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Hopf algebras constructed by the FRT-construction
Journal of Pure and Applied Algebra 213 (2009) 772–782
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Hopf algebras constructed by the FRT-construction Jacob Towber a , Sara Westreich b,∗ a
Department of Mathematics, De Paul University, Chicago, IL, United States
b
Interdisciplinary Department of the Social Sciences, Bar-Ilan University, Ramat-Gan, Israel
article
info
Article history: Received 10 January 2008 Received in revised form 31 August 2008 Available online 13 November 2008 Communicated by C. Kassel This article is dedicated to the memory of Jack Towber, a good friend and colleague MSC: Primary: 16W30 17B37
a b s t r a c t Given any bialgebra A and a braiding product h|i on A, a bialgebra Uh|i was constructed in [R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of ‘‘quantum group’’ and ‘‘quantum Lie algebra’’, Comm. Algebra 19 (1991) 3295–3345], contained in the finite dual of A. This construction generalizes a (not very well known) construction of Fadeev, Reshetikhin and Takhtajan [L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97–110]. In the present paper it is proved that when Uh|i is finite-dimensional (even if A is not), then it is a quasitriangular Hopf algebra. © 2008 Elsevier B.V. All rights reserved.
0. Introduction One of the most fundamental constructions in the theory of Lie groups, is that which associates to a Lie or algebraic group G, its Lie algebra g. Expressing this in Hopf-algebraic terms, we have a construction
Φ : A(G) 7→ U (g) which inputs the bialgebra A(G) of representative functions on an affine algebraic group G (over the ground field K) and constructs from it the Hopf K-algebra U (g). It is remarkable that in the last 20 or so years, when so much attention has been devoted to generalizing results from Lie groups and Lie algebras to quantum groups, there seems to have been no proposed generalization of Φ to quantum groups. One purpose of this paper is to investigate such a proposed generalization. This paper completes a train of thought begun 15 years ago in [1], where a construction Φ , closely related to the remarkable constructions of Faddeev, Reshetikhin and Takhtajan [2,3] was given.1 Let us recall some facts about the constructions in [2,3] and the related LT-construction. Given an invertible Yang–Baxter operator R, Faddeev, Reshetikhin and Takhtajan associate to it two bialgebras, which we shall here denote by AFRT (R) and UFRT (R). It was proved in [1, Th. 5.1] that AFRT (R) has a braiding structure h|iR depending only on R. Unlike the AFRT (R)-construction which is quite known, the second one, UFRT (R), seems to have been largely forgotten. Going further it was proved in [1, Section 6] that for any bialgebra A with a braiding structure h|i, there exists a general Φ : (A, h|i) → Uh|i , where Uh|i is a bialgebra contained in the finite dual of (A, h|i). When (A, h|i) = AFRT (R) then the bialgebra Uh|i is naturally isomorphic to UFRT (R) and thus the LT-construction is a generalization of the UFRT (R)-construction. All the constructions mentioned above are of bialgebras. However, known specific cases of UFRT (R) are all Hopf algebras. Such are for example the multiparameter deformations of U (sln ) described in [4]. Thus, a natural question to ask is whether Uh|i is always a Hopf algebra. In this paper we give a partial positive answer to this question. We prove:
∗
Corresponding author. E-mail addresses: [email protected] (J. Towber), [email protected] (S. Westreich).
1 In this connection, L. Takhtajan was kind enough to supply some very valuable help, which guided us into the related literature. 0022-4049/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2008.09.012
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Theorem 2.4. Let (A, h|i) be a bialgebra with a braiding structure h|i. If Uh|i is finite-dimensional then it is a Hopf algebra. Let it be emphasized that Uh|i can be finite-dimensional even when (A, h|i) is not. The Uh|i -construction utilizes the braiding product in a simple but not quite straightforward way; it is NOT merely a question of taking the finite dual of A. As a result, neither the dimension nor quasitriangularity of Uh|i follow just by dualizing the structure of (A, h|i) (Example 1.4). We also prove: Theorem 3.3. Let (A, h|i) be a bialgebra with a braiding structure h|i. If Uh|i is finite-dimensional then it is a Hopf algebra quotient of a Drinfeld double and thus a quasitriangular Hopf algebra. An important class of Hopf algebras obtained by the Uh|i -construction is the family of Hopf algebras denoted by U (RQ ), where RQ is the Yang–Baxter operator associated with the multiparameter deformation of GLn supplied in [5,6]. These Hopf algebras were described and discussed in detail in [4,7]. In this paper we characterize all finite-dimensional Hopf algebras (thus necessarily quasitriangular) obtained by this construction for the case n = 2. Though finite-dimensional pointed Hopf algebras were fully characterized by [8], the characterization does not tell quasitriangular Hopf algebras from the others. Aside from the one-parameter deformations of Jimbo, Drinfeld, Lusztig et al., evaluated at roots of unity [9], (which are not always quasitriangular), there have it seems hitherto been no general examples of finite-dimensional quasitriangular Hopf algebras. Thus, it may be of interest that the Uh|i -construction produces infinitely many new such examples, non-isomorphic and unrelated by a cocycle twist [10]. 1. Preliminaries Throughout this paper (except where otherwise specified), all bialgebras considered will be taken over a fixed groundfield K. ⊗ will then mean ⊗K . Let A be a bialgebra over K. We denote by P (A) the set of all K-linear maps
h|i : A ⊗ A → K,
a ⊗ b 7→ ha|bi.
and note the natural associative product ∗A on P (A), defined for all h|i1 and h|i2 in P (A) and a, b ∈ A by
(h|i1 ∗A h|i2 )(a ⊗ b) =
X
ha(1) | b(1) i1 ha(2) | b(2) i2 .
(a),(b)
Note that the 2-sided identity element 1(∗A ) with respect to ∗A is given by 1(∗A )(a ⊗ b) := ε(ab) . We say that h|i ∈ P (A) is (convolution) invertible if it has a 2-sided inverse with respect to ∗A (which we shall denote by h|i(−1) ), i.e. a product which satisfies
h|i ∗A h|i(−1) = h|i(−1) ∗A h|i = 1(∗A ). If h|i is (convolution) invertible we shall also denote by h|i∗ the element of P (A) defined (for all a, b ∈ A) by
ha|bi∗ := hb|ai(−1) .
(1.1)
Definition 1.1. An element h|i in P (A) will be called a left braiding product for A, precisely when it satisfies the following four conditions, for all a, b and c in A: (B1) h|i P is (convolution) invertible. P (B2) (a),(b) ha(1) |b(1) ib(2) a(2) = (a),(b) ha(2) |b(2) ia(1) b(1) . P (B3) ha|bc i = h a(1) |biha(2) |c i. ( a ) P (B4) hab|c i = (c ) hb|c(1) iha|c(2) i. We define the braiding maps on A, to be the following four maps
λ+ : A → A0 , λ− : A → A0 ,
a 7→ ha|−i ∗
a 7→ ha|−i
ρ + : A → A0 , a 7→ h−|ai; ρ − : A → A0 , a → 7 h−|ai∗ .
Note that the definition of h|i∗ also implies immediately that
λ+ ∗ ρ − = 1L = ρ − ∗ λ+ and
λ− ∗ ρ + = 1L = ρ + ∗ λ− .
(1.2)
0
It is known that these four maps land in the finite dual A . Moreover, Proposition 1.2. λ± are algebra anti-homomorphisms and coalgebra homomorphisms, while ρ ± are algebra homomorphisms and coalgebra anti-homomorphisms. Retaining the above assumptions, we are ready to define the following bialgebra: Definition 1.3. Define Uh|i to be the subalgebra of A0 generated by the set of all λ+ (a), λ− (a), ρ + (a) and ρ − (a) with a ∈ A.
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To emphasize the fact that Uh|i is more then just taking the finite dual we give the following example. Example 1.4. Let M = {g i , i ≥ 0} be an abelian monoid and let A = kM be the monoid algebra. A has a bialgebra structure by letting ∆(g i ) = g i ⊗ g i and ε(g i ) = 1 for all i ≥ 0. Fix some q ∈ k. We put a braiding structure on A by defining
hg i |g j i := qi+j . It is straightforward that this is indeed a braiding structure on the bialgebra A and that h|i(−1) is given by:
hg i |g j i(−1) = q−(i+j) . Since the braiding is symmetric we have λ+ = µ+ and λ− = µ− . The construction of Uh|i implies now that λ+ (g i ) is the algebra character given by λ+ (g i )(g j ) = qi+j while λ− (g i ) is the algebra character given by λ− (g i )(g j ) = q−(i+j) . Thus, λ+ (g i ) and λ− (g i ) are group inverses of each other inside Uh|i , and so Uh|i , is the group algebra of the the group consisting of the algebra characters defined above. In particular Uh|i is a Hopf algebra. Observe also that if q is a root of unity then we obtain a finite set of characters. In this case Uh|i is a finite-dimensional Hopf algebra while A is an infinite-dimensional bialgebra. 2. Finite-dimensional Hopf algebras obtained as Uh|i The purpose of this section is the proof that finite-dimensional bialgebras of the form Uh|i are always Hopf algebras. In order to prove this, we shall define and utilize the ‘‘quantum Borel subalgebras’’ B+ and B− of Uh|i which will be shown to be Hopf algebras satisfying B+ B− = B− B+ when they are finite-dimensional. Proposition 2.1. Let Uh|i the sub-bialgebra of A0 generated by the images of the four maps λ+ , λ− , ρ + , ρ − as defined in Definition 1.3. Then
(Im ρ + )(Im ρ − ) = (Im ρ − )(Im ρ + ) and
(Im λ+ )(Im λ− ) = (Im λ− )(Im λ+ ) where (Im ρ + )(Im ρ − ) denotes the K-span of all products of the form ρ + (a)ρ − (a0 ), with a and a0 in A, and similarly for the other three products involved. Proof. It was proved [1, Prop. 2.8] that h|i∗ also satisfies the braiding axioms. Thus, for all a, b, c ∈ A,
(B∗ 2)
X
a(2) c(2) ha(1) |c(1) i(−1) =
X
c(1) a(1) ha(2) |c(2) i(−1)
(obtained from (B2) upon replacing h|i by h|i∗ , a by c and b by a). Also
(B∗ 4)
X X ha|bc i(−1) = ha(2) |bi(−1) ha(1) |c i(−1) .
Hence for all a, b, c ∈ A we have
(ρ + (a)ρ − (b))(c ) = = = = = =
X (ρ + (a))(c(1) )(ρ − (b))(c(2) ) X hc(1) |aihb|c(2) i(−1) X ε(a(2) b(2) )hc(1) |a(1) ihb(1) |c(2) i(−1) X hb(2) |a(2) i(−1) hb(3) |a(3) ihc(1) |a(1) ihb(1) |c(2) i(−1) X hc(1) |a(1) ihb(1) |a(2) c(2) i(−1) hb(2) |a(3) i by (B∗ 4) X hc(2) |a(2) ihb(1) |c(1) a(1) i(−1) hb(2) |a(3) i in (B2) replace a by c, b by a;
then apply hb(1) |−i(−1) to both sides X = hc(2) |a(2) ihb(2) |c(1) i(−1) hb(1) |a(1) i(−1) hb(3) |a(3) i X = hb(1) |a(1) i(−1) hb(3) |a(3) i(ρ − (b(2) )ρ + (a(2) ))(c ).
by (B∗ 4)
Hence
ρ + (a)ρ − (b) =
X
hb(1) |a(1) i(−1) hb(3) |a(3) iρ − (b(2) )ρ + (a(2) ).
(2.1)
Thus, (Im ρ + )(Im ρ − ) ⊆ (Im ρ − )(Im ρ + ). The converse inclusion follows similarly and we have:
ρ − (a)ρ + (b) =
X
ha(1) |b(1) iha(3) |b(3) i(−1) ρ + (b(2) )ρ − (a(2) )
(2.2)
J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
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and so finally
(Im ρ + )(Im ρ − ) = (Im ρ − )(Im ρ + ). It is left to the reader to supply the precisely similar proof that
(Im λ+ )(Im λ− ) = (Im λ− )(Im λ+ ). We assume for the remainder of this paper (unless otherwise stated) that U := Uh|i is a finite-dimensional vector space over K. Set B+ := Im λ+
B− := Im λ− .
By Proposition 1.2, B+ and B− are K-sub-bialgebras of A0 , finite-dimensional as vector-spaces over K. Proposition 2.2. If U is finite-dimensional then B+ and B− are Hopf algebras with invertible antipodes. Proof. Let M denote the convolution algebra (HomK (B+ , B+ ), ∗). We shall first show that IdB+ is a left non-zero-divisor in M: Assume, P on the contrary, that IdB+ ∗ f = 0 for some non-zero f ∈ M. Thus, (x) x(1) f (x(2) ) = 0 for all x ∈ Im λ+ . Applying this with x = λ+ (a), we have (since λ+ is a coalgebra homomorphism, and using (1.2)) f (x) = f (λ+ (a)) =
X
X ε(a(1) )f (λ+ (a(2) )) = ρ − (a(1) )λ+ (a(2) )f (λ+ (a(3) )) X X = ρ − (a(1) ) (λ+ (a(2) )(1) f ((λ+ (a(2) )(2) ))) =
X
(a)
(λ+ (a(2) ))
ρ (a(1) )(I ∗ f )(λ+ (a(2) )) = 0. −
Hence f = 0, contradiction. Thus, IdB+ is a left non-zero divisor in M; a precisely symmetrical argument shows that it is a right non-zero divisor. Since M is finite-dimensional over K, it follows that IdB+ is a unit in M; i.e., that B+ is a Hopf algebra. A precisely similar argument shows that B− = Im λ− is a Hopf algebra. Finally, we apply to B+ and B− the theorem ([11, Cor.5.1.6]) that the antipode of a finite-dimensional Hopf algebra is invertible. Proposition 2.3. Let U be finite-dimensional over K, and let S + , S − denote the antipodes of B+ , B− , respectively. Then: 1. S + ◦ λ+ = ρ − and S − ◦ λ− = ρ + . 2. Im λ+ = Im ρ − and Im λ− = Im ρ + . Proof. 1. To prove the assertion that S + ◦ λ+ and ρ − are equal, we shall next show that each is a two-sided inverse to λ+ in the convolution algebra L := (HomK ((A, U ), ∗)). For all a in A, since λ+ is a coalgebra homomorphism,
((S + ◦ λ+ ) ∗ λ+ )(a) =
X
S + (λ+ (a(1) ))λ+ (a(2) )
(a)
=
X
S + ((λ+ (a))(1) )(λ+ (a)(2) )
(λ+ (a))
= ε(λ+ (a)) = ε(a) so that (using (1.2)),
(S + ◦ λ+ ) ∗ λ+ = 1L = ρ − ∗ λ+ . Similarly, we prove
λ+ ∗ (S + ◦ λ+ ) = 1L = λ+ ∗ ρ − . Hence S + ◦ λ+ and ρ − are equal. The proof that S − ◦ λ− = ρ + (since each is inverse to λ− ) is precisely similar. 2. From part 1 and the fact that S + : Im λ+ → Im λ+ is an invertible map, it follows immediately that Im λ+ = Im(S ◦ λ+ ) = Im ρ − . Similarly, Im ρ + = Im λ− . As a result we obtain Theorem 2.4. Let U = Uh|i the sub-bialgebra of A0 generated by the images of the four maps λ+ , λ− , ρ + , ρ − as defined in Definition 1.3. Assume U is finite-dimensional. Then U = B+ B− = B− B+ is a Hopf algebra.
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Proof. The fact that U = B+ B− = B− B+ follows from part 2 of Proposition 2.3 and from Proposition 2.1. To see that U is a Hopf algebra, it suffices as before to show that IdU is neither a left zero-divisor, nor a right zero-divisor in (hom(U , U ), ∗). The proofs of both parts are precisely similar; we shall here show that IdU is not a right zero-divisor in (hom(U , U ), ∗). Suppose f : U → