows from Remark 1 and the above example that, ag is uniquely determined by the height one elements in G/H. Thus, we have the following proposition. 1 Proposition 8. When the graph of ≤ on G/H is a tree, weak-1-cocycles, Z≤ (G, (K n )× ), are in one-to-one correspondence with the functions from ground elements, ht(1) = {σ2 H, σ3 H, ..., σr H}, to K n .
H = Z 1 (G, (K n )× ) ∼ =
(K n )×
(r-1) times
Example 3. The Waterhouse idempotent: Consider the partial order ≤w on G with 1G as the minimum element and with no other nontrivial relations. That is, the only relations are 1 ≤w g and g ≤w g for all g ∈ G. The corresponding idempotent cocycle, called the Waterhouse idempotent, is w : G ×G → {0, 1} where w(g, h) = 0 ⇐⇒ g = 1g or h = 1g . The corresponding weak crossed product algebra is Aw = K ⊕ J with J 2 = 0. One can |G|−1 ˆ∼ easily verify that in this case Λ = SG , H = g∈G,g=1G (K n ) and H SG . = (K n ) |G|−1 Thus, AutK (K ⊗ Aw ) ∼ SG . = (1 + JK ) (K n ) Example 4. Consider the following lower subtractive partial order, shown in the left as a graph, on the cyclic group of order 5, C 5 , generated by α. The dual of ≤, ≤† , is shown in the right. α3
≤
α
α4
α2
1G
≤†
α2
α
α4
α3
1G
˙ i.e., in Λ and Because Λ = Λ† , one can easily see that the only permutation in Λ, ∼ preserves the order, is the identity permutation. Thus, Λ = LG = G. Furthermore,
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H = K n × K n . Let Ae be the weak crossed product algebra that corresponds to the ˆ where idempotent cocycle e arising from ≤. Then AutK (K ⊗ Ae ) ∼ = (1 + JK ) H n n ∼ ˆ H = (K × K ) G. Example 5. Consider lower subtractive partial order on the cyclic group of order 5 generated by α, C 5 , with respect to the generating set S = {α, α4 }. α2
α3
α
α4
1G Note that there are only two order preserving maps, the identity map and the map ˙ Thus, Λ = which sends α → α−1 . One can easily verify that both of these are in Λ. G.Λ˙ ∼ = G C 2 , where C 2 ∼ = Λ˙ is the cyclic group of order 2. Moreover, H = K n × K n as ˆ of the K-automorphism group of the corresponding before. Therefore, the subgroup H n algebra, is isomorphic to (K × K n ) (G C 2 ). As before, if Ae is the weak crossed product algebra that corresponds to the idempotent cocycle e arising from the lower subtractive partial order we are considering, then AutK (K ⊗ Ae ) ∼ = (1 + JK ) ((K n × K n ) (G C 2 )). Example 6. Weak Bruhat order of a Coxeter system. Let (W, S) be a finite Coxeter group. Associated to this is its diagram DS . By a diagram automorphism we mean an automorphism of the graph DS . Every such diagram automorphism gives rise to a unique automorphism of W preserving S as a set. In what follows, we will denote these diagram automorphism as well as their extension to automorphism of W by Aut(DS ). The left weak order of W is a lower subtractive partial order which we denote by ≤. Proposition 9. Λ˙ = Aut(DS ) Proof. Suppose φ ∈ Aut(DS ). Recall that an automorphism of W is in Λ if and only if it preserves the order. The fact that φ preserves the order follows from the fact that φ is a homomorphism and for w ∈ W if w = s1 s2 · · · sk is a reduced word for w then ˙ φ(w) = φ(s1 )φ(s2 ) · · · φ(sk ) is a reduced word for φ(w). Thus φ ∈ Λ. ˙ To show the converse, suppose φ ∈ Λ. Then φ in particular, preserve the weak order ≤ and hence φ maps S to S. Recall that for a subset X of S, the subgroup generated by X, WX , forms a lower interval in the weak order. Moreover, this interval is the weak order on WX and WX S = X (see [10,11]).
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Consider two element s1 , s2 with O(s1 s2 ) = m and gm = s1 s2 s1 · · · = s2 s1 s2 · · · where each product has m terms. Because s1 ≤ gm and s2 ≤ gm and only s1 and s2 are the elements below gm , it follows that φ(s1 ) ≤ φ(gm ) and φ(s2 ) ≤ φ(gm ) and only φ(s1 ) and φ(s2 ) are below φ(gm ). It follows by considering X = {φ(s1 ), φ(s2 )}, φ(gm ) = φ(s1 )φ(s2 )φ(s1 ) · · · = φ(s2 )φ(s1 )φ(s2 ) · · · , with m terms and hence O(φ(s1 )φ(s2 )) = m. Thus, φ is a diagram automorphism. The fact that φ is the extension of the diagram automorphism also follows similarly. 2 Proposition 10. W Λ and Λ = W Aut(DS ) Proof. Since W Λ˙ = {id}, we only need to show that W is normal in Λ. Let lw be the left multiplication by an element in W . Then for φ ∈ Λ˙ and with the help of Proposition 9, we have φ lg φ−1 = lφ(g) . 2 It is also worth noting that conjugation with respect to the unique largest element m, φm , preserves the ordering ≤ (see [4]) and φm (τ )−1 φm (τ x) = φm (x). Thus, φm ∈ Aut(DS ) ≤ Λ. If AS is the weak crossed product algebra arising from the associated idempotent cocycle, then AutK (K ⊗ AS ) ∼ = (1 + JK ) (H (W Aut(DS ))). Example 7. Weak Bruhat order on Sn , the symmetric group on n letters (continuation of Example 6). In this case there is only one nontrivial diagram automorphism and it is φm . Thus, Λ˙ = {id, φm }, where φm is the conjugation by the unique maximal element. ˆ∼ Therefore, Λ = C 2 Sn and H = H (C 2 Sn ). If AS is the algebra associated with the idempotent cocycle, AutK (K ⊗ AS ) ∼ = (1 + JK ) (H (C 2 Sn )). Acknowledgments The author would like to express his deepest gratitude to Professor Darrell Haile for his support and comments given throughout this work. The anonymous referee is also acknowledged for his valuable comments. References [1] R. Stimets, Weak Galois cohomology and group extensions, Comm. Algebra 28 (3) (2000) 1285–1308. [2] D. Haile, R. Larson, M. Sweedler, A new invariant of C over R: almost invertible co-homology theory and the classification of idempotent co-homology classes and algebras by partially ordered sets with a Galois group action, Amer. J. Math. 105 (1983) 689–814. [3] A. Bjorner, F. Brenti, Combinatorics of Coxeter Groups, vol. 231, Springer Science & Business Media, 2006. [4] A. Aljouiee, On weak crossed products, Frobenius algebras, and the weak Bruhat ordering, J. Algebra 287 (2005) 88–102. [5] D. Haile, Brauer monoid of a field, J. Algebra 81 (1983) 521–539. [6] D. Haile, On crossed product algebras arising from weak cocycles, J. Algebra 74 (1982) 270–279. [7] D. Haile, L. Rowen, Weakly Azumaya algebras, J. Algebra 250 (2002) 134–177.
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[8] F. Guil-Asensio, M. Saorin, The automorphism group and the Picard group of a monomial algebra, Comm. Algebra 27 (1999) 857–887. [9] R.S. Pierce, Associative Algebras, Grad. Texts in Math., vol. 88, Springer, Berlin, 1982. [10] N. Bourbaki, Lie Groups and Lie Algebras, Springer-Verlag, 2002, Chapters 4–6, translated from the 1968 French original by Andrew Pressley. [11] A. Bjorner, Orderings of coxeter groups, in: C. Greene (Ed.), Combinatorics and Algebra, Amer. Math. Soc, 1984, pp. 175–195.
