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Twisted Calabi–Yau property of right coideal subalgebras of quantized enveloping algebras
Journal of Algebra 399 (2014) 1073–1085
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Journal of Algebra www.elsevier.com/locate/jalgebra
Twisted Calabi–Yau property of right coideal subalgebras of quantized enveloping algebras L.-Y. Liu, Q.-S. Wu ∗ School of Mathematical Sciences, Fudan University, Shanghai 200433, China
a r t i c l e
i n f o
Article history: Received 19 July 2012 Available online 13 December 2013 Communicated by J.T. Stafford MSC: primary 16E40, 16W30, 16S35
a b s t r a c t Suppose that U is the quantized enveloping algebra of some finitedimensional semisimple Lie algebra g and C is a right coideal subalgebra of U such that the group-like elements contained in C form a group. Then C is Artin–Schelter regular and twisted Calabi– Yau. The Nakayama automorphism of C is also determined if C is contained in the Borel part U 0 . © 2013 Elsevier Inc. All rights reserved.
Keywords: Quantized enveloping algebra Right coideal subalgebra Twisted Calabi–Yau algebra Nakayama automorphism
0. Introduction Calabi–Yau algebras [8] are a class of algebras appearing in the fields such as noncommutative algebraic geometry, representation theory and mathematical physics. There is a long list of examples of Calabi–Yau algebras. For any Calabi–Yau algebra A, there is a Poincaré duality connecting the Hochschild homologies and cohomologies, namely, there exists an integer d 0, such that H • ( A , M ) ∼ = H d−• ( A , M ) for any A-bimodule M. However, there are many other algebras A for which the Poincaré duality should be twisted by some algebra automorphism σ of A, i.e., H •( A, M ) ∼ = H d−• ( A , σ M ), which is a special case of Van den Bergh duality [29]. One such example is any noetherian Artin–Schelter regular (AS-regular, for short) Hopf algebra, which is proved to be rigid Gorenstein by Brown and Zhang [4]. Such algebras are examples of twisted Calabi–Yau algebras (see Definition 1.1). A twisted Calabi–Yau algebra is Calabi–Yau in the sense of Ginzburg [8] if and only if the associated Nakayama automorphism is an inner automorphism. Twisted Calabi–Yau algebras are closely related to AS-regular algebras. Generally speaking, a twisted Calabi–Yau augmented algebra is
*
Corresponding author. E-mail addresses: [email protected] (L.-Y. Liu), [email protected] (Q.-S. Wu).
0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.10.019
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always AS-regular. The other direction is true for noetherian Hopf algebras and noetherian connected graded algebras. As an analogue of a homogeneous space in the noncommutative setting, a quantum homogeneous space of a Hopf algebra/quantum group H is a right coideal subalgebra of H over which H is a faithfully flat extension (see [27,24,25], etc.). If H is a pointed Hopf algebra, then a right coideal subalgebra C of H is a quantum homogeneous space if and only if the group-like elements of H contained in C form a group [23]. Recently, Krähmer studied Van den Bergh duality for homologically smooth AS-Gorenstein quantum homogeneous spaces [15]. Motivated by [4,15] and the above discussion, one question is to study the twisted Calabi–Yau property of quantum homogeneous spaces. The quantized enveloping algebra U q (g) of a finite-dimensional complex semisimple Lie algebra g is Calabi–Yau [5]. Unlike in the U (g) case, for a Lie subalgebra h ⊂ g, even when U q (h) is defined, it is not necessarily isomorphic to a Hopf subalgebra of U q (g) in general. Instead, a large class of right coideal subalgebras, as quantum analogues of U (h), are studied in the literature. In the past few years, the classification of right coideal subalgebras of the Borel part U 0 of U q (g) containing all of the group-like elements has been given in the case that q is not a root of unity. When g is of type A n , B n and G 2 , it is given in [14], [13] and [26] respectively. According to the earlier results, Kharchenko conjectured that the number of such right coideal subalgebras coincides with the order of the Weyl group if q is not a root of unity [13]. Later on, the fact was confirmed by Heckenberger and Schneider [11]. Let W be the Weyl group of g, U = U q (g) and U 0 be the coradical of U . A recent work of Heckenberger and Kolb [9] established a bijection between the set of all right coideal subalgebras C of U 0 for which C ∩ U 0 is a group algebra and the set of all triples ( w , ϕ , L ) where w ∈ W , ϕ : U + [ w ] → k is a character, and L is a subgroup of the root lattice such that ϕ and L satisfy an additional compatibility condition. In their latest paper [10], the classification of right coideal subalgebras of U q (g) containing U 0 is completed. Using Heckenberger and Kolb’s result in [9] and a result in [20] which says that the twisted Calabi– Yau property is preserved under Ore extensions, it is proved in this paper that any such right coideal subalgebra of U 0 is an iterated Ore extension of some group algebra, thus twisted Calabi–Yau (see Section 3). An explicit expression of the Nakayama automorphism is given also (see Theorem 3.4). Theorem 0.1. Let C be a right coideal subalgebra of U 0 such that C ∩ U 0 is a group algebra. Then C = k[ K γ±11 , . . . , K γ±M1 ][ X β1 ; σ1 ][ X β2 ; σ2 , δ2 ] · · · [ X βN ; σ N , δ N ] is an iterated Ore extension and C is twisted Calabi–Yau. The Nakayama automorphism ν of C , uniquely up to an inner automorphism, is given by
ν ( K γi ) = q
N
l=1 (βl ,γi )
ν ( X β j ) = q−
j −1 l =1
K γi ,
N
(βl ,β j )+
l= j +1 (βl ,β j )
Xβ j ,
where the parentheses stand for the bilinear form on the root lattice. A bimodule resolution for C is given in Proposition 3.8. If C ⊆ U 0 is connected (see Definition 2.3), then C is similar to a quantum space in some sense. In the general case that C ⊆ U , we use a filtration introduced by G. Letzter [16] and study the associated graded ring. Such right coideal subalgebras C are proved to be AS-regular, which are twisted Calabi–Yau by [19, Theorem 3.6] (see Theorem 4.3). Theorem 0.2. Let C be a right coideal subalgebra of U such that C ∩ U 0 is a group algebra. Then C is AS-regular and twisted Calabi–Yau. The paper is organized as follows. In Section 1, we recall the definition of twisted Calabi–Yau algebras and some notions related to quantized enveloping algebras. In Section 2, we mainly introduce the results of Heckenberger and Kolb [9] such as the algebra structures, PBW bases, gradings of the right coideal subalgebras C of U 0 , as well as the relations between C and the Weyl group W . In Section 3, for the case C ⊆ U 0 we compute the Nakayama automorphism and the homological
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˜ (C ) for the right coideal subalgebras C . In integral of C , and we construct a bimodule complex K Section 4, we prove that any right coideal subalgebra C of U such that C ∩ U 0 is a group algebra is AS-regular and twisted Calabi–Yau. 1. Preliminaries 1.1. Twisted Calabi–Yau algebras Throughout, k is a field and all algebras are k-algebras. Unadorned ⊗ means ⊗k and Hom means Homk . Suppose that A is an algebra. Let A op be the opposite ring of A and A e = A ⊗ A op . The term A e -modules is used for A-bimodules sometimes. For any A e -module M and any endomorphisms ν , σ of A, denote by ν M σ the A e -module whose ground vector space is M and the A e -action is given by a · m · b = ν (a)mσ (b) for all a, b ∈ A and m ∈ M. If ν or σ is the identity map, it is usually omitted. Definition 1.1. An algebra A is said to be of A and for some integer d 0 if
ν -twisted Calabi–Yau of dimension d for some automorphism ν
(1) A is homologically smooth, that is, as an A e -module, A has a finitely generated projective resolution of finite length; (2)
ExtiA e ( A , A e ) ∼ =
0, Aν ,
i = d, i =d
as A e -modules, where the A e -module structure on the Ext-group is induced by the right A e -module structure of A e . In this case ν is uniquely determined up to an inner automorphism and is called the Nakayama automorphism of A. Note that A is Calabi–Yau in the sense of Ginzburg [8] if and only if ν is an inner automorphism of A. Definition 1.2. Suppose that A is an augmented algebra with the augmentation map ε : A → k. Then A is called Artin–Schelter Gorenstein (for short, AS-Gorenstein) if (1) inj.dim A A = d < ∞, (2) dimk ExtdA (k, A ) = 1 and ExtiA (k, A ) = 0, for all i = d, (3) the right A-module versions of (1) and (2) hold. If further, A has finite global dimension, then A is called Artin–Schelter regular (for short, ASregular). For any augmented algebra A, if A is twisted Calabi–Yau of dimension d, then A is AS-regular with global dimension d [20, Lemma 1.3]. 1.2. Quantized enveloping algebra U q (g) From now on, let k be of characteristic zero and so Q can be viewed as a subfield of k. We first recall some related material about quantized enveloping algebras. For the notations, we mainly refer to [12,3]. Let g be a finite-dimensional complex semisimple Lie algebra and Φ be the root system with respect to a fixed Cartan subalgebra. Denote by Π = {α1 , α2 , . . . , αn } the sets of simple roots, and by (ai j )n×n the corresponding Cartan matrix of g. Let di = (αi , αi )/2, then (di ai j )n×n is symmetric, positive definite. Let W be the Weyl group of g, and let Q = ZΠ be the root lattice, Q + be the positive root lattice.The parentheses (·,·) are extended to a bilinearform on the Q-space spanned by Π . For any β = i li αi ∈ Q , the height ht β of β is defined to be i li .
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Associated to every root β ∈ Φ there is a reflection sβ defined by
sβ (ω) = ω −
2(ω, β)
(β, β)
ω ∈ Φ.
β,
One can show that W is generated by {sα | α ∈ Π}. Every element w of W has a reduced expression as a product of sα . The minimal number of factors in sα in the reduced expressions of w is called the length of w, denoted by ( w ). Note that an element w may have different reduced expressions. Assume that q ∈ k \ {0} is not a root of unity. Denote q i = qdi . Let U = U q (g) be the quantized
enveloping algebra of g with generators K i , K i−1 , E i , F i (1 i n). We refer to [12] for the generating relations. The algebra U is a Hopf algebra with comultiplication , counit ε and antipode S defined by
( K i ) = K i ⊗ K i ,
ε ( K i ) = 1,
S ( K i ) = K i−1 ,
( E i ) = E i ⊗ 1 + K i ⊗ E i ,
ε ( E i ) = 0,
S ( E i ) = − K i−1 E i ,
( F i ) = F i ⊗ K i−1 + 1 ⊗ F i ,
ε ( F i ) = 0,
S(Fi) = −Fi Ki.
In the following, K i , E i and F i are denoted by K αi , E αi and F αi respectively for 1 i n, and the product
n
i =1
l
K αi i is denoted by K β where β =
n
i =1 l i
αi ∈ Q .
2. Structures of right coideal subalgebras in the Borel parts Let U + , U − , U 0 , U 0 and U 0 be the subalgebras of U generated by the sets { E αi | αi ∈ Π}, { F αi | αi ∈ Π}, { K ±αi | αi ∈ Π}, { E αi , K ±αi | αi ∈ Π} and { F αi , K ±αi | αi ∈ Π}, respectively. U 0 and U 0 are called the positive Borel part and the negative Borel part of U respectively. Lusztig defined a series of algebra automorphisms T i of U [22] (also see [12,3]), by which one obtains the Poincaré–Birkhoff–Witt bases (PBW bases, for short) for U . −ai j
Ti K j = K j Ki
Ti E j =
,
T i F i = − K i−1 E i ,
Ti Ei = −Fi Ki,
−ai j (r ) r (−ai j −r ) (−1)r q− Ei E j Ei if i = j , i r =0
Ti F j =
−ai j (−a −r ) (r ) (−1)r qri F i F j F i i j if i = j , r =0
(m)
(m)
where E i is E m . i divided by the q-factorial [m]q ! and similarly for F i Let si = sαi . Now, fix an element w ∈ W and a reduced expression w = si 1 si 2 · · · si N of w. Set β j = si 1 si 2 · · · si j−1 (αi j ) and define
E β j = T i 1 T i 2 · · · T i j −1 ( E α i ) j
and
F β j = T i 1 T i 2 · · · T i j −1 ( F α i ) j
for j = 1, . . . , N. Proposition 2.1. (1) [6, §2] The subspace
r r r U + [ w ] := spank E β11 E β22 · · · E βNN r1 , r2 , . . . , r N ∈ N
r r r resp. U − [ w ] := spank F βNN · · · F β22 F β11 r1 , r2 , . . . , r N ∈ N
(2.1)
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is a subalgebra of U + (resp. U − ) with the monomials E β11 E β22 · · · E βNN (resp. F βNN · · · F β22 F β11 ) as the PBW bases, and U + [ w ] (resp. U − [ w ]) is independent of the reduced expression of w. (2) [17, Proposition 5.5.2] For i < j, one has r
r
E βi E β j − q(β j ,βi ) E β j E βi =
r
r
r
cl E l ,
r
(2.2)
l∈N N l
l
l
where cl ∈ k, E l = E β11 E β22 · · · E βNN , and cl = 0 only when l = (l1 , . . . , l N ) is such that lt = 0 for t i or t j, and
F βi F β j − q−(β j ,βi ) F β j F βi =
dl F l ,
(2.3)
l∈N N l
l
l
where dl ∈ k, F l = F βNN · · · F β22 F β11 , and dl = 0 only when l = (l N , . . . , l1 ) is such that lt = 0 for t i or t j. (3) [22, Proposition 4.2] Multiplication defines an isomorphism of k-spaces
U− ⊗ U0 ⊗ U+ ∼ = U. The subscripts βi give Q + -graded algebra structures on U + [ w ] and U − [ w ]. Remark 2.2. By [6, Proposition in §2], U + [ w ] can be obtained by an iterated Ore extension, that is, U + [ w ] = k[ E β1 ][ E β2 ; σ2 , δ2 ] · · · [ E βN ; σ N , δ N ], where the automorphisms σ j and derivations δ j are implicitly given by (2.2). So we call the generating relations (2.2) the derivation relations of U + [ w ] and denote them by DR( E β j ). With the same argument to U − [ w ], we denote the relations (2.3) by DR( F βi ), by abuse of notation. In the following, we assume that the right coideal subalgebra C of U always satisfies the condition that C ∩ U 0 is a group algebra. For any h ∈ U , there is a left adjoint action ad h on U , defined by
(ad h)(u ) =
h(1) u S (h(2) ),
u ∈ U.
Suppose A is a subalgebra of U and R is a subspace of U . If R is stable under the left adjoint action of ad a for all a ∈ A, R is called an ad A-module. Definition 2.3. (See [9, Definition 2.6].) Let C be a right coideal subalgebra of U . C is called connected if C ∩ U 0 = k1. For any right coideal subalgebra C of U 0 , we write L (C ) for the subgroup of Q corresponding to the group algebra C ∩ U 0 . Clearly, C is connected if and only if L (C ) = 0. Conversely, for any subgroup L of Q , write T L for the group algebra spanned by { K α | α ∈ L }. For any right coideal subalgebra C of U 0 , let C β = U + K β ∩ C , and let
I (C ) =
u ∈ C −β (ad K γ )(u ) = q(γ ,β) u for all γ ∈ L (C ) .
(2.4)
β∈ Q +
The elements in each component I (C )β are called weight vectors in some literature. It was shown that any right coideal subalgebra C of U 0 can be decomposed into a tensor product of a group algebra and a connected right coideal subalgebra as a k-space.
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Proposition 2.4. (See [9, Proposition 2.8].) Suppose that C is a right coideal subalgebra of U 0 such that C ∩ U 0 is a group algebra. (1) I (C ) is a connected right coideal subalgebra of U 0 as well as an ad T L (C ) -module. (2) The decomposition (2.4) gives a Q + -graded algebra structure on I (C ). (3) The multiplication map I (C ) ⊗ T L (C ) → C is bijective. The following theorem illustrates the relation between the right coideal subalgebras of U 0 and the Weyl group W of g. Theorem 2.5. (See [9, Theorem 2.17].) There is a one-to-one correspondence between the set of all right coideal subalgebras C of U 0 such that C ∩ U 0 is a group algebra and the set of all triples ( w , ϕ , L ), where w ∈ W , ϕ : U + [ w ] → k is a character and L is a subgroup of Q such that ϕ and L satisfy an additional compatibility condition. More precisely, for a given C , there exists a unique w ∈ W and a unique isomorphism of Q + -graded algebras F : U + [ w ] → I (C ), ϕ = ε F and L = L (C ). Conversely, for any appropriate triple ( w , ϕ , L ), there exists a connected right coideal subalgebra D ( w , ϕ ) of U 0 which is isomorphic to U + [ w ] as an algebra and has an ad T L -module structure, such that C is given by D ( w , ϕ ) T L . Let G + be the subalgebra of U generated by { E α K −α | α ∈ Π}. For any w ∈ W , the assignment xβ → q−(β,β)/2 xβ K −β , where xβ ∈ U + [ w ]β , gives an injective algebra map U + [ w ] → G + , whose image is a right coideal subalgebra of U 0 , denoted by G + [ w ]. Remark 2.6. (1) Clearly, U 0 is an N-graded algebra with deg( E i ) = 1 and deg( K i ) = 0. If C is an N-graded subalgebra, then C corresponds to a triple ( w , ε , L ). In this case, I (C ) = G + [ w ]. (2) For any C , I (C ) is a subalgebra of G + . In particular, if C is connected, C is a subalgebra of G + . (3) It follows from Proposition 2.4 and Theorem 2.5 that C = I (C ) # T L (C ) as algebras. (4) U + [ w ] is an ad T L (C ) -module. By the proof of [9, Theorem 2.17], F is also an isomorphism of ad T L (C ) -modules. If C is a right coideal subalgebra of the negative Borel part U 0 , then all the results are parallel to the positive Borel part with a few modifications. We state them as follows. The subgroup L (C ) of Q is defined completely the same as in the positive Borel part. Let G − be the subalgebra of U generated by { F α K α | α ∈ Π} and C β = G − K β ∩ C . Let
I (C ) =
u ∈ C −β (ad K γ )(u ) = q−(γ ,β) u for all γ ∈ L (C ) .
β∈ Q +
Then I (C ) is a connected right coideal subalgebra of U 0 as well as an ad T L (C ) -module, contained in U − , and C = I (C ) # T L (C ) . There is an anti-isomorphism of Q + -graded algebras G : U − [ w ] → I (C ) for some w ∈ W , which is also an isomorphism of ad T L (C ) -modules. If C is an N-graded subalgebra of U 0 , then G ( y β ) = q(β,β)/2 S ( y β ) K −β where y β ∈ U − [ w ]β . 3. Nakayama automorphisms and bimodule resolutions In this section, we focus on the right coideal subalgebras C of U 0 . By Theorem 2.5 and Remark 2.6, C corresponds to a triple ( w , ϕ , L ), and F : U + [ w ] → I (C ) is an isomorphism of ad T L (C ) -module algebras. Given a reduced expression w = si 1 si 2 · · · si N , let E β j be as given by (2.1) and for j = 1, 2, . . . , N let
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X β j = F ( E β j ). The algebra structure of U + [ w ] is illustrated by Proposition 2.1. Suppose that the rank of L = L (C ) is M. Pick γ1 , . . . , γ M to be the generators of L. Let us consider the algebra structure of C . Since C = I (C ) # T L (C ) and
(ad K γi )( X β j ) = F (ad K γi ) E β j = q(γi ,β j ) X β j for any 1 i M and 1 j N, it follows that C is generated by K ±γi (1 i M) and X β j (1 j N), with the relations
K γi K γi = K γi K γi ,
K γ i K −γ i = 1,
K γi X β j K −γi = q(γi ,β j ) X β j ,
1 i, i M , 1 j N,
DR( X β j ),
where the notation DR is given by Remark 2.2. 3.1. Computation of Nakayama automorphisms We construct an algebra D as follows,
D = k g 1 , . . . , g M , x1 , . . . , x N
g i g i − g i g i , g i x j − q(β j ,γi ) x j g i , DR(x j ) .
It is easy to see that C is isomorphic to the localization of D at { g 1 , . . . , g M }. First of all, let us prove that D is twisted Calabi–Yau and compute the Nakayama automorphism ν D . For 1 r t N, denote by D r ,t the subalgebra of D generated by xr , . . . , xt over k[ g 1 , . . . , g M ]. It is easy to verify that D r ,t = D r ,t −1 [xt ; σ , δ] is an Ore extension if r < t, where
σ ( g i ) = q−(βt ,γi ) g i , i = 1, . . . , M , σ (x j ) = q−(βt ,β j ) x j ,
j = r , . . . , t − 1.
Furthermore, D r ,t = D r +1,t [xr ; σ , δ ] is also an Ore extension, where
σ ( g i ) = q−(βr ,γi ) g i , i = 1, . . . , M , σ (x j ) = q(βr ,β j ) x j ,
j = r + 1, . . . , t .
The twisted Calabi–Yau property is preserved by Ore extensions [20]. Theorem 3.1. (See [20, Theorem 3.3].) Suppose that B = A [x; σ , δ] is an Ore extension with σ an automorphism of A. If A is ν A -twisted Calabi–Yau of dimension d, then B is twisted Calabi–Yau of dimension d + 1 whose Nakayama automorphism ν B satisfies ν B | A = σ −1 ν A . Lemma 3.2. For 1 r t N, D r ,t is twisted Calabi–Yau of dimension M + t − r + 1. The Nakayama automorphism νr ,t of D r ,t is uniquely given by
νr ,t ( g i ) = q
t
l=r (βl ,γi )
j −1
νr ,t (x j ) = q−
for i = 1, 2, . . . , M and j = r , r + 1, . . . , t.
l=r
gi ,
(βl ,β j )+
t
M
l= j +1 (βl ,β j )−
s=1 (γs ,β j )
xj
(3.1)
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Proof. The first assertion follows immediately from Theorem 3.1. The Nakayama automorphism is unique since the identity map is the only inner automorphism of D r ,t . We prove it by induction on t − r. If t − r = 0, then D r ,t = D r ,r is a quantum ( M + 1)-space, namely,
D r ,r = k g 1 , . . . , g M , xr The Nakayama automorphism
g i g i − g i g i , g i xr − q(βr ,γi ) xr g i .
νr ,r is given by (see also [20, Proposition 4.1]) M
νr ,r (xr ) = q−
νr ,r ( g i ) = q(βr ,γi ) g i ,
s=1 (γs ,βr )
xr .
Hence (3.1) holds. Suppose t − r 1. By the inductive hypothesis, (3.1) holds for D r ,t −1 . By Theorem 3.1, the Nakayama automorphism νr ,t satisfies that for i = 1, . . . , M and j = r , . . . , t − 1,
νr ,t ( g i ) = σ −1 νr ,t −1 ( g i ) = q
t
l=r (βl ,γi )
j −1
νr ,t (x j ) = σ −1 νr ,t −1 (x j ) = q−
l=r
gi , t
(βl ,β j )+
M
s=1 (γs ,β j )
l= j +1 (βl ,β j )−
x j.
Similarly, the above deduction can be applied to the Ore extension D r ,t = D r +1,t [ X βr ; ρ , δ ]. Then
− 1
νr ,t (xt ) = σ
j −1
νr +1,t (xt ) = q−
l=r
t
(βl ,βt )+
M
l= j +1 (βl ,βt )−
s=1 (γs ,βt )
xt .
2
Therefore, (3.1) holds for any r, t.
Let d = M + N. Since D = D 1, N , we have Lemma 3.3. The algebra D is twisted Calabi–Yau of dimension d. The Nakayama automorphism ν D is uniquely given by
ν D ( gi ) = q
N
l=1 (βl ,γi )
j −1
ν D (x j ) = q −
l =1
gi ,
(βl ,β j )+
N
M
l= j +1 (βl ,β j )−
s=1 (γs ,β j )
xj
for i = 1, 2, . . . , M and j = 1, 2, . . . , N. The following theorem is the main result in this section. Theorem 3.4. Let C be a right coideal subalgebra of U 0 such that C ∩ U 0 is a group algebra. Then C = I (C ) # T L (C ) is ν -twisted Calabi–Yau of dimension d, where the Nakayama automorphism ν is given, up to an inner automorphism, by
ν ( K γi ) = q
N
l=1 (βl ,γi )
j −1
ν ( X β j ) = q− for i = 1, 2, . . . , M and j = 1, 2, . . . , N.
l =1
K γi , N
(βl ,β j )+
l= j +1 (βl ,β j )
Xβ j
(3.2)
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Proof. As we mentioned, C is isomorphic to the localization of D at { g 1 , . . . , g M }. By [7, Theorem 6 and Example 8], C is twisted Calabi–Yau of dimension d provided that D is, also. In this case, the Nakayama automorphism ν of C is nothing but the extension of ν D . So by Lemma 3.3, we obtain
ν ( K γi ) = q
N
l=1 (βl ,γi )
j −1
ν ( X β j ) = q−
l =1
K γi , N
(βl ,β j )+
M
l= j +1 (βl ,β j )−
s=1 (γs ,β j )
Xβ j .
(3.3)
Since the inner automorphisms of C are those adjoint actions ad K γ for some γ ∈ L (C ), in particular, γ = sM=1 γs and replace ν by (ad K −γ )ν in (3.3). Then the new map satisfies (3.2), as desired. 2
let
3.2. Homological integrals Homological integrals of AS-Gorenstein Hopf algebras were introduced by Lu, Zhang and the second author [21, Definition 1.1], as a generalization of the classical integrals of finite-dimensional Hopf algebras. The homological integral is a useful tool in studying infinite-dimensional Hopf algebras. The definition was generalized for any augmented AS-Gorenstein algebra [19, Definition 3.1]. Roughly speaking, a left homological integral of an augmented AS-Gorenstein algebra A of dimen-
l
l
sion d is the 1-dimensional k-space ExtdA (k, A ), denoted by A . It is easy to show A is an A e -module with trivial left A-action. The right A-action determines a character χ : A → k such that x · a = χ (a)x
l for all x ∈ A . Lemma 3.5. (See [20, Lemma 1.3].) Let A be a ν -twisted Calabi–Yau augmented algebra with the augmentation map ε : A → k. Then A is AS-regular, and the character χ determined by the left homological integral of A is εν . Proposition 3.6. Let C be a right coideal subalgebra of U 0 corresponding to a triple ( w , ϕ , L ). Then the character χ determined by the left homological integral of C satisfies
χ ( K γi ) = q
N
l=1 (βl ,γi )
Proof. By Lemma 3.5 and (3.2),
and
χ ( K γi ) = q j −1
χ ( X β j ) = q−
l =1
j −1
= q−
l =1
j −1
= q−
j −1
χ ( X β j ) = q−
l =1
N
l=1 (βl ,γi )
(βl ,β j )+ (βl ,β j )+ (βl ,β j )+
l =1
l= j +1 (βl ,β j )
ϕ ( E β j ).
and
N
l= j +1 (βl ,β j )
N
l= j +1 (βl ,β j )
N
N
(βl ,β j )+
l= j +1 (βl ,β j )
ε( X β j ) εF ( E β j ) ϕ ( E β j ).
2
3.3. Bimodule resolutions In this subsection, we first consider connected right coideal subalgebras C ⊆ U 0 . Suppose C corresponds to a triple ( w , ϕ , 0) with ( w ) = N. Then C can be viewed as a deformation of some quantum affine space
OΛ k N := k X 1 , . . . , X N /( X j X i − λi j X i X j ; 1 i < j N ) with λi j ∈ k \ {0}. These quantum spaces are Koszul algebras. In order to compute the Hochschild (co)homology for Koszul algebras A, Van den Bergh constructed the so-called Koszul bimodule complex K( A ) [28].
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According to his construction, K(OΛ (k N )) is of length N and K(OΛ (k N ))i is a free OΛ (k N )e -module N
of rank i for 0 i N. We will show that C is similar to a quantum space in some sense. Proposition 3.7. Let C ⊆ U 0 be a connected right coideal subalgebra corresponding to a triple ( w , ϕ , 0) with
( w ) = N. Then there exists a C -bimodule complex K˜ (C ) such that
˜ (C ) is of length N, (1) K
˜ (C )i is a free C e -module of rank N for 0 i N, and K˜ (C )0 = C ⊗ C , (2) K i μ
˜ (C ) −−→ C −→ 0 is exact via the multiplication μ. (3) K
˜ (C ) can be constructed inductively. Proof. By [20], the complex K ∼ ˜ (C ) clearly exists. If N = 1, C = k[x] and such complex K If N > 1, suppose w = si 1 · · · si N . Let w = si 1 · · · si N −1 , ϕ be the restriction of ϕ to U + [ w ] and C be the right coideal subalgebra corresponding to ( w , ϕ , 0). So C = C [ X βN ; σ , δ] is an Ore extension
˜ (C ) satisfying the three of C . By the inductive hypothesis, there exists a C -bimodule complex K ˜ (C ) is conditions. Assume K C ⊗ V N −1 ⊗ C −→ · · · −→ C ⊗ V 1 ⊗ C −→ C ⊗ C , where V i is a k-space of dimension
N −1
i
. −1
˜ ( C ) ⊗C σ Since C is free over C , C ⊗C K −1 σ C ⊗C C and C ⊗C C , respectively. The map
ξ : C ⊗C σ
−1
˜ (C ) ⊗C C are C e -free resolutions of C and C ⊗C K
C −→ C ⊗C C ,
c 1 ⊗ c 2 −→ c 1 ⊗ X βN c 2 − c 1 X βN ⊗ c 2 is a C e -module homomorphism. By the comparison lemma, ξ can be lifted to a morphism of com˜ (C ) ⊗C σ −1 C → C ⊗C K˜ (C ) ⊗C C . Since plexes θ : C ⊗C K −1 ξ mult 0 −→ C ⊗C σ C −→ C ⊗C C −−→ C −→ 0
is exact, cone(θ) is a C e -free resolution of C via N −1 N −1 N
+ i −1 = i . i
˜ (C ) = cone(θ), as desired. Let K
μ and the rank of cone(θ)i is dimk V i + dimk V i−1 =
2
Next, we consider non-connected cases. Suppose C corresponds to a triple ( w , ϕ , L ) with ( w ) = N and the rank of L equal to M. Notice that the Koszul bimodule complex K( P ) of the polynomial ring P = k[ g 1 , . . . , g M ] is of
M length M and the rank of K( P )i is i . Since T L is a localization of P , by the proof of [7, Theorem 6], T L ⊗ P K( P ) ⊗ P T L is a T L -bimodule complex of T L . Using the same argument as the proof of Proposition 3.7, we have Proposition 3.8. Let C ⊆ U 0 be a right coideal subalgebra corresponding to a triple ( w , ϕ , L ) with ( w ) = N ˜ (C ) such that and the rank of L equal to M. Let d = M + N. Then there exists a C -bimodule complex K
˜ (C ) is of length d, (1) K
˜ (C )i is a free C e -module of rank d for 0 i d, and K˜ (C )0 = C ⊗ C , (2) K i
μ ˜ (C ) −− → C −→ 0 is exact via the multiplication μ. (3) K
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4. Twisted Calabi–Yau property in general case In this section, we consider the right coideal subalgebra C of U , and prove that C is AS-regular and twisted Calabi–Yau. First of all, let us introduce a filtration on U which was defined by G. Letzter [16]. Let w ∈ W be of the longest length, say T , and fix a reduced expression of w. Then U + [ w ] = U + [22,17] and we have a family of generators E βi and F βi (i = 1, 2, . . . , T ) for U as given in (2.1). The set consisting of l r l r all monomials Ml,α ,r := F βTT · · · F β11 K α E β11 · · · E βTT , where l = (l T , . . . , l1 ), r = (r1 , . . . , r T ) ∈ N T , α ∈ Q , is a PBW basis for U . For any Ml,α ,r , equip it with a bi-degree
bideg(Ml,α ,r ) =
T
li (ht βi ),
i =1
T
r i (ht βi ) ∈ N2 .
i =1
We will view N2 as a totally ordered monoid with the lexicographical order from right to left. Let Γt U be the subspace spanned by the monomials Ml,α ,r such that bideg(Ml,α ,r ) t for any t ∈ N2 . Then we obtain a filtration on U . The triangular decomposition
U∼ = U − ⊗ U 0 ⊗ G +, proved in [16, (1.12)] is an alternative expression of Proposition 2.1 (3). Analogous with [16, (4.14)], we have gr U ∼ = (U − ⊗ G + ) # U 0 , where ⊗ stands for the braided tensor product with braiding
G + ⊗ U − −→ U − ⊗ G + , E β K −β ⊗ F α −→ q(α ,β) F α ⊗ E β K −β . It is clear that gr U − = U − and gr G + = G + . Remark 4.1. Observe that gr U can be realized iteratively. In fact, one may define a positive filtration on U via the second component of the bi-degree. The associated graded ring U (1) together with the first component of the bi-degree, is a positive filtered ring whose filtration is compatible with the graded structure. Thus taking the associated graded ring again, one obtains an N2 -graded ring U (2) . It is routine to check that the N2 -graded ring U (2) is nothing but gr U . Next, we consider the induced filtration on C and the associated graded ring. By the right coideal version of [16, Theorem 4.9],
gr C ∼ =
U − ∩ gr C ⊗ G + ∩ gr C
# T L (C ) ,
and U − ∩ gr C (resp. G + ∩ gr C ) is a connected right coideal subalgebra of U 0 (resp. U 0 ). Since U − ∩ gr C and G + ∩ gr C are both N-graded algebras, by Remark 2.6, the corresponding triples must be of the form ( w 1 , ε , 0), ( w 2 , ε , 0) for some w 1 , w 2 ∈ W . Fix their reduced expressions. Thus the PBW bases for U − ∩ grΓ C , G + ∩ grΓ C are determined. In addition, we may assume the reduced expression of w 2 is an initial segment of that of w, for simplicity (see [2, Chapter 3]). So their generator sets are
Y β Y β = q(βi ,βi )/2 S ( F β ) K −β , i = 1, . . . , L 1 ,
i
i
i
i
X β j X β j = q−(β j ,β j )/2 E β j K −β j , j = 1, . . . , L 2 ,
where L 1 = ( w 1 ) and L 2 = ( w 2 ).
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Remark 4.2. In ring theory and homological algebra, using a positive filtration (or more generally, Zariskian filtration) one can lift information from the associated graded ring/module to the filtered ring/module. By Remark 4.1, gr C can also be realized iteratively. Hence we may investigate the properties of C by studying gr C . Recall the relations DR in Remark 2.2. Obviously, the generators X β j of G + ∩ gr C satisfy DR( X β j ). Similarly, DR( F β ) are transferred to the generating relations of U − ∩ gr C via the anti-isomorphism G , i
denoted by DRop (Y β ). i Let L 3 be the rank of L (C ) and d = L 1 + L 2 + L 3 . It follows that gr C is generated by these Y β , X β j i and K γ (γ ∈ L (C )), subject to
K γ K γ = K γ +γ , K γ Y β K −γ = q
K 0 = 1,
−(γ ,β ) i
i
X β j Y β = q(βi ,β j ) Y β X β j , i
Y β , i
op
DR (Y β ), i
i
(4.1)
K γ X β j K −γ = q(γ ,β j ) X β j ,
(4.2)
DR( X β j ).
(4.3)
Consequently, gr C is an iterated graded Ore extension of the group algebra T L (C ) . By Theorem 3.1, gr C is graded twisted Calabi–Yau of dimension d. Hence it is graded AS-regular. Theorem 4.3. Let C be a right coideal subalgebra of U such that C ∩ U 0 is a group algebra. Then C is AS-regular and twisted Calabi–Yau. Proof. Endow k with the obvious good filtration. Then by Remark 4.2 and by [1, Proposition 3.1] or [18, Proposition 4, §2.2, Chapter III], ExtiC (k, C ) has a filtration and gr ExtiC (k, C ) is isomorphic to i a subquotient of Extgr C (gr k, gr C ) for every i. Since gr k = k and gr C is graded AS-regular of dimension d, C has finite global dimension and ExtiC (k, C ) = 0 for all i = d and dimk ExtdC (k, C ) = 0 or 1. By the double-Ext spectral sequence pq
p
−q
E 2 = ExtC op ExtC (k, C ), C
⇒
H p +q (k),
where
Hn (k) =
0, n = 0, k , n = 0,
one has ExtdC (k, C ) = 0. Therefore, dimk ExtdC (k, C ) = 1. Similarly, ExtiC op (k, C ) = 0 for all i = d and dimk ExtdC op (k, C ) = 1. Since U is pointed, it follows from [19, Theorem 3.6] that
ExtiC e C , C e ∼ =
0, Cν,
i = d, i =d
for some automorphism ν . It remains to check its homological smoothness. To this end, first by [19, Lemma 3.7], C admits a finitely generated projective resolution as a C e -module, and further,
proj.dim C e C proj.dim gr C e gr C = proj.dim (gr C )e gr C = d. Therefore, C is twisted Calabi–Yau.
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Acknowledgments Both authors thank the referee for the valuable comments. This research is supported by the NSFC (key projects 10731070 and 11171067), and STCSM (Science and Technology Committee, Shanghai Municipality, project 11XD1400500), and a training program for innovative talents of key disciplines, Fudan University. References [1] J.-E. Björk, The Auslander condition on noetherian rings, in: Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année, Paris, 1987/1988, in: Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 137–173. [2] A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Grad. Texts in Math., vol. 231, Springer, New York, 2005. [3] K.A. Brown, K.R. Goodearl, Lectures on Algebraic Quantum Groups, Adv. Courses Math. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. [4] K.A. Brown, J.J. Zhang, Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras, J. Algebra 320 (2008) 1814–1850. [5] S. Chemla, Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators, J. Algebra 276 (2004) 80–102. [6] C. De Concini, V.G. Kac, C. Procesi, Some quantum analogues of solvable Lie groups, in: Geometry and Analysis, Bombay, 1992, Tata Inst. Fund. Res., Bombay, 1995, pp. 41–65. [7] M. Farinati, Hochschild duality, localization, and smash products, J. Algebra 284 (2005) 415–434. [8] V. Ginzburg, Calabi–Yau algebras, preprint, arXiv:math/0612139v3, 2006, 79 pp. [9] I. Heckenberger, S. Kolb, Right coideal subalgebras of the Borel part of a quantized enveloping algebra, Int. Math. Res. Not. (2011) 419–451. [10] I. Heckenberger, S. Kolb, Homogeneous right coideal subalgebras of quantized enveloping algebras, Bull. Lond. Math. Soc. 44 (2012) 837–848. [11] I. Heckenberger, H.-J. Schneider, Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid, Israel J. Math. 197 (1) (2013) 139–187. [12] J.C. Jantzen, Lectures on Quantum Groups, Grad. Stud. Math., vol. 6, American Mathematical Society, Providence, RI, 1996. [13] V. Kharchenko, Right coideal subalgebras in U q+ (so2n+1 ), J. Eur. Math. Soc. 13 (2011) 1677–1735. [14] V.K. Kharchenko, A.V. Lara Sagahon, Right coideal subalgebras in U q (sln+1 ), J. Algebra 319 (2008) 2571–2625. [15] U. Krähmer, On the hochschild (co)homology of quantum homogeneous spaces, Israel J. Math. 189 (2012) 237–266. [16] G. Letzter, Coideal subalgebras and quantum symmetric pairs, in: New Directions in Hopf Algebras, in: Math. Sci. Res. Inst. Publ., vol. 43, Cambridge University Press, Cambridge, 2002, pp. 117–165. [17] S. Levendorski˘ıand, Y. Soibelman, Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Comm. Math. Phys. 139 (1991) 141–170. [18] Huishi Li, F. Van Oystaeyen, Zariskian Filtrations, K -Monogr. Math., vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. [19] L.-Y. Liu, Q.-S. Wu, Rigid dualizing complexes over quantum homogeneous spaces, J. Algebra 353 (2012) 121–141. [20] L.-Y. Liu, S.-Q. Wang, Q.-S. Wu, Twisted Calabi–Yau property of Ore extensions, J. Noncommut. Geom. (2013), in press, preprint, arXiv:1205.0893, 2012, 19 pp. [21] D.-M. Lu, Q.-S. Wu, J.J. Zhang, Homological integral of Hopf algebras, Trans. Amer. Math. Soc. 359 (2007) 4945–4975. [22] G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990) 89–113. [23] A. Masuoka, On Hopf algebras with cocommutative coradicals, J. Algebra 144 (1991) 451–466. [24] A. Masuoka, D. Wigner, Faithful flatness of Hopf algebras, J. Algebra 170 (1994) 156–164. [25] E.F. Müller, H.-J. Schneider, Quantum homogeneous spaces with faithfully flat module structures, Israel J. Math. 111 (1999) 157–190. [26] B. Pogorelsky, Right coideal subalgebras of the quantum Borel algebra of type G 2 , J. Algebra 322 (2009) 2335–2354. [27] M. Takeuchi, Relative Hopf modules—equivalences and freeness criteria, J. Algebra 60 (1979) 452–471. [28] M. Van den Bergh, Noncommutative homology of some three-dimensional quantum spaces, K -Theory 8 (1994) 213–230. [29] M. Van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998) 1345–1348, Erratum: Proc. Amer. Math. Soc. 130 (2002) 2809–2810 (electronic).
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
Twisted Calabi–Yau property of right coideal subalgebras of quantized enveloping algebras L.-Y. Liu, Q.-S. Wu ∗ School of Mathematical Sciences, Fudan University, Shanghai 200433, China
a r t i c l e
i n f o
Article history: Received 19 July 2012 Available online 13 December 2013 Communicated by J.T. Stafford MSC: primary 16E40, 16W30, 16S35
a b s t r a c t Suppose that U is the quantized enveloping algebra of some finitedimensional semisimple Lie algebra g and C is a right coideal subalgebra of U such that the group-like elements contained in C form a group. Then C is Artin–Schelter regular and twisted Calabi– Yau. The Nakayama automorphism of C is also determined if C is contained in the Borel part U 0 . © 2013 Elsevier Inc. All rights reserved.
Keywords: Quantized enveloping algebra Right coideal subalgebra Twisted Calabi–Yau algebra Nakayama automorphism
0. Introduction Calabi–Yau algebras [8] are a class of algebras appearing in the fields such as noncommutative algebraic geometry, representation theory and mathematical physics. There is a long list of examples of Calabi–Yau algebras. For any Calabi–Yau algebra A, there is a Poincaré duality connecting the Hochschild homologies and cohomologies, namely, there exists an integer d 0, such that H • ( A , M ) ∼ = H d−• ( A , M ) for any A-bimodule M. However, there are many other algebras A for which the Poincaré duality should be twisted by some algebra automorphism σ of A, i.e., H •( A, M ) ∼ = H d−• ( A , σ M ), which is a special case of Van den Bergh duality [29]. One such example is any noetherian Artin–Schelter regular (AS-regular, for short) Hopf algebra, which is proved to be rigid Gorenstein by Brown and Zhang [4]. Such algebras are examples of twisted Calabi–Yau algebras (see Definition 1.1). A twisted Calabi–Yau algebra is Calabi–Yau in the sense of Ginzburg [8] if and only if the associated Nakayama automorphism is an inner automorphism. Twisted Calabi–Yau algebras are closely related to AS-regular algebras. Generally speaking, a twisted Calabi–Yau augmented algebra is
*
Corresponding author. E-mail addresses: [email protected] (L.-Y. Liu), [email protected] (Q.-S. Wu).
0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.10.019
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always AS-regular. The other direction is true for noetherian Hopf algebras and noetherian connected graded algebras. As an analogue of a homogeneous space in the noncommutative setting, a quantum homogeneous space of a Hopf algebra/quantum group H is a right coideal subalgebra of H over which H is a faithfully flat extension (see [27,24,25], etc.). If H is a pointed Hopf algebra, then a right coideal subalgebra C of H is a quantum homogeneous space if and only if the group-like elements of H contained in C form a group [23]. Recently, Krähmer studied Van den Bergh duality for homologically smooth AS-Gorenstein quantum homogeneous spaces [15]. Motivated by [4,15] and the above discussion, one question is to study the twisted Calabi–Yau property of quantum homogeneous spaces. The quantized enveloping algebra U q (g) of a finite-dimensional complex semisimple Lie algebra g is Calabi–Yau [5]. Unlike in the U (g) case, for a Lie subalgebra h ⊂ g, even when U q (h) is defined, it is not necessarily isomorphic to a Hopf subalgebra of U q (g) in general. Instead, a large class of right coideal subalgebras, as quantum analogues of U (h), are studied in the literature. In the past few years, the classification of right coideal subalgebras of the Borel part U 0 of U q (g) containing all of the group-like elements has been given in the case that q is not a root of unity. When g is of type A n , B n and G 2 , it is given in [14], [13] and [26] respectively. According to the earlier results, Kharchenko conjectured that the number of such right coideal subalgebras coincides with the order of the Weyl group if q is not a root of unity [13]. Later on, the fact was confirmed by Heckenberger and Schneider [11]. Let W be the Weyl group of g, U = U q (g) and U 0 be the coradical of U . A recent work of Heckenberger and Kolb [9] established a bijection between the set of all right coideal subalgebras C of U 0 for which C ∩ U 0 is a group algebra and the set of all triples ( w , ϕ , L ) where w ∈ W , ϕ : U + [ w ] → k is a character, and L is a subgroup of the root lattice such that ϕ and L satisfy an additional compatibility condition. In their latest paper [10], the classification of right coideal subalgebras of U q (g) containing U 0 is completed. Using Heckenberger and Kolb’s result in [9] and a result in [20] which says that the twisted Calabi– Yau property is preserved under Ore extensions, it is proved in this paper that any such right coideal subalgebra of U 0 is an iterated Ore extension of some group algebra, thus twisted Calabi–Yau (see Section 3). An explicit expression of the Nakayama automorphism is given also (see Theorem 3.4). Theorem 0.1. Let C be a right coideal subalgebra of U 0 such that C ∩ U 0 is a group algebra. Then C = k[ K γ±11 , . . . , K γ±M1 ][ X β1 ; σ1 ][ X β2 ; σ2 , δ2 ] · · · [ X βN ; σ N , δ N ] is an iterated Ore extension and C is twisted Calabi–Yau. The Nakayama automorphism ν of C , uniquely up to an inner automorphism, is given by
ν ( K γi ) = q
N
l=1 (βl ,γi )
ν ( X β j ) = q−
j −1 l =1
K γi ,
N
(βl ,β j )+
l= j +1 (βl ,β j )
Xβ j ,
where the parentheses stand for the bilinear form on the root lattice. A bimodule resolution for C is given in Proposition 3.8. If C ⊆ U 0 is connected (see Definition 2.3), then C is similar to a quantum space in some sense. In the general case that C ⊆ U , we use a filtration introduced by G. Letzter [16] and study the associated graded ring. Such right coideal subalgebras C are proved to be AS-regular, which are twisted Calabi–Yau by [19, Theorem 3.6] (see Theorem 4.3). Theorem 0.2. Let C be a right coideal subalgebra of U such that C ∩ U 0 is a group algebra. Then C is AS-regular and twisted Calabi–Yau. The paper is organized as follows. In Section 1, we recall the definition of twisted Calabi–Yau algebras and some notions related to quantized enveloping algebras. In Section 2, we mainly introduce the results of Heckenberger and Kolb [9] such as the algebra structures, PBW bases, gradings of the right coideal subalgebras C of U 0 , as well as the relations between C and the Weyl group W . In Section 3, for the case C ⊆ U 0 we compute the Nakayama automorphism and the homological
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˜ (C ) for the right coideal subalgebras C . In integral of C , and we construct a bimodule complex K Section 4, we prove that any right coideal subalgebra C of U such that C ∩ U 0 is a group algebra is AS-regular and twisted Calabi–Yau. 1. Preliminaries 1.1. Twisted Calabi–Yau algebras Throughout, k is a field and all algebras are k-algebras. Unadorned ⊗ means ⊗k and Hom means Homk . Suppose that A is an algebra. Let A op be the opposite ring of A and A e = A ⊗ A op . The term A e -modules is used for A-bimodules sometimes. For any A e -module M and any endomorphisms ν , σ of A, denote by ν M σ the A e -module whose ground vector space is M and the A e -action is given by a · m · b = ν (a)mσ (b) for all a, b ∈ A and m ∈ M. If ν or σ is the identity map, it is usually omitted. Definition 1.1. An algebra A is said to be of A and for some integer d 0 if
ν -twisted Calabi–Yau of dimension d for some automorphism ν
(1) A is homologically smooth, that is, as an A e -module, A has a finitely generated projective resolution of finite length; (2)
ExtiA e ( A , A e ) ∼ =
0, Aν ,
i = d, i =d
as A e -modules, where the A e -module structure on the Ext-group is induced by the right A e -module structure of A e . In this case ν is uniquely determined up to an inner automorphism and is called the Nakayama automorphism of A. Note that A is Calabi–Yau in the sense of Ginzburg [8] if and only if ν is an inner automorphism of A. Definition 1.2. Suppose that A is an augmented algebra with the augmentation map ε : A → k. Then A is called Artin–Schelter Gorenstein (for short, AS-Gorenstein) if (1) inj.dim A A = d < ∞, (2) dimk ExtdA (k, A ) = 1 and ExtiA (k, A ) = 0, for all i = d, (3) the right A-module versions of (1) and (2) hold. If further, A has finite global dimension, then A is called Artin–Schelter regular (for short, ASregular). For any augmented algebra A, if A is twisted Calabi–Yau of dimension d, then A is AS-regular with global dimension d [20, Lemma 1.3]. 1.2. Quantized enveloping algebra U q (g) From now on, let k be of characteristic zero and so Q can be viewed as a subfield of k. We first recall some related material about quantized enveloping algebras. For the notations, we mainly refer to [12,3]. Let g be a finite-dimensional complex semisimple Lie algebra and Φ be the root system with respect to a fixed Cartan subalgebra. Denote by Π = {α1 , α2 , . . . , αn } the sets of simple roots, and by (ai j )n×n the corresponding Cartan matrix of g. Let di = (αi , αi )/2, then (di ai j )n×n is symmetric, positive definite. Let W be the Weyl group of g, and let Q = ZΠ be the root lattice, Q + be the positive root lattice.The parentheses (·,·) are extended to a bilinearform on the Q-space spanned by Π . For any β = i li αi ∈ Q , the height ht β of β is defined to be i li .
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Associated to every root β ∈ Φ there is a reflection sβ defined by
sβ (ω) = ω −
2(ω, β)
(β, β)
ω ∈ Φ.
β,
One can show that W is generated by {sα | α ∈ Π}. Every element w of W has a reduced expression as a product of sα . The minimal number of factors in sα in the reduced expressions of w is called the length of w, denoted by ( w ). Note that an element w may have different reduced expressions. Assume that q ∈ k \ {0} is not a root of unity. Denote q i = qdi . Let U = U q (g) be the quantized
enveloping algebra of g with generators K i , K i−1 , E i , F i (1 i n). We refer to [12] for the generating relations. The algebra U is a Hopf algebra with comultiplication , counit ε and antipode S defined by
( K i ) = K i ⊗ K i ,
ε ( K i ) = 1,
S ( K i ) = K i−1 ,
( E i ) = E i ⊗ 1 + K i ⊗ E i ,
ε ( E i ) = 0,
S ( E i ) = − K i−1 E i ,
( F i ) = F i ⊗ K i−1 + 1 ⊗ F i ,
ε ( F i ) = 0,
S(Fi) = −Fi Ki.
In the following, K i , E i and F i are denoted by K αi , E αi and F αi respectively for 1 i n, and the product
n
i =1
l
K αi i is denoted by K β where β =
n
i =1 l i
αi ∈ Q .
2. Structures of right coideal subalgebras in the Borel parts Let U + , U − , U 0 , U 0 and U 0 be the subalgebras of U generated by the sets { E αi | αi ∈ Π}, { F αi | αi ∈ Π}, { K ±αi | αi ∈ Π}, { E αi , K ±αi | αi ∈ Π} and { F αi , K ±αi | αi ∈ Π}, respectively. U 0 and U 0 are called the positive Borel part and the negative Borel part of U respectively. Lusztig defined a series of algebra automorphisms T i of U [22] (also see [12,3]), by which one obtains the Poincaré–Birkhoff–Witt bases (PBW bases, for short) for U . −ai j
Ti K j = K j Ki
Ti E j =
,
T i F i = − K i−1 E i ,
Ti Ei = −Fi Ki,
−ai j (r ) r (−ai j −r ) (−1)r q− Ei E j Ei if i = j , i r =0
Ti F j =
−ai j (−a −r ) (r ) (−1)r qri F i F j F i i j if i = j , r =0
(m)
(m)
where E i is E m . i divided by the q-factorial [m]q ! and similarly for F i Let si = sαi . Now, fix an element w ∈ W and a reduced expression w = si 1 si 2 · · · si N of w. Set β j = si 1 si 2 · · · si j−1 (αi j ) and define
E β j = T i 1 T i 2 · · · T i j −1 ( E α i ) j
and
F β j = T i 1 T i 2 · · · T i j −1 ( F α i ) j
for j = 1, . . . , N. Proposition 2.1. (1) [6, §2] The subspace
r r r U + [ w ] := spank E β11 E β22 · · · E βNN r1 , r2 , . . . , r N ∈ N
r r r resp. U − [ w ] := spank F βNN · · · F β22 F β11 r1 , r2 , . . . , r N ∈ N
(2.1)
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is a subalgebra of U + (resp. U − ) with the monomials E β11 E β22 · · · E βNN (resp. F βNN · · · F β22 F β11 ) as the PBW bases, and U + [ w ] (resp. U − [ w ]) is independent of the reduced expression of w. (2) [17, Proposition 5.5.2] For i < j, one has r
r
E βi E β j − q(β j ,βi ) E β j E βi =
r
r
r
cl E l ,
r
(2.2)
l∈N N l
l
l
where cl ∈ k, E l = E β11 E β22 · · · E βNN , and cl = 0 only when l = (l1 , . . . , l N ) is such that lt = 0 for t i or t j, and
F βi F β j − q−(β j ,βi ) F β j F βi =
dl F l ,
(2.3)
l∈N N l
l
l
where dl ∈ k, F l = F βNN · · · F β22 F β11 , and dl = 0 only when l = (l N , . . . , l1 ) is such that lt = 0 for t i or t j. (3) [22, Proposition 4.2] Multiplication defines an isomorphism of k-spaces
U− ⊗ U0 ⊗ U+ ∼ = U. The subscripts βi give Q + -graded algebra structures on U + [ w ] and U − [ w ]. Remark 2.2. By [6, Proposition in §2], U + [ w ] can be obtained by an iterated Ore extension, that is, U + [ w ] = k[ E β1 ][ E β2 ; σ2 , δ2 ] · · · [ E βN ; σ N , δ N ], where the automorphisms σ j and derivations δ j are implicitly given by (2.2). So we call the generating relations (2.2) the derivation relations of U + [ w ] and denote them by DR( E β j ). With the same argument to U − [ w ], we denote the relations (2.3) by DR( F βi ), by abuse of notation. In the following, we assume that the right coideal subalgebra C of U always satisfies the condition that C ∩ U 0 is a group algebra. For any h ∈ U , there is a left adjoint action ad h on U , defined by
(ad h)(u ) =
h(1) u S (h(2) ),
u ∈ U.
Suppose A is a subalgebra of U and R is a subspace of U . If R is stable under the left adjoint action of ad a for all a ∈ A, R is called an ad A-module. Definition 2.3. (See [9, Definition 2.6].) Let C be a right coideal subalgebra of U . C is called connected if C ∩ U 0 = k1. For any right coideal subalgebra C of U 0 , we write L (C ) for the subgroup of Q corresponding to the group algebra C ∩ U 0 . Clearly, C is connected if and only if L (C ) = 0. Conversely, for any subgroup L of Q , write T L for the group algebra spanned by { K α | α ∈ L }. For any right coideal subalgebra C of U 0 , let C β = U + K β ∩ C , and let
I (C ) =
u ∈ C −β (ad K γ )(u ) = q(γ ,β) u for all γ ∈ L (C ) .
(2.4)
β∈ Q +
The elements in each component I (C )β are called weight vectors in some literature. It was shown that any right coideal subalgebra C of U 0 can be decomposed into a tensor product of a group algebra and a connected right coideal subalgebra as a k-space.
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Proposition 2.4. (See [9, Proposition 2.8].) Suppose that C is a right coideal subalgebra of U 0 such that C ∩ U 0 is a group algebra. (1) I (C ) is a connected right coideal subalgebra of U 0 as well as an ad T L (C ) -module. (2) The decomposition (2.4) gives a Q + -graded algebra structure on I (C ). (3) The multiplication map I (C ) ⊗ T L (C ) → C is bijective. The following theorem illustrates the relation between the right coideal subalgebras of U 0 and the Weyl group W of g. Theorem 2.5. (See [9, Theorem 2.17].) There is a one-to-one correspondence between the set of all right coideal subalgebras C of U 0 such that C ∩ U 0 is a group algebra and the set of all triples ( w , ϕ , L ), where w ∈ W , ϕ : U + [ w ] → k is a character and L is a subgroup of Q such that ϕ and L satisfy an additional compatibility condition. More precisely, for a given C , there exists a unique w ∈ W and a unique isomorphism of Q + -graded algebras F : U + [ w ] → I (C ), ϕ = ε F and L = L (C ). Conversely, for any appropriate triple ( w , ϕ , L ), there exists a connected right coideal subalgebra D ( w , ϕ ) of U 0 which is isomorphic to U + [ w ] as an algebra and has an ad T L -module structure, such that C is given by D ( w , ϕ ) T L . Let G + be the subalgebra of U generated by { E α K −α | α ∈ Π}. For any w ∈ W , the assignment xβ → q−(β,β)/2 xβ K −β , where xβ ∈ U + [ w ]β , gives an injective algebra map U + [ w ] → G + , whose image is a right coideal subalgebra of U 0 , denoted by G + [ w ]. Remark 2.6. (1) Clearly, U 0 is an N-graded algebra with deg( E i ) = 1 and deg( K i ) = 0. If C is an N-graded subalgebra, then C corresponds to a triple ( w , ε , L ). In this case, I (C ) = G + [ w ]. (2) For any C , I (C ) is a subalgebra of G + . In particular, if C is connected, C is a subalgebra of G + . (3) It follows from Proposition 2.4 and Theorem 2.5 that C = I (C ) # T L (C ) as algebras. (4) U + [ w ] is an ad T L (C ) -module. By the proof of [9, Theorem 2.17], F is also an isomorphism of ad T L (C ) -modules. If C is a right coideal subalgebra of the negative Borel part U 0 , then all the results are parallel to the positive Borel part with a few modifications. We state them as follows. The subgroup L (C ) of Q is defined completely the same as in the positive Borel part. Let G − be the subalgebra of U generated by { F α K α | α ∈ Π} and C β = G − K β ∩ C . Let
I (C ) =
u ∈ C −β (ad K γ )(u ) = q−(γ ,β) u for all γ ∈ L (C ) .
β∈ Q +
Then I (C ) is a connected right coideal subalgebra of U 0 as well as an ad T L (C ) -module, contained in U − , and C = I (C ) # T L (C ) . There is an anti-isomorphism of Q + -graded algebras G : U − [ w ] → I (C ) for some w ∈ W , which is also an isomorphism of ad T L (C ) -modules. If C is an N-graded subalgebra of U 0 , then G ( y β ) = q(β,β)/2 S ( y β ) K −β where y β ∈ U − [ w ]β . 3. Nakayama automorphisms and bimodule resolutions In this section, we focus on the right coideal subalgebras C of U 0 . By Theorem 2.5 and Remark 2.6, C corresponds to a triple ( w , ϕ , L ), and F : U + [ w ] → I (C ) is an isomorphism of ad T L (C ) -module algebras. Given a reduced expression w = si 1 si 2 · · · si N , let E β j be as given by (2.1) and for j = 1, 2, . . . , N let
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X β j = F ( E β j ). The algebra structure of U + [ w ] is illustrated by Proposition 2.1. Suppose that the rank of L = L (C ) is M. Pick γ1 , . . . , γ M to be the generators of L. Let us consider the algebra structure of C . Since C = I (C ) # T L (C ) and
(ad K γi )( X β j ) = F (ad K γi ) E β j = q(γi ,β j ) X β j for any 1 i M and 1 j N, it follows that C is generated by K ±γi (1 i M) and X β j (1 j N), with the relations
K γi K γi = K γi K γi ,
K γ i K −γ i = 1,
K γi X β j K −γi = q(γi ,β j ) X β j ,
1 i, i M , 1 j N,
DR( X β j ),
where the notation DR is given by Remark 2.2. 3.1. Computation of Nakayama automorphisms We construct an algebra D as follows,
D = k g 1 , . . . , g M , x1 , . . . , x N
g i g i − g i g i , g i x j − q(β j ,γi ) x j g i , DR(x j ) .
It is easy to see that C is isomorphic to the localization of D at { g 1 , . . . , g M }. First of all, let us prove that D is twisted Calabi–Yau and compute the Nakayama automorphism ν D . For 1 r t N, denote by D r ,t the subalgebra of D generated by xr , . . . , xt over k[ g 1 , . . . , g M ]. It is easy to verify that D r ,t = D r ,t −1 [xt ; σ , δ] is an Ore extension if r < t, where
σ ( g i ) = q−(βt ,γi ) g i , i = 1, . . . , M , σ (x j ) = q−(βt ,β j ) x j ,
j = r , . . . , t − 1.
Furthermore, D r ,t = D r +1,t [xr ; σ , δ ] is also an Ore extension, where
σ ( g i ) = q−(βr ,γi ) g i , i = 1, . . . , M , σ (x j ) = q(βr ,β j ) x j ,
j = r + 1, . . . , t .
The twisted Calabi–Yau property is preserved by Ore extensions [20]. Theorem 3.1. (See [20, Theorem 3.3].) Suppose that B = A [x; σ , δ] is an Ore extension with σ an automorphism of A. If A is ν A -twisted Calabi–Yau of dimension d, then B is twisted Calabi–Yau of dimension d + 1 whose Nakayama automorphism ν B satisfies ν B | A = σ −1 ν A . Lemma 3.2. For 1 r t N, D r ,t is twisted Calabi–Yau of dimension M + t − r + 1. The Nakayama automorphism νr ,t of D r ,t is uniquely given by
νr ,t ( g i ) = q
t
l=r (βl ,γi )
j −1
νr ,t (x j ) = q−
for i = 1, 2, . . . , M and j = r , r + 1, . . . , t.
l=r
gi ,
(βl ,β j )+
t
M
l= j +1 (βl ,β j )−
s=1 (γs ,β j )
xj
(3.1)
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Proof. The first assertion follows immediately from Theorem 3.1. The Nakayama automorphism is unique since the identity map is the only inner automorphism of D r ,t . We prove it by induction on t − r. If t − r = 0, then D r ,t = D r ,r is a quantum ( M + 1)-space, namely,
D r ,r = k g 1 , . . . , g M , xr The Nakayama automorphism
g i g i − g i g i , g i xr − q(βr ,γi ) xr g i .
νr ,r is given by (see also [20, Proposition 4.1]) M
νr ,r (xr ) = q−
νr ,r ( g i ) = q(βr ,γi ) g i ,
s=1 (γs ,βr )
xr .
Hence (3.1) holds. Suppose t − r 1. By the inductive hypothesis, (3.1) holds for D r ,t −1 . By Theorem 3.1, the Nakayama automorphism νr ,t satisfies that for i = 1, . . . , M and j = r , . . . , t − 1,
νr ,t ( g i ) = σ −1 νr ,t −1 ( g i ) = q
t
l=r (βl ,γi )
j −1
νr ,t (x j ) = σ −1 νr ,t −1 (x j ) = q−
l=r
gi , t
(βl ,β j )+
M
s=1 (γs ,β j )
l= j +1 (βl ,β j )−
x j.
Similarly, the above deduction can be applied to the Ore extension D r ,t = D r +1,t [ X βr ; ρ , δ ]. Then
− 1
νr ,t (xt ) = σ
j −1
νr +1,t (xt ) = q−
l=r
t
(βl ,βt )+
M
l= j +1 (βl ,βt )−
s=1 (γs ,βt )
xt .
2
Therefore, (3.1) holds for any r, t.
Let d = M + N. Since D = D 1, N , we have Lemma 3.3. The algebra D is twisted Calabi–Yau of dimension d. The Nakayama automorphism ν D is uniquely given by
ν D ( gi ) = q
N
l=1 (βl ,γi )
j −1
ν D (x j ) = q −
l =1
gi ,
(βl ,β j )+
N
M
l= j +1 (βl ,β j )−
s=1 (γs ,β j )
xj
for i = 1, 2, . . . , M and j = 1, 2, . . . , N. The following theorem is the main result in this section. Theorem 3.4. Let C be a right coideal subalgebra of U 0 such that C ∩ U 0 is a group algebra. Then C = I (C ) # T L (C ) is ν -twisted Calabi–Yau of dimension d, where the Nakayama automorphism ν is given, up to an inner automorphism, by
ν ( K γi ) = q
N
l=1 (βl ,γi )
j −1
ν ( X β j ) = q− for i = 1, 2, . . . , M and j = 1, 2, . . . , N.
l =1
K γi , N
(βl ,β j )+
l= j +1 (βl ,β j )
Xβ j
(3.2)
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1081
Proof. As we mentioned, C is isomorphic to the localization of D at { g 1 , . . . , g M }. By [7, Theorem 6 and Example 8], C is twisted Calabi–Yau of dimension d provided that D is, also. In this case, the Nakayama automorphism ν of C is nothing but the extension of ν D . So by Lemma 3.3, we obtain
ν ( K γi ) = q
N
l=1 (βl ,γi )
j −1
ν ( X β j ) = q−
l =1
K γi , N
(βl ,β j )+
M
l= j +1 (βl ,β j )−
s=1 (γs ,β j )
Xβ j .
(3.3)
Since the inner automorphisms of C are those adjoint actions ad K γ for some γ ∈ L (C ), in particular, γ = sM=1 γs and replace ν by (ad K −γ )ν in (3.3). Then the new map satisfies (3.2), as desired. 2
let
3.2. Homological integrals Homological integrals of AS-Gorenstein Hopf algebras were introduced by Lu, Zhang and the second author [21, Definition 1.1], as a generalization of the classical integrals of finite-dimensional Hopf algebras. The homological integral is a useful tool in studying infinite-dimensional Hopf algebras. The definition was generalized for any augmented AS-Gorenstein algebra [19, Definition 3.1]. Roughly speaking, a left homological integral of an augmented AS-Gorenstein algebra A of dimen-
l
l
sion d is the 1-dimensional k-space ExtdA (k, A ), denoted by A . It is easy to show A is an A e -module with trivial left A-action. The right A-action determines a character χ : A → k such that x · a = χ (a)x
l for all x ∈ A . Lemma 3.5. (See [20, Lemma 1.3].) Let A be a ν -twisted Calabi–Yau augmented algebra with the augmentation map ε : A → k. Then A is AS-regular, and the character χ determined by the left homological integral of A is εν . Proposition 3.6. Let C be a right coideal subalgebra of U 0 corresponding to a triple ( w , ϕ , L ). Then the character χ determined by the left homological integral of C satisfies
χ ( K γi ) = q
N
l=1 (βl ,γi )
Proof. By Lemma 3.5 and (3.2),
and
χ ( K γi ) = q j −1
χ ( X β j ) = q−
l =1
j −1
= q−
l =1
j −1
= q−
j −1
χ ( X β j ) = q−
l =1
N
l=1 (βl ,γi )
(βl ,β j )+ (βl ,β j )+ (βl ,β j )+
l =1
l= j +1 (βl ,β j )
ϕ ( E β j ).
and
N
l= j +1 (βl ,β j )
N
l= j +1 (βl ,β j )
N
N
(βl ,β j )+
l= j +1 (βl ,β j )
ε( X β j ) εF ( E β j ) ϕ ( E β j ).
2
3.3. Bimodule resolutions In this subsection, we first consider connected right coideal subalgebras C ⊆ U 0 . Suppose C corresponds to a triple ( w , ϕ , 0) with ( w ) = N. Then C can be viewed as a deformation of some quantum affine space
OΛ k N := k X 1 , . . . , X N /( X j X i − λi j X i X j ; 1 i < j N ) with λi j ∈ k \ {0}. These quantum spaces are Koszul algebras. In order to compute the Hochschild (co)homology for Koszul algebras A, Van den Bergh constructed the so-called Koszul bimodule complex K( A ) [28].
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According to his construction, K(OΛ (k N )) is of length N and K(OΛ (k N ))i is a free OΛ (k N )e -module N
of rank i for 0 i N. We will show that C is similar to a quantum space in some sense. Proposition 3.7. Let C ⊆ U 0 be a connected right coideal subalgebra corresponding to a triple ( w , ϕ , 0) with
( w ) = N. Then there exists a C -bimodule complex K˜ (C ) such that
˜ (C ) is of length N, (1) K
˜ (C )i is a free C e -module of rank N for 0 i N, and K˜ (C )0 = C ⊗ C , (2) K i μ
˜ (C ) −−→ C −→ 0 is exact via the multiplication μ. (3) K
˜ (C ) can be constructed inductively. Proof. By [20], the complex K ∼ ˜ (C ) clearly exists. If N = 1, C = k[x] and such complex K If N > 1, suppose w = si 1 · · · si N . Let w = si 1 · · · si N −1 , ϕ be the restriction of ϕ to U + [ w ] and C be the right coideal subalgebra corresponding to ( w , ϕ , 0). So C = C [ X βN ; σ , δ] is an Ore extension
˜ (C ) satisfying the three of C . By the inductive hypothesis, there exists a C -bimodule complex K ˜ (C ) is conditions. Assume K C ⊗ V N −1 ⊗ C −→ · · · −→ C ⊗ V 1 ⊗ C −→ C ⊗ C , where V i is a k-space of dimension
N −1
i
. −1
˜ ( C ) ⊗C σ Since C is free over C , C ⊗C K −1 σ C ⊗C C and C ⊗C C , respectively. The map
ξ : C ⊗C σ
−1
˜ (C ) ⊗C C are C e -free resolutions of C and C ⊗C K
C −→ C ⊗C C ,
c 1 ⊗ c 2 −→ c 1 ⊗ X βN c 2 − c 1 X βN ⊗ c 2 is a C e -module homomorphism. By the comparison lemma, ξ can be lifted to a morphism of com˜ (C ) ⊗C σ −1 C → C ⊗C K˜ (C ) ⊗C C . Since plexes θ : C ⊗C K −1 ξ mult 0 −→ C ⊗C σ C −→ C ⊗C C −−→ C −→ 0
is exact, cone(θ) is a C e -free resolution of C via N −1 N −1 N
+ i −1 = i . i
˜ (C ) = cone(θ), as desired. Let K
μ and the rank of cone(θ)i is dimk V i + dimk V i−1 =
2
Next, we consider non-connected cases. Suppose C corresponds to a triple ( w , ϕ , L ) with ( w ) = N and the rank of L equal to M. Notice that the Koszul bimodule complex K( P ) of the polynomial ring P = k[ g 1 , . . . , g M ] is of
M length M and the rank of K( P )i is i . Since T L is a localization of P , by the proof of [7, Theorem 6], T L ⊗ P K( P ) ⊗ P T L is a T L -bimodule complex of T L . Using the same argument as the proof of Proposition 3.7, we have Proposition 3.8. Let C ⊆ U 0 be a right coideal subalgebra corresponding to a triple ( w , ϕ , L ) with ( w ) = N ˜ (C ) such that and the rank of L equal to M. Let d = M + N. Then there exists a C -bimodule complex K
˜ (C ) is of length d, (1) K
˜ (C )i is a free C e -module of rank d for 0 i d, and K˜ (C )0 = C ⊗ C , (2) K i
μ ˜ (C ) −− → C −→ 0 is exact via the multiplication μ. (3) K
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4. Twisted Calabi–Yau property in general case In this section, we consider the right coideal subalgebra C of U , and prove that C is AS-regular and twisted Calabi–Yau. First of all, let us introduce a filtration on U which was defined by G. Letzter [16]. Let w ∈ W be of the longest length, say T , and fix a reduced expression of w. Then U + [ w ] = U + [22,17] and we have a family of generators E βi and F βi (i = 1, 2, . . . , T ) for U as given in (2.1). The set consisting of l r l r all monomials Ml,α ,r := F βTT · · · F β11 K α E β11 · · · E βTT , where l = (l T , . . . , l1 ), r = (r1 , . . . , r T ) ∈ N T , α ∈ Q , is a PBW basis for U . For any Ml,α ,r , equip it with a bi-degree
bideg(Ml,α ,r ) =
T
li (ht βi ),
i =1
T
r i (ht βi ) ∈ N2 .
i =1
We will view N2 as a totally ordered monoid with the lexicographical order from right to left. Let Γt U be the subspace spanned by the monomials Ml,α ,r such that bideg(Ml,α ,r ) t for any t ∈ N2 . Then we obtain a filtration on U . The triangular decomposition
U∼ = U − ⊗ U 0 ⊗ G +, proved in [16, (1.12)] is an alternative expression of Proposition 2.1 (3). Analogous with [16, (4.14)], we have gr U ∼ = (U − ⊗ G + ) # U 0 , where ⊗ stands for the braided tensor product with braiding
G + ⊗ U − −→ U − ⊗ G + , E β K −β ⊗ F α −→ q(α ,β) F α ⊗ E β K −β . It is clear that gr U − = U − and gr G + = G + . Remark 4.1. Observe that gr U can be realized iteratively. In fact, one may define a positive filtration on U via the second component of the bi-degree. The associated graded ring U (1) together with the first component of the bi-degree, is a positive filtered ring whose filtration is compatible with the graded structure. Thus taking the associated graded ring again, one obtains an N2 -graded ring U (2) . It is routine to check that the N2 -graded ring U (2) is nothing but gr U . Next, we consider the induced filtration on C and the associated graded ring. By the right coideal version of [16, Theorem 4.9],
gr C ∼ =
U − ∩ gr C ⊗ G + ∩ gr C
# T L (C ) ,
and U − ∩ gr C (resp. G + ∩ gr C ) is a connected right coideal subalgebra of U 0 (resp. U 0 ). Since U − ∩ gr C and G + ∩ gr C are both N-graded algebras, by Remark 2.6, the corresponding triples must be of the form ( w 1 , ε , 0), ( w 2 , ε , 0) for some w 1 , w 2 ∈ W . Fix their reduced expressions. Thus the PBW bases for U − ∩ grΓ C , G + ∩ grΓ C are determined. In addition, we may assume the reduced expression of w 2 is an initial segment of that of w, for simplicity (see [2, Chapter 3]). So their generator sets are
Y β Y β = q(βi ,βi )/2 S ( F β ) K −β , i = 1, . . . , L 1 ,
i
i
i
i
X β j X β j = q−(β j ,β j )/2 E β j K −β j , j = 1, . . . , L 2 ,
where L 1 = ( w 1 ) and L 2 = ( w 2 ).
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Remark 4.2. In ring theory and homological algebra, using a positive filtration (or more generally, Zariskian filtration) one can lift information from the associated graded ring/module to the filtered ring/module. By Remark 4.1, gr C can also be realized iteratively. Hence we may investigate the properties of C by studying gr C . Recall the relations DR in Remark 2.2. Obviously, the generators X β j of G + ∩ gr C satisfy DR( X β j ). Similarly, DR( F β ) are transferred to the generating relations of U − ∩ gr C via the anti-isomorphism G , i
denoted by DRop (Y β ). i Let L 3 be the rank of L (C ) and d = L 1 + L 2 + L 3 . It follows that gr C is generated by these Y β , X β j i and K γ (γ ∈ L (C )), subject to
K γ K γ = K γ +γ , K γ Y β K −γ = q
K 0 = 1,
−(γ ,β ) i
i
X β j Y β = q(βi ,β j ) Y β X β j , i
Y β , i
op
DR (Y β ), i
i
(4.1)
K γ X β j K −γ = q(γ ,β j ) X β j ,
(4.2)
DR( X β j ).
(4.3)
Consequently, gr C is an iterated graded Ore extension of the group algebra T L (C ) . By Theorem 3.1, gr C is graded twisted Calabi–Yau of dimension d. Hence it is graded AS-regular. Theorem 4.3. Let C be a right coideal subalgebra of U such that C ∩ U 0 is a group algebra. Then C is AS-regular and twisted Calabi–Yau. Proof. Endow k with the obvious good filtration. Then by Remark 4.2 and by [1, Proposition 3.1] or [18, Proposition 4, §2.2, Chapter III], ExtiC (k, C ) has a filtration and gr ExtiC (k, C ) is isomorphic to i a subquotient of Extgr C (gr k, gr C ) for every i. Since gr k = k and gr C is graded AS-regular of dimension d, C has finite global dimension and ExtiC (k, C ) = 0 for all i = d and dimk ExtdC (k, C ) = 0 or 1. By the double-Ext spectral sequence pq
p
−q
E 2 = ExtC op ExtC (k, C ), C
⇒
H p +q (k),
where
Hn (k) =
0, n = 0, k , n = 0,
one has ExtdC (k, C ) = 0. Therefore, dimk ExtdC (k, C ) = 1. Similarly, ExtiC op (k, C ) = 0 for all i = d and dimk ExtdC op (k, C ) = 1. Since U is pointed, it follows from [19, Theorem 3.6] that
ExtiC e C , C e ∼ =
0, Cν,
i = d, i =d
for some automorphism ν . It remains to check its homological smoothness. To this end, first by [19, Lemma 3.7], C admits a finitely generated projective resolution as a C e -module, and further,
proj.dim C e C proj.dim gr C e gr C = proj.dim (gr C )e gr C = d. Therefore, C is twisted Calabi–Yau.
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Acknowledgments Both authors thank the referee for the valuable comments. This research is supported by the NSFC (key projects 10731070 and 11171067), and STCSM (Science and Technology Committee, Shanghai Municipality, project 11XD1400500), and a training program for innovative talents of key disciplines, Fudan University. References [1] J.-E. Björk, The Auslander condition on noetherian rings, in: Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année, Paris, 1987/1988, in: Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 137–173. [2] A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Grad. Texts in Math., vol. 231, Springer, New York, 2005. [3] K.A. Brown, K.R. Goodearl, Lectures on Algebraic Quantum Groups, Adv. Courses Math. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. [4] K.A. Brown, J.J. Zhang, Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras, J. Algebra 320 (2008) 1814–1850. [5] S. Chemla, Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators, J. Algebra 276 (2004) 80–102. [6] C. De Concini, V.G. Kac, C. Procesi, Some quantum analogues of solvable Lie groups, in: Geometry and Analysis, Bombay, 1992, Tata Inst. Fund. Res., Bombay, 1995, pp. 41–65. [7] M. Farinati, Hochschild duality, localization, and smash products, J. Algebra 284 (2005) 415–434. [8] V. Ginzburg, Calabi–Yau algebras, preprint, arXiv:math/0612139v3, 2006, 79 pp. [9] I. Heckenberger, S. Kolb, Right coideal subalgebras of the Borel part of a quantized enveloping algebra, Int. Math. Res. Not. (2011) 419–451. [10] I. Heckenberger, S. Kolb, Homogeneous right coideal subalgebras of quantized enveloping algebras, Bull. Lond. Math. Soc. 44 (2012) 837–848. [11] I. Heckenberger, H.-J. Schneider, Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid, Israel J. Math. 197 (1) (2013) 139–187. [12] J.C. Jantzen, Lectures on Quantum Groups, Grad. Stud. Math., vol. 6, American Mathematical Society, Providence, RI, 1996. [13] V. Kharchenko, Right coideal subalgebras in U q+ (so2n+1 ), J. Eur. Math. Soc. 13 (2011) 1677–1735. [14] V.K. Kharchenko, A.V. Lara Sagahon, Right coideal subalgebras in U q (sln+1 ), J. Algebra 319 (2008) 2571–2625. [15] U. Krähmer, On the hochschild (co)homology of quantum homogeneous spaces, Israel J. Math. 189 (2012) 237–266. [16] G. Letzter, Coideal subalgebras and quantum symmetric pairs, in: New Directions in Hopf Algebras, in: Math. Sci. Res. Inst. Publ., vol. 43, Cambridge University Press, Cambridge, 2002, pp. 117–165. [17] S. Levendorski˘ıand, Y. Soibelman, Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Comm. Math. Phys. 139 (1991) 141–170. [18] Huishi Li, F. Van Oystaeyen, Zariskian Filtrations, K -Monogr. Math., vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. [19] L.-Y. Liu, Q.-S. Wu, Rigid dualizing complexes over quantum homogeneous spaces, J. Algebra 353 (2012) 121–141. [20] L.-Y. Liu, S.-Q. Wang, Q.-S. Wu, Twisted Calabi–Yau property of Ore extensions, J. Noncommut. Geom. (2013), in press, preprint, arXiv:1205.0893, 2012, 19 pp. [21] D.-M. Lu, Q.-S. Wu, J.J. Zhang, Homological integral of Hopf algebras, Trans. Amer. Math. Soc. 359 (2007) 4945–4975. [22] G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990) 89–113. [23] A. Masuoka, On Hopf algebras with cocommutative coradicals, J. Algebra 144 (1991) 451–466. [24] A. Masuoka, D. Wigner, Faithful flatness of Hopf algebras, J. Algebra 170 (1994) 156–164. [25] E.F. Müller, H.-J. Schneider, Quantum homogeneous spaces with faithfully flat module structures, Israel J. Math. 111 (1999) 157–190. [26] B. Pogorelsky, Right coideal subalgebras of the quantum Borel algebra of type G 2 , J. Algebra 322 (2009) 2335–2354. [27] M. Takeuchi, Relative Hopf modules—equivalences and freeness criteria, J. Algebra 60 (1979) 452–471. [28] M. Van den Bergh, Noncommutative homology of some three-dimensional quantum spaces, K -Theory 8 (1994) 213–230. [29] M. Van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998) 1345–1348, Erratum: Proc. Amer. Math. Soc. 130 (2002) 2809–2810 (electronic).