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Journal of Algebra www.elsevier.com/locate/jalgebra
K-automorphisms of a weak-crossed product F -algebra over a Galois extension K/F Jayampathy Ratnayake Department of Mathematics, University of Colombo, Colombo 3, Sri Lanka
a r t i c l e
i n f o
Article history: Received 4 October 2018 Available online 22 August 2019 Communicated by Louis Rowen Keywords: Weak crossed products Lower subtractive partial order Weak Bruhat order
a b s t r a c t We compute the K-automorphism group of AK = K ⊗F Af of a weak crossed product algebra Af for a weak 2-cocycle f over a Galois extension K/F with Galois group G. The K-automorphism group of AK decomposes into its unipotent ˆ Automorphisms in H ˆ are part and the reductive part H. computed via their restriction to semi-simple part of AK . ˆ and the lowerThere is a strong relationship between H subtractive relation (≤) induced by f on G. We introduce ˆ namely Λ, which contains interesting a subgroup of H, combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lowersubtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Throughout this paper, F ⊆ K is a Galois extension of fields with Galois group G. We will write 1G for the identity element of G. A weak 2-cocycle is a function f : G ×G → K, E-mail address: [email protected]. https://doi.org/10.1016/j.jalgebra.2019.08.011 0021-8693/© 2019 Elsevier Inc. All rights reserved.
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which satisfies the 2-cocycle condition f (g, h)f (gh, k) = f g (h, k)f (g, hk) for all g, h, k ∈ G, written without the inverses. The important distinction between a weak 2-cocycle and a classical 2-cocycle is that a weak 2-cocycle can take the value 0 ∈ K. Two weak 2-cocycles f1 and f2 are cohomologous, written f1 ∼ f2 , if there is a function α : G → K × α(g)α(h)g f2 (g, h). We will denote the set of equivalence classes of such that f1 (g, h) = α(gh) weak 2-cocycles, with respect to this equivalence relation, by M 2 (G, K). The pointwise product of two weak 2-cocycles is again a weak 2-cocycle, which one can verify easily. This product induces a product in M 2 (G, K), which gives it a structure of a monoid. We call M 2 (G, K) the Brauer Monoid of K/F . The Brauer monoid is a disjoint union of abelian groups, indexed by the set of idempotents, where the identity element in each component corresponds to an idempotent in the monoid. Weak 2-cocycles that corresponds to idempotents of the Brauer monoid are the idempotent cocycles, i.e. the cocycles that take only the values 0 or 1 in K. Given a weak idempotent 2-cocycle e, the abelian subgroup determined by e is denoted by Me2 (G, K). Associated to a weak 2-cocycle f there is a relation on G, determined by g ≤f h iff f (g, g −1 h) = 0 (see [1]). This relation satisfies the following left lower subtractive property: if g ≤f h then g ≤f x ≤f h ⇐⇒ g −1 x ≤f g −1 h (right lower subtractive property can be defined similarly). The Inertial subgroup of f , Hf , is the subgroup of G defined by Hf = {g ∈ G : g ≤f 1G } = {g ∈ G : f (g, g −1 ) = 0}. The relation ≤f induces a lower subtractive partial order on the cosets G/Hf in the obvious way: g1 H ≤f g2 H ⇐⇒ g1 ≤ g2 . It was shown in [2] that the partial order on G/H is well-defined. For two weak 2-cocycles f and g, ≤f =≤g if and only if f and g are in the same component Me2 (G, K) for some idempotent e. Moreover, for every lower subtractive relation ≤ on G with the property that 1G ≤ g for all g in G, there is a corresponding idempotent weak 2-cocycle e, given by e(g1 , g2 ) = 1 ⇔ g1 ≤ g1 g2 , such that ≤e =≤. The lower subtractive property guarantees that it is a 2-cocycle. Thus, idempotent weak 2-cocycles are naturally in a one-to-one correspondence with the set of lower subtractive relations on G/H, where H varies over all subgroups of G, and with the idempotents of M 2 (G, K) (see [2] for more details on this). Weak Bruhat orders (see [3]) are important examples of lower subtractive partial orders. A more general example, which includes weak Bruhat orders, is when a group G is given with a generating set S. In this case there is a natural lower subtractive relation, which is also a partial order, on G given by the minimal words in S as follows. For x, y ∈ G, we define x ≤ y if there is a minimal word for y starting with a word (necessarily minimal) for x. This is called the standard graph of G with respect to the generating set S and we invite the reader to refer [4] for more details. Every weak 2-cocycle is cohomologous to a normalized weak 2-cocycle f satisfying the condition f (g, 1G ) = f (1G , g) = 1 for all g ∈ G. As in the classical case, such a weak 2-cocycle gives rise to a crossed product algebra Af = g∈G Kxg , where the
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multiplication is given by axg1 bxg2 = ag1 (b)f (g1 , g2 )xg1 g2 for all a, b ∈ K and g1 , g2 ∈ G. We will denote this algebra, associated to a normalized weak 2-cocycle f , by Af . Weak 2-cocycles and weak crossed products were first introduced and studied in [2]. We recall some basic properties of Af bellow and invite readers to refer [2,4–6] for further details on weak 2-cocycles, weak crossed products and weakly Azumaya algebras. The center of Af is F and hence it is an F -algebra, but in general it is not simple. The nilpotent radical of a weak crossed product algebra is zero if and only if the associated weak 2-cocycle is a classical 2-cocycle. Modulo the radical, weak crossed product algebras are central simple algebras over an intermediate extension of K/F . The equivalence relation on weak 2-cocycles given by 1-coboundaries is such that f ∼ f if and only if there is an F -algebra isomorphism α : Af → Af which is identity on K. Specifically, Af = Bf Jf where Bf = h∈Hf Kxh is the semisimple part of Af and Jf = h∈H / f Kxh is the Jacobson radical of Af consisting of nilpotent elements. Thus Af = B ⊕ J is a Wedderburn-Malcev decomposition. Moreover, Bf is a classical crossed product algebra with respect to the cocycle f |Hf ×Hf , f restricted to Hf × Hf and the center of Bf is L = K Hf . The idempotent cocycles are useful for two main reasons. First, they have a role in weakly Azumaya algebras which is similar to the role matrix algebras have in central simple algebras. It is a well-known fact that every central simple F algebra is a form of a matrix algebra. Even though it is not true that every weakly Azumaya algebra is a form of an algebra associated to a weak idempotent cocycle, one would like to know up to what extent a similar result holds for weakly Azumaya algebras. The second reason is that they present a way to study lower subtractive relations on G, such as the weak Bruhat order of a Coxeter Group (W, S), which are interesting combinatorial objects in their own right. In this paper, we compute the K-automorphism group of K ⊗F Af . We were mainly motivated towards this study because it lays the foundation to study the K-forms of a weak crossed product algebra Af , via Descent Theory, by finding H 1 (G, AutK (AK )), which we wish to present in a sequel to this paper. In Section 2 of this paper we will compute the K-automorphism group of K ⊗ Af , where Af is a weak crossed product ˜ ((13) of Definition 3), two interesting algebra. We introduce Γ (Definition 2) and Λ subgroups that carry combinatorial information about the lower subtractive relation. In Section 3 we investigate the automorphism group of an idempotent cocycle, with the help of the associated lower subtractive partial order on G/Hf . We exhibit a duality on the lower subtractive partial orders and exhibit how it is relevant to the study of automorphisms. Finally, in Section 4 we give some examples including the automorphism group of the algebra associated to the Weak-Bruhat order of a Coxeter system (Example 6 and 7) and show it is related to the automorphisms of Coxeter diagrams.
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2. K-automorphisms of K
F
Af
Let Af = g∈G Kxg be the weak crossed product algebra with respect to the normalized weak 2-cocycle f : G × G → K and Hf the inertial subgroup of f . Set n = |G|. All tensor products, unless otherwise stated, are over the field F . Consider K as an F -algebra. Then K ⊗ K is a K-algebra where the K-vector space structure comes from the first factor. Thus, for k ∈ K and x ⊗ y ∈ K ⊗ K, k.(x ⊗ y) = kx ⊗ y. The map α : K ⊗ K → K n where α(a ⊗ b) = (aσ(b))σ∈G is an isomorphism of K-algebras, where K n = {(aσ )σ∈G : aσ ∈ K} is the set of n-tuples indexed by the elements of G with the diagonal K-action. Lemma 1. AK = K⊗Af ∼ =K g∈G K n xg , where the K-algebra structure on g∈G K n xg is given by xg (aσ )σ∈G = (aσg )σ∈G xg and xg1 xg2 = (f σ (g1 , g2 ))σ∈G xg1 g2 In particular xg eτ = eτ g−1 xg , where eτ = (δτ,σ )σ∈G is the idempotent in K n with 1 in the τ -place and zeros in the other co-ordinates. n Proof. The map λ : K ⊗ Af → (K ⊗ K)xg = K xg given by λ(p ⊗ qxg ) = (pσ(q))σ∈G xg is the required K-algebra isomorphism, which we leave for the reader to verify. 2 n xg . Note that BK = Throughout this paper we will identify AK with g∈G K n n ∼ ∼ K x K ⊗B is the semisimple part of A and J = = h K K K h∈Hf g ∈H / f K xg =K K ⊗J is the Jacobson radical of AK . The decomposition AK = BK ⊕ JK is a WedderburnMalcev decomposition of AK . The following lemma is a consequence of AK being a weakly Azumaya algebra with respect to BK (see Theorem 2.1 of [7]), which can also be directly verified.
Lemma 2. The centralizer of BK in AK is the center of BK . The following lemma follows from the above. Lemma 3. If conjugation by 1 + j ∈ 1 + JK preserves BK then j = 0. Proof. If (1 + j)BK (1 + j)−1 = BK , then for any b ∈ BK there is some b ∈ BK such that b + jb = (1 + j)b = b (1 + j) = b + b j. Thus, b = b and jb = b j. Therefore, j commutes with all the elements in BK . From the previous lemma, j ∈ BK . Therefore, j = 0. 2 The following definitions were motivated by [8].
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Definition 1. ˆ = {ϕ ∈ AutK (AK ) : ϕ(BK ) ⊆ BK }, be the subgroup of automorphisms in 1. Let H AutK (AK ) which preserve BK . ˆ consisting of automorphisms ˆ : ϕ|B = idB }, be the subgroup of H 2. Let H = {ϕ ∈ H K K which restrict to the identity on BK . ˆ → AutK (BK ) Thus, H = AutBK (AK ), and it is the kernel of the homomorphism r : H ˆ to ϕ|B , the restriction of ϕ to BK . Hence H is a normal subgroup which sends ϕ ∈ H K ˆ of H and there is a short exact sequence of groups as follows. r
ˆ− 1→H→H → Im(r) → 1
(1)
For any K-automorphism ϕ of AK , ϕ(BK ) gives another Wedderburn-Malcev decomposition of AK , AK = ϕ(BK ) + JK . Because BK is uniquely determined up to a conjugation by an element in 1 + J(AK ) ([9] Theorem 11.6), there is an element 1 + j ∈ 1 + JK such that BK = (1 + j)−1 ϕ(BK )(1 + j). Furthermore, if 1 + j ∈ 1 + JK is another element such that BK = (1 + j )−1 ϕ(BK )(1 + j ), then conjugation by (1 + j)(1 + j )−1 ∈ 1 + JK takes BK to itself. Then it follows from Lemma 3 that 1 + j = 1 + j . Let Inn∗ (AK ) be the subgroup of inner automorphisms of AK determined by elements in 1 + JK . By the observations above we see that Inn∗ (AK ) ∼ = 1 + JK . Furthermore, up to a unique inner automorphism given by an element in 1 + JK , every K-automorphism of AK preserves BK . Thus, we have the following proposition. Proposition 1. Inn∗ (AK ) ∼ = 1 +JK is a normal subgroup of AutK (AK ) and AutK (AK ) = ˆ In particular Inn∗ (AK ) H. ˆ→1 1 → Inn∗ (AK ) → AutK (AK ) → H
(2)
is a split exact sequence of groups. The rest of this section (as well as of this paper) is devoted to understanding the ˆ nature of H. When the inertial subgroup is the entire Galois group G, f is a classical cocycle, ˆ∼ AK ∼ = P GLn (K). The simplest non-classical situation is = Mn (K) and AutK (AK ) = H when f has a trivial inertial group, in which case BK = K n . Even in the general case, knowing the structure of K-automorphisms of AK taking K n to itself turns out to be useful. Every K-automorphism of K n is determined by the permutation it induces on the set of minimal idempotents {eτ = (δτ,σ )σ∈G : τ ∈ G} of K n , where, δτ,σ = 1 if τ = σ and 0 otherwise. Since each minimal idempotent is identified by an element τ in G, we have AutK (K n ) ∼ = P erm(G) = Sn . We use this identification throughout.
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Let ϕ ∈ AutK (AK ) be a K-automorphism which takes K n to K n . Then, ϕ|K n : K n → K n is a K-automorphism of K n . We will denote the permutation of G obtained by restricting ϕ to K n by the same letter, ϕ. With this terminology we may write ϕ(eτ ) = eϕ(τ ) , where on the right hand side ϕ refers to the permutation induced by ϕ on G. The following proposition identifies the structure of K-automorphisms of AK that preserve K n . Proposition 2. Every ϕ ∈ AutK (AK ) that preserve K n is of the form ϕ (eτ xg ) = aτ,g eϕ(τ ) xϕ(τ )−1 ϕ(τ g)
(3)
where aτ,g ∈ K × and ϕ ∈ SG = P erm(G) satisfy the following conditions for all τ, g, g ∈ G. g ≤f g ⇐⇒ ϕ(τ )−1 ϕ(τ g) ≤f ϕ(τ )−1 ϕ(τ g )
aτ,g aτ g,g f ϕ(τ ) (ϕ(τ )−1 ϕ(τ g), ϕ(τ g)−1 ϕ(τ gg )) = aτ,gg f τ (g, g )
(4) (5)
Note that ϕ in the right hand side of (3) is r(ϕ) = ϕ|K n considered as a permutation of idempotents. Conversely, if ϕ ∈ SG satisfies condition (4) and we choose elements aτ,g ∈ K × satisfying (5) for all τ, g ∈ G, then the map ϕ obtained by extending formula (3) linearly is a K-automorphism of AK preserving K n . g g n Proof. Suppose ϕ(xg ) = h∈G ah xh where, ah ∈ K . Then, one can easily verify τ,g τ,g that ϕ(eτ xg ) = is the ϕ(τ )-coordinate of ahg . Because h∈G ah eϕ(τ ) xh , where ah ϕ is a K-algebra homomorphism ϕ(eτ xg eτ xg ) = ϕ(eτ xg )ϕ(eτ xg ). On the other τ,g σ −1 hand, (g, g ))σ∈G xgg and thus we have h∈G ah eϕ(τ ) xh × eτ xg eτ xg = eτ eτ g (f τ ,g σ h ∈G ah eϕ(τ ) xh = ϕ eτ eτ g −1 (f (g, g ))σ∈G xgg . τ,g In particular, when g = 1G (i.e. xg = 1) and simplifying we have ah eϕ(τ ) × τ,g eϕ(τ )h−1 xh = ah eϕ(τ ) eϕ(τ g−1 ) xh . The unique h ∈ G for which aτ,g = 0 is given by h −1 h = ϕ(τ ) ϕ(τ g). This obtains (3). Using (3) one can easily get that for all τ , g, g in G,
aτ,g aτ g,g f ϕ(τ ) (ϕ(τ )−1 ϕ(τ g), ϕ(τ g)−1 ϕ(τ gg )) = aτ,gg f τ (g, g ). In particular, we have g ≤f gg ⇐⇒ ϕ(τ )−1 ϕ(τ g) ≤f ϕ(τ )−1 ϕ(τ gg ) and both sides of the equality are nonzero precisely when g ≤f gg . The proof of the converse is similar and we leave the verification to the reader. 2 Consider the case where the inertial subgroup of f is trivial, i.e., Hf = {1G }. In this case, B = Kx1G ∼ J, where J = g=1G Kxg . Furthermore, = K. Therefore, Af = K BK ∼ =K K n is a commutative algebra and JK = g=1G K n xg . The lower subtractive
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partial order on G defined by f as defined in Section 1 is given by g ≤ gh ⇐⇒ f (g, h) = 0. As an application of Proposition 2, we may record the following corollary. ˆ is of the form Corollary 1. If f has trivial inertial group, then every automorphism in H given by (3) satisfying (4) and (5). Returning to the general case, let H = Hf be the inertial group of f , L = K H the fixed field of H, m = |H| = deg(K/L) and l = n/m = |G/H|. Moreover, let T = {σ1 , ..., σl }, with σ1 = 1G , be a complete set of coset representatives of H. Then Li = σi (L), for i = 1, .., l, are the distinct embeddings of L into K. Let Ki be the field K with L vector space structure twisted by σi . That is, for l ∈ L and x ∈ K, l.x = σi (l)x. The map l l l α : K ⊗F L → i=1 Kσi which sends x ⊗ l to (l.x)i=1 = (xσi (l))i=1 and extended linearly is an isomorphism of K-algebras. We will first compute the K-automorphism group of BK (Proposition 3) and then use it to investigate K-automorphisms of AK . The K- automorphisms of AK that stabilize BK (for a general inertial group) are described in Theorem 1, and is analogous to that of a trivial inertial group (Corollary 1). Recall that B is a classical crossed product algebra with respect to the Galois group H and the cocycle f1 = f |H×H . Thus, by using the crossed product notation we can write B = (K/L, H, f1 ). The Galois group of K/Li is Hi = σi Hσi−1 and there is an associated cocycle fi and a crossed product algebra Bi given by the G action on the second cohomology of G. Explicitly, fi : Hi × Hi → K is the cocycle given by fi (σi h1 σi−1 , σi h2 σi−1 ) = f σi (h1 , h2 ) = σi (f (h1 , h2 )), for h1 and h2 in H, and Bi = (K/Li , Hi , fi ). The algebra Bi is a central simple Li algebra and there is a K-isomorphism of algebras Ki ⊗L B → K ⊗Li Bi which sends w⊗vxh to w⊗σi (v)xσi hσ−1 . i Thus, all the algebras Ki ⊗L B are split and K-isomorphic to the algebra of m × m matrices, Mm (K). Therefore, we have the following isomorphisms of K-algebras. BK
∼ =K =K (K ⊗F L) ⊗L B ∼
l
i=1
Ki ⊗L B ∼ =K
l
(K ⊗Li
Bi ) ∼ =K
i=1
l
Mm (K)
i=1
Lemma 4. AutK (BK ) ∼ = Inn(BK ) Sl , where Inn(BK ) ∼ = P GLm (K)l . Hence there is a split short exact sequence of groups as follows. 1 → P GLm (K)l → AutK (BK ) → Sl → 1
(6)
l Proof. We need to compute the K-automorphism group of M := i=1 Mm (K). Any permutation of the coordinates is a K-automorphism of M and any automorphism of M up to a unique permutation preserves coordinates. Any K-automorphism which preserves the coordinates is a product of K- automorphisms of Mm (K). Because AutK (Mm (K)) = P GLm (K), the group of K- automorphisms of M that preserve the coordinates × × l l l l × × is P GL (K) = (M (K) /K ) = M (K) / K) = m m m i=1 i=1 i=1 i=1
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M × / (Z(M )× ) = Inn(M ) ∼ = Inn(BK ). Also note that P GLm (K)l is normal subgroup. In fact, for (ϕi ) ∈ P GLm (K)l and π ∈ Sl , π(ϕi )π −1 = (ϕπ(i) ). 2 We use a set T = {σ1 , ..., σl } of distinct coset representatives to compute AutK (BK ) explicitly. The set T is naturally in bijection with G/H where the latter is a G set. This gives the set T a G structure, given by g.σi = σj ⇐⇒ g.σi H = σj H. In this section we will identify the permutation group on l letters as the group of permutations of the set T . So, Sl = P erm (T ) and there is a G action on Sl given by g ϕ(σi ) = g.ϕ(g −1 .σi ), for g ∈ G. Because B is a central simple L-algebra, the center of BK is K l ∼ = K ⊗F L, and l has K-dimension l = dim(L/F ). Let e˜σi = k∈H eσi k and e˜h = i=1 eσi h . We have eσi h = e˜σi e˜h . Note that for i = 1, .., l, e˜σi commutes with every xh for h ∈ H, and hence is in the center of BK . By considering the dimension, it follows that the center of BK is the K-span of {˜ eσi |i = 1, ..., l}. Every permutation ϕ ∈ Sl = P erm(T ) lifts to a permutation on G, say ϕ, ˜ given by ϕ(σ ˜ i h) = ϕ(σi )h. For ϕ ∈ Sl , ϕ˜ ∈ P erm(G) satisfies the property ϕ(gh) ˜ = ϕ(g)h ˜ for any g ∈ G and h ∈ H. This permutation on G corresponds to the K-automorphism of K |G| = K n , which sends e˜σi to e˜ϕ(σi ) for 1 ≤ i ≤ l and fixes e˜h ’s element-wise for h ∈ H. We will extend ϕ˜ to a K-linear map on BK , which we will also denote by ϕ. ˜ Consider the K-basis {eτ xk |τ ∈ G, k ∈ H} of BK and express every τ ∈ G in the unique form τ = σi h for σi ∈ T and h ∈ H. We define ϕ˜ : BK → BK to be the K-linear extension of the following map: ϕ˜ : eσi h xk →
f σi (h, k) eϕ(σi )h xk f ϕ(σi ) (h, k)
(7)
where h, k ∈ H. We claim ϕ˜ is a K-algebra automorphism of BK . To verify this, first observe that Proposition 2 is applicable here. For k ∈ H, f σi (h,k) ϕ(τ ˜ )−1 ϕ(τ ˜ k) = k, and (7) has the form given in Proposition 2 with aσi h,k = f ϕ(σ . i ) (h,k) Condition (4) in Proposition 2 is trivially satisfied (for this case g and g in (4) are in H). Therefore, what we need to verify is (5), which reads as f σi (h, kl) σi h f σi (h, k) f σi (hk, l) ϕ(σi )h f f (k, l), (k, l) = f ϕ(σi ) (h, k) f ϕ(σi ) (hk, l) f ϕ(σi ) (h, kl) and follows from the cocycle condition on f . Moreover, for ϕ1 and ϕ2 in Sl , one can verify the equality ϕ˜2 ◦ ϕ˜1 = ϕ 2 ◦ ϕ1 by checking it on the K basis eσi h xk . It follows from this that S˜l is a subgroup of AutK (BK ) that is isomorphic to Sl . The intersection of S˜l with Inn(BK ) is trivial, because any inner automorphism of BK is identity on its center. Then from Lemma 4 it follows that AutK (BK )/InnK (BK ) ∼ = S˜l and |AutK (BK )/InnK (BK )| = l!. We summarize these observations, and a few other easy to verify facts, in the following proposition:
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Proposition 3. 1. The K- linear endomorphism, ϕ˜ : BK → BK given by (7) is a K-automorphism of BK , which takes K n back to itself. 2. The restriction of ϕ˜ to the center is the identity if and only if ϕ is the identity permutation. In fact, ϕ(˜ ˜ eσi ) = e˜ϕ(σi ) . 3. S˜l := {ϕ˜ ∈ AutK (BK )|ϕ ∈ Sl } ∼ = Sl is a subgroup of K-automorphisms of BK whose intersection with Inn(BK ) is trivial. 4. AutK (BK ) = Inn(BK ) S˜l . Equivalently, there is a split exact sequence as follows. 1 → Inn(BK ) → AutK (BK ) → S˜l → 1 × 5. Given a K-automorphism φ of BK , there is bφ ∈ BK , unique up to a central unit, −1 ˜ such that x → bφ φ(x)bφ ∈ Sl .
Thus, we have realized a subgroup of automorphisms of BK which corresponds to Sl in ˆ in the general setting, let us first recall Proposition 2 Lemma 4. To prove a result on H and note that every K-automorphism of AK which preserves BK and restricts to an automorphism of BK in S˜l also preserves K n . We will denote the conjugation by an invertible element u by Cu . ˆ consider its restriction to BK , r(φ). There exist b ∈ B × (unique up to a Given φ ∈ H, K central unit in BK ) and ϕ˜ ∈ S˜l such that r(φ)= Cb ◦ ϕ. ˜ Let φ = Cb−1 ◦φ. Then r(φ ) = ϕ˜ and hence, φ is of the form in (7). Because ϕ˜ preserves K n , so does φ and its restriction to K n is given by the permutation ϕ ˜ ∈ P erm(G), which permutes the idempotents eτ ’s. Thus, from Proposition 2, φ (eτ xg ) = aτ,g eϕ(τ ˜ ) xϕ(τ ˜ )−1 ϕ(τ ˜ g) , which satisfies conditions f σi (h,k) (4), (5) and aσi h,k = f ϕ(σ for all τ = σ h ∈ G and h, k ∈ H. i i ) (h,k)
× Note that the choice of b ∈ BK is not unique and can be replaced by vb for any × v ∈ Z(BK ) . To understand the effect of choosing vb instead of b, we need to understand the nature of the conjugation automorphism, Cv , by an element v ∈ Z(BK )× . Because −1 Z(BK ) = i=1,..,l K e˜σi ∼ ui e˜σi where ui ∈ K × and v −1 = ui e˜σi . = K G/H , v = For g ∈ G, let ug¯ = ui where gH = σi H. Then one can easily verify that Cv (eτ xg ) = uτ¯ uτ−1 ¯g eτ xg . Thus we have the following theorem.
ˆ there is a unit b ∈ B × and a unique ϕ ∈ Sl such that Theorem 1. Given φ ∈ H, K b φ (eτ xg ) b−1 = aτ,g eϕ(τ ˜ ) xϕ(τ ˜ )−1 ϕ(τ ˜ g)
(8)
where aτ,g (b) = aτ,g ∈ K ∗ and ϕ˜ ∈ SG = P erm(G) satisfies the following conditions for all τ, g, g ∈ G, h, k ∈ H and σi ∈ T : g ≤f g ⇐⇒ ϕ(τ ˜ )−1 ϕ(τ ˜ g) ≤f ϕ(τ ˜ )−1 ϕ(τ ˜ g )
(9)
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˜ ) aτ,g aτ g,g f ϕ(τ (ϕ(τ ˜ )−1 ϕ(τ ˜ g), ϕ(τ ˜ g)−1 ϕ(τ ˜ gg )) = aτ,gg f τ (g, g )
aσi h,k =
(10)
σi
f (h, k) f ϕ(σi ) (h, k)
(11)
Furthermore, b can be replaced by vb for any v ∈ Z(BK )× . If v = ui e˜σi ∈ Z(BK )× , −1 τ,g τ,g changing b to vb will change the coefficients a to c = uϕ(τ aτ,g . ˜ ) uϕ(τ ˜ g) ˆ that have the form given Definition 2. Let Γ denote the subgroup of automorphisms in H by the right hand side of Equation (8) in Theorem 1. ˆ | φ (eτ xg ) = aτ,g eϕ(τ Γ = {φ ∈ H ˜ ) xϕ(τ ˜ )−1 ϕ(τ ˜ g) , aτ,g ∈ K × , ϕ ∈ Sl and satisfies conditions (9) , (10) and (11)} Clearly Γ = r−1 (S˜l ) and hence is a subgroup. Also, let us denote the subgroup of × ˆ given by the units of BK by B ¯∼ inner automorphisms in H /K × . Because φ ◦ Cb = = BK ¯ is a normal subgroup of H. ˆ Furthermore, it follows from the last part of Cφ(b) ◦ φ, B ¯ Proposition 3 that φ ∈ B ∩ Γ if and only if it is a conjugation by a central element. Thus ¯ ∩ Γ = Z(BK )× /K × ∼ B = (K l )× /K × . Thus, we have the following corollary. ˆ = BΓ, ¯ B ¯ H ˆ and B ¯ ∩Γ∼ Corollary 2. H = (K l )× /K × . ˜ ≤ S˜l and Due to condition (9), it is natural to consider the following subgroups, Λ Λ ≤ Sl . Definition 3. Λ := {ϕ ∈ Sl |∀g1 , g2 , τ ∈ G, g ≤ g ⇐⇒ ϕ(τ ˜ )−1 ϕ(τ ˜ g) ≤ ϕ(τ ˜ )−1 ϕ(τ ˜ g )}
(12)
˜ := {ϕ˜ ∈ S˜l |ϕ ∈ Λ} Λ
(13)
We wish to remind the reader that ϕ ˜ stands for the lift of ϕ ∈ Sl to an element either in P erm(G) or in S˜l , which could be understood without ambiguity from the context. ˜ Moreover, the isomorphism Sl ∼ = S˜l , given by (7), restricts to an isomorphism Λ ∼ = Λ. ˜ ˜ Before we discuss more about Λ, it is worth noting that r(Γ) < Λ. 2.1. H : BK -automorphisms of AK ˆ (or of AutK (AK )) consisting of all automorphisms Recall that H is the subgroup of H which restrict to the identity on BK . Thus every φ ∈ H (by assumption) acts trivially on the elements xh for h ∈ H. Each φ ∈ H is of the form φ(eτ xg ) = aτ,g eτ xg , where the aτ,g τ,g satisfy conditions (10) and (11). Thus, φ(xg ) = φ( τ eτ xg ) = a eτ xg = (aτ,g )τ ∈G xg and the corresponding permutation ϕ˜ on G is the identity. Hence the condition (10)
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can be rewritten as aτ,g aτ g,g f τ (g, g ) = aτ,gg f τ (g, g ) whenever g ≤ gg . However, f (g, g ) = 0 ⇐⇒ g ≤ gg . Therefore, the condition (10) amounts to
aτ,g aτ g,g = aτ,gg whenever g ≤ gg .
(14)
Given a G-module M , written multiplicatively, we will write the action of g ∈ G on m ∈ M by g m. Definition 4 (Weak 1-cocycle (see [1])). Given a G-module M , a function a : G → M is a weak 1-cocycle, with respect to a given lower subtractive relation ≤, if a satisfies the condition a(gg ) = a(g) g a(g ) whenever g ≤ gg
(15)
a(h) = 1 for h ≤ 1G (i.e. h ∈ H)
(16)
Weak 1-cocycles with pointwise multiplication form a group, which we denote by 1 Z≤ (G, M ). Thus, 1 Z≤ (G, M ) = {a : G → M | a satisfies conditions (15) and (16)}
For each a = (aτ )τ ∈G ∈ K n and g ∈ G, define g a by g
(aτ )τ ∈G = (aτ g )τ ∈G
(17)
This defines an action of G on K n . In fact this is the action of xg ’s on K n in Proposition 1. Furthermore, conditions (14) and (11) imply that the function a : G → (K n )× , where a(g) = ag = (aτ,g )τ ∈G is a weak-1-cocycle 1 Proposition 4. The group H is isomorphic to the group of weak 1-cocycles Z≤ (G, (K n )× ), n 1 where G acts on K according to (17). Given a weak 1-cocycle a ∈ Z≤ (G, (K n )× ), the associated automorphism is given by
φ(xg ) = (aτ,g )τ ∈G xg and φ(eτ ) = eτ
(18)
Proof. We saw that for each ϕ ∈ H, there is an associated weak 1-cocycle. Moreover, since the action of G on K n is precisely the action of xg ’s on the idempotents of K n (in Proposition 1), every such weak 1-cocycle determines a K-automorphism of AK as given. 2 Remark 1. The partial order on G/H induces a partial order on T (choosing a set of coset representatives). Let ht(1) = {σ2 , σ3 , ..., σr } be the height one elements of this partial order. Then, G is generated by ht(1) ∪ H. In fact, every element of G is of the form σi1 σi2 ...σir h, where σi1 , ..., σir ∈ ht(1) (see [2]) for details). One can easily verify
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that the value of any weak 1-cocycle a at g is uniquely determined by the values of a on σi1 , σi2 , ..., σir (see (24) in Example 1). Hence, every weak 1-cocycle is determined by its values on the ht(1) elements and therefore H < |ht(1)| (K n )× . Because P GLm (K)l ∼ = Inn(BK ) ⊆ Img(r) and Img(r) ∩ S˜l = r(Γ), we can now rewrite the short exact sequence (1) as follows. Corollary 3. The exact sequence (1) can be rewritten as follows. 1 ˆ− 1 → Z≤ (G, (K n )× ) → H → Inn(BK ) r(Γ) → 1 r
(19)
3. The automorphism group of an idempotent cocycle In this section we will consider the special case where f is an idempotent cocycle. Recall that f : G × G → K is an idempotent cocycle if it takes only the values 0 and 1 in K. Such cocycles are in one-to-one correspondence with lower subtractive relations ≤ on G with the property that 1G ≤ g for all g ∈ G. ˜ We know that Inn(BK ) ≤ r(H) ˆ is a subgroup of Inn(BK ) Λ. ˆ Recall that r(H) ˜ and r(Γ) ≤ Λ. However it is not clear, in the general case, whether every permutation ˜ lifts to an automorphism of AK . Nonetheless, when the cocycle is an idempotent in Λ ˜ the map cocycle, it is clear that we can choose aτ,g = 1 in (8). Thus, given ϕ˜ ∈ Λ, ˜ ϕ˜ : eτ xg → eϕ(τ ˜ ) xϕ(τ ˜ )−1 ϕ(τ ˜ g) is a K-automorphism of AK . Therefore, r(Γ) = Λ and r ˜ ˆ onto Inn(BK ) Λ. The short exact sequence (19) for this case is maps H 1 ˜ →1 ˆ → InnK (BK ) Λ 1 → Z≤ (G, K n ) → H
(20)
Remark 2. Recall the G action on AutK (AK ) given by g ϕ = (g ⊗1) ◦ϕ ◦(g −1 ⊗1) for ϕ ∈ ˜ to an element in H, ˆ gives AutK (K ⊗ A). The lifting described above, of an element in Λ ˜ ˆ rise to a group homomorphism from Λ to H. However, it should be noted that this group homomorphism is not a G map, with respect to the G action on automorphisms. In fact ˜ is not even G stable in general. For example in the case of the Waterhouse idempotent Λ ˜ = S˜l and S˜l is not G invariant. Moreover, even though every element (see Example 3), Λ ˆ there is no canonical way of doing it. in Inn(BK ) can be lifted to an element in H, 3.1. Lower subtractive partial orders and some observations about Λ Recall that T is a set of distinct coset representatives of H, S˜l ∼ = Sl = P erm(T ) and Λ is defined by (12). The definition of Λ depends only on the lower subtractive relation defined on G. Hence it is determined by the corresponding idempotent cocycle. The group G acts from the left on G/H and hence for each γ ∈ G, there is an associated permutation of cosets in G/H which induces a permutation on T . We denote the permutation induced by γ ∈ G on T by γ¯ and it is given by γ¯ (σi ) = σj ⇐⇒ γσi H =
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σj H ⇐⇒ σj−1 γσi ∈ H. In this way, we get a group homomorphism from G to Sl , γ → γ¯ . ¯ and image of H by H. ¯ Also, Let us denote the image of G in Sl under this map by G let us write γ.¯ g for γ¯ (σi ) where g ∈ σi H. Lemma 5. Λ = {ϕ ∈ Sl | for all g1 , g2 , τ ∈ G, g1 ≤ g2 ⇐⇒
ϕ(τ ˜ )−1 . ϕ(τ.g¯1 ) ≤ ϕ(τ ˜ )−1 . ϕ(τ.g¯2 )},
¯ < Λ. and G Proof. We will leave it for the reader to verify that the condition that defines Λ can be ¯ < Λ. expressed as given here. We will outline the proof of G Recall that h ≤ g for all g ∈ G and h ∈ H. For any g, g ∈ G and h ∈ H, hg ≤ hg ⇔ (h ≤ hg ≤ hg and h ≤ hg ) ⇔ h−1 hg ≤ h−1 hg ⇔ g ≤ g , where the second if and only if is due to the lower subtractive property. Thus, left multiplication by elements of the inertial subgroup preserves the partial order. Let γ ∈ G. Note that for any τ ∈ G, γ˜¯ (τ ) = γτ h for some h ∈ H. Thus, for any g1 , g2 , τ ∈ G, there are h, h1 , h2 ∈ H, such that −1 γ˜¯ (τ )−1 γ˜¯ (τ gi ) = (γτ h) (γτ gi hi ) = h−1 gi hi
Thus; γ˜¯ (τ )−1 γ˜¯ (τ g1 ) ≤ γ˜¯ (τ )−1 γ˜¯ (τ g2 ) ⇐⇒ h−1 g1 h1 ≤ h−1 g2 h2 ⇐⇒ g1 h1 ≤ g2 h2 ⇐⇒ g1 ≤ g2 The last equivalence follows from g1 h1 ≤ g2 h2 ⇐⇒ g1 h1 ≤ g2 h2 ⇐⇒ g¯1 ≤ g¯2 ⇐⇒ g 1 ≤ g2 . 2 The ordering on G restricts to a partial order on T , which is isomorphic as partial orders to G/H. Let us denote the subgroup of permutations in Sl that preserve the ordering by Aut(≤). One can easily show that ϕ ∈ Sl is in Aut(≤) if and only if ϕ˜ preserve the ordering on G. Let us also define Λ˙ to be the order preserving permutations, that is, Λ˙ = Aut(≤) ∩ Λ. Suppose ϕ ∈ Sl is in Λ. Let σ1 ∈ T be the element such that σ1 H = H. For any γ ∈ Hϕ(σ1 )−1 , γ¯ ϕ is a permutation in Λ, which takes σ1 back to itself. Now, by taking
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˙ τ = 1 in the definition of Λ, we see that γ¯ϕ ∈ Λ˙ also preserves the order. Thus, γ¯ϕ ∈ Λ. ˙ Hence, we may conclude that ¯ and γ¯ ϕ ∈ Λ. Therefore, ϕ = γ −1 (¯ γ ϕ), where γ −1 ∈ G ˙ ˙ ¯ ¯ Λ = GΛ. Furthermore, if γ¯ ∈ G ∩ Λ, then it preserves the root σ1 , and hence γ ∈ H ˙ Therefore, we have the following ¯ Conversely any element in H ¯ is in Λ. and γ¯ ∈ H. proposition. ¯ Λ˙ and G ¯ ∩ Λ˙ = H¯f Proposition 5. Λ = G Associated to ≤, there are two right lower-subtractive relations with the same inertial group, the right-graph ≤∗ and the inverse-graph ≤− . The right graph, ≤∗ , was first introduced in [6] to study the ideal structure of Af . Definition 5. Relations ≤∗ and ≤− are defined by the following rules. 1. g ≤∗ h ⇐⇒ hg −1 ≤ h (or equivalently x ≤ y ⇐⇒ x−1 y ≤∗ y). − −1 −1 2. g ≤ h ⇐⇒ g ≤ h . The following lemma follows directly from the definition. We leave the verification to the reader. Lemma 6. ≤∗ and ≤− are right lower subtractive relations on G with the same inertial group as ≤. Dually, given a right lower-subtractive relation , there are two associated left lower subtractive relations, ∗ and − . The following properties can be easily verified. Proposition 6. If ≤ is a lower subtractive relation on G, then the following holds. 1. (≤∗ )∗ = ≤ i.e., x(≤∗ )∗ y ⇐⇒ x ≤ y − 2. (≤ )− = ≤ 3. (≤∗ )− = (≤− )∗ . Definition 6. The “dual” of a left lower subtractive relation ≤, denoted by ≤† , is the left lower subtractive relation (≤− )∗ =(≤∗ )− . Corollary 4. 1. x ≤† y ⇐⇒ y −1 x ≤ y −1 . Equivalently, x ≤ y ⇐⇒ y −1 x ≤† y −1 † 2. ≤† = ≤. The relations ≤† and ≤ share several important properties. For example the graph of ≤ on G/H is a tree (acyclic) if and only if the graph of ≤† on G/H is a tree. Moreover, for a standard graph with respect to the generating set S, the dual graph ≤† is also
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standard with generating set S −1 , the set of inverses of the elements in S. In particular if the set of height one elements is closed under taking inverses (for example all the generators considered in a Coxeter group have order 2) the dual relation is the same as the original. Let us denote the corresponding Λ for ≤† by Λ† . The statement of the following proposition also relates ≤ and ≤† and is particularly useful in the setting of this paper. Proposition 7. Λ = Λ† . Proof. Let ϕ˜ ∈ Λ. To show that ϕ˜ ∈ Λ† , we need to show g ≤† g if and only if ϕ(τ ˜ )−1 ϕ(τ ˜ g) ≤† ϕ(τ ˜ )−1 ϕ(τ ˜ g ), ∀τ ∈ G. From the definition of ≤† , the condition above amounts to showing g −1 g ≤ g −1 ⇐⇒ ϕ(τ ˜ g )−1 ϕ(τ ˜ g) ≤ ϕ(τ ˜ g )−1 ϕ(τ ˜ ), ∀τ ∈ G.
(21)
Because ϕ˜ is in Λ, we have g −1 g ≤ g −1 ⇐⇒ ϕ(τ ˜ )−1 ϕ(τ ˜ g −1 g) ≤ ϕ(τ ˜ )−1 ϕ(τ ˜ g −1 )∀τ ∈ G
(22)
Because τ is arbitrary, the equation (21) is obtained by replacing τ in (22) by τ g . Thus, we have that ϕ˜ ∈ Λ† . Thus Λ ⊆ Λ† . Applying this to Λ† , we also have Λ† ⊆ (Λ† )† = Λ. Therefore, Λ = Λ† . 2 Proposition 7 is particularly useful in computing Λ (see Example 4). It implies that, if e is the idempotent cocycle associated with ≤ and e† the idempotent cocycle associated to ≤† , then the K-automorphism groups of K ⊗F Ae and K ⊗F Ae† share Λ as a common subgroup. The group Λ will also be useful in computing the K-forms of Ae and Ae† . 3.2. Idempotent cocycles with trivial inertial group In this subsection we consider idempotent cocycles with trivial inertial subgroups. In this case BK is commutative and hence there are no nontrivial inner automorphisms. Let Λ be the subgroup of P erm(G) = Sn which satisfies the condition given in (4). Thus: Λ = {ϕ ∈ P erm(G)|∀g, g , τ ∈ G, g ≤ g ⇐⇒ ϕ(τ )−1 ϕ(τ g) ≤ ϕ(τ )−1 ϕ(τ g )} As observed just before Remark 2, ϕ ∈ Λ lifts (canonically) to a K- automorphism of ˆ by choosing aτ,g = 1 in (3) of Proposition 2. As a consequence, the image of AK , ϕ˜ in H, ˜ (1) is a split short exact sequence and H ˆ∼ r is Λ, = H Λ. We record this in the following lemma.
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Lemma 7. Lef f be an idempotent cocycle with trivial inertial subgroup. Then the sequence ˆ→Λ→1 1→H→H
(23)
ˆ∼ is split. Therefore, H = H Λ. Consider the left regular representation of G in SG . Every element γ ∈ G defines a permutation of G by left multiplication. We will denote this permutation by lγ . The map l : γ → lγ , embeds G in SG . In this way we identify G as a subgroup of SG . We have Λ = G.Λ˙ and G ∩ Λ˙ = {id}. Also, if ψ ∈ Aut(G) then for every τ ∈ G, ψ(τ )−1 ψ(tg) = ψ(g) and hence ψ ∈ Λ if and only if ψ ∈ Aut(≤). Thus, every order preserving automorphism of G is in Λ. I.e., Aut(G) Aut(≤) is a subgroup of Λ. Remark 3. Short exact sequence (23) is in fact a short exact sequence of G groups with respect to the G action on AutK (AK ) mentioned in Remark 2. The copy of G in SG is also G invariant with respect to this action, which reduces to the conjugation action of G on itself. 4. Examples Example 1. When the inertial group of the weak 2-cocycle f is the trivial subgroup {1G }. In this case B = K and hence BK is a commutative algebra. The relation ≤ is a lower subtractive partial order on G and AutK (BK ) ∼ = P erm(G). Furthermore, Λ = {ϕ ∈ SG = P erm(G)| ∀g, g , τ ∈ G, g ≤ g ⇐⇒ ϕ(τ )−1 ϕ(τ g) ≤ ϕ(τ )−1 ϕ(τ g )} ˆ < Λ and H ˆ = Γ. Note that the condition (11) is trivial. In is a subgroup of SG , r(H) ˆ fact, H, as explained in Corollary 1, is of the form (3) subject to conditions (4) and (5) in Proposition 2. Thus, AutK (AK ) ∼ = (1 + JK ) Γ. Consider the subgroup H in this case. Let ht(1) = {g1 , ..., gr } be the set of height 1 elements (i.e., the elements immediately above 1G ). Recall that G is generated by ht(1). Consider an element g ∈ G and a path 1G < gk1 < gk1 gk2 < gk1 gk2 gk3 < ... <
t
gki = g
i=1
starting from 1G and ending at g. Then one can use the cocycle condition to find the value of ag in terms of the values of agi ’s as follows. ag = a ti=1 gk = agk1 gk1 a ti=2 gk = agk1 gk1 ak2 ...( i
i
t−1 i=1
gk i )
akt
(24)
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Therefore, the value of a at g is determined by the values of a on the height 1 elements. Hence H is a subgroup of the group of functions from ht(1) to K n , M aps (ht(1), (K n )× ) = |ht(1)| (K n )× , where a is in H if and only if for every g ∈ G and for all distinct paths to g, computation in (24) evaluates to the same value. Here, G act on |ht(1)| (K n )× diagonally where the action on each K n is as in (17). Example 2. When the graph of the partial order on G/H induced by a weak 2-cocycle f is a tree. Let a : G → K n a weak 1-cocycle. Given any g ∈ G, because there is a unique path from 1G H to gH, it follows from Remark 1 and the above example that, ag is uniquely determined by the height one elements in G/H. Thus, we have the following proposition. 1 Proposition 8. When the graph of ≤ on G/H is a tree, weak-1-cocycles, Z≤ (G, (K n )× ), are in one-to-one correspondence with the functions from ground elements, ht(1) = {σ2 H, σ3 H, ..., σr H}, to K n .
H = Z 1 (G, (K n )× ) ∼ =
(K n )×
(r-1) times
Example 3. The Waterhouse idempotent: Consider the partial order ≤w on G with 1G as the minimum element and with no other nontrivial relations. That is, the only relations are 1 ≤w g and g ≤w g for all g ∈ G. The corresponding idempotent cocycle, called the Waterhouse idempotent, is w : G ×G → {0, 1} where w(g, h) = 0 ⇐⇒ g = 1g or h = 1g . The corresponding weak crossed product algebra is Aw = K ⊕ J with J 2 = 0. One can |G|−1 ˆ∼ easily verify that in this case Λ = SG , H = g∈G,g=1G (K n ) and H SG . = (K n ) |G|−1 Thus, AutK (K ⊗ Aw ) ∼ SG . = (1 + JK ) (K n ) Example 4. Consider the following lower subtractive partial order, shown in the left as a graph, on the cyclic group of order 5, C 5 , generated by α. The dual of ≤, ≤† , is shown in the right. α3
≤
α
α4
α2
1G
≤†
α2
α
α4
α3
1G
˙ i.e., in Λ and Because Λ = Λ† , one can easily see that the only permutation in Λ, ∼ preserves the order, is the identity permutation. Thus, Λ = LG = G. Furthermore,
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H = K n × K n . Let Ae be the weak crossed product algebra that corresponds to the ˆ where idempotent cocycle e arising from ≤. Then AutK (K ⊗ Ae ) ∼ = (1 + JK ) H n n ∼ ˆ H = (K × K ) G. Example 5. Consider lower subtractive partial order on the cyclic group of order 5 generated by α, C 5 , with respect to the generating set S = {α, α4 }. α2
α3
α
α4
1G Note that there are only two order preserving maps, the identity map and the map ˙ Thus, Λ = which sends α → α−1 . One can easily verify that both of these are in Λ. G.Λ˙ ∼ = G C 2 , where C 2 ∼ = Λ˙ is the cyclic group of order 2. Moreover, H = K n × K n as ˆ of the K-automorphism group of the corresponding before. Therefore, the subgroup H n algebra, is isomorphic to (K × K n ) (G C 2 ). As before, if Ae is the weak crossed product algebra that corresponds to the idempotent cocycle e arising from the lower subtractive partial order we are considering, then AutK (K ⊗ Ae ) ∼ = (1 + JK ) ((K n × K n ) (G C 2 )). Example 6. Weak Bruhat order of a Coxeter system. Let (W, S) be a finite Coxeter group. Associated to this is its diagram DS . By a diagram automorphism we mean an automorphism of the graph DS . Every such diagram automorphism gives rise to a unique automorphism of W preserving S as a set. In what follows, we will denote these diagram automorphism as well as their extension to automorphism of W by Aut(DS ). The left weak order of W is a lower subtractive partial order which we denote by ≤. Proposition 9. Λ˙ = Aut(DS ) Proof. Suppose φ ∈ Aut(DS ). Recall that an automorphism of W is in Λ if and only if it preserves the order. The fact that φ preserves the order follows from the fact that φ is a homomorphism and for w ∈ W if w = s1 s2 · · · sk is a reduced word for w then ˙ φ(w) = φ(s1 )φ(s2 ) · · · φ(sk ) is a reduced word for φ(w). Thus φ ∈ Λ. ˙ To show the converse, suppose φ ∈ Λ. Then φ in particular, preserve the weak order ≤ and hence φ maps S to S. Recall that for a subset X of S, the subgroup generated by X, WX , forms a lower interval in the weak order. Moreover, this interval is the weak order on WX and WX S = X (see [10,11]).
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Consider two element s1 , s2 with O(s1 s2 ) = m and gm = s1 s2 s1 · · · = s2 s1 s2 · · · where each product has m terms. Because s1 ≤ gm and s2 ≤ gm and only s1 and s2 are the elements below gm , it follows that φ(s1 ) ≤ φ(gm ) and φ(s2 ) ≤ φ(gm ) and only φ(s1 ) and φ(s2 ) are below φ(gm ). It follows by considering X = {φ(s1 ), φ(s2 )}, φ(gm ) = φ(s1 )φ(s2 )φ(s1 ) · · · = φ(s2 )φ(s1 )φ(s2 ) · · · , with m terms and hence O(φ(s1 )φ(s2 )) = m. Thus, φ is a diagram automorphism. The fact that φ is the extension of the diagram automorphism also follows similarly. 2 Proposition 10. W Λ and Λ = W Aut(DS ) Proof. Since W Λ˙ = {id}, we only need to show that W is normal in Λ. Let lw be the left multiplication by an element in W . Then for φ ∈ Λ˙ and with the help of Proposition 9, we have φ lg φ−1 = lφ(g) . 2 It is also worth noting that conjugation with respect to the unique largest element m, φm , preserves the ordering ≤ (see [4]) and φm (τ )−1 φm (τ x) = φm (x). Thus, φm ∈ Aut(DS ) ≤ Λ. If AS is the weak crossed product algebra arising from the associated idempotent cocycle, then AutK (K ⊗ AS ) ∼ = (1 + JK ) (H (W Aut(DS ))). Example 7. Weak Bruhat order on Sn , the symmetric group on n letters (continuation of Example 6). In this case there is only one nontrivial diagram automorphism and it is φm . Thus, Λ˙ = {id, φm }, where φm is the conjugation by the unique maximal element. ˆ∼ Therefore, Λ = C 2 Sn and H = H (C 2 Sn ). If AS is the algebra associated with the idempotent cocycle, AutK (K ⊗ AS ) ∼ = (1 + JK ) (H (C 2 Sn )). Acknowledgments The author would like to express his deepest gratitude to Professor Darrell Haile for his support and comments given throughout this work. The anonymous referee is also acknowledged for his valuable comments. References [1] R. Stimets, Weak Galois cohomology and group extensions, Comm. Algebra 28 (3) (2000) 1285–1308. [2] D. Haile, R. Larson, M. Sweedler, A new invariant of C over R: almost invertible co-homology theory and the classification of idempotent co-homology classes and algebras by partially ordered sets with a Galois group action, Amer. J. Math. 105 (1983) 689–814. [3] A. Bjorner, F. Brenti, Combinatorics of Coxeter Groups, vol. 231, Springer Science & Business Media, 2006. [4] A. Aljouiee, On weak crossed products, Frobenius algebras, and the weak Bruhat ordering, J. Algebra 287 (2005) 88–102. [5] D. Haile, Brauer monoid of a field, J. Algebra 81 (1983) 521–539. [6] D. Haile, On crossed product algebras arising from weak cocycles, J. Algebra 74 (1982) 270–279. [7] D. Haile, L. Rowen, Weakly Azumaya algebras, J. Algebra 250 (2002) 134–177.
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[8] F. Guil-Asensio, M. Saorin, The automorphism group and the Picard group of a monomial algebra, Comm. Algebra 27 (1999) 857–887. [9] R.S. Pierce, Associative Algebras, Grad. Texts in Math., vol. 88, Springer, Berlin, 1982. [10] N. Bourbaki, Lie Groups and Lie Algebras, Springer-Verlag, 2002, Chapters 4–6, translated from the 1968 French original by Andrew Pressley. [11] A. Bjorner, Orderings of coxeter groups, in: C. Greene (Ed.), Combinatorics and Algebra, Amer. Math. Soc, 1984, pp. 175–195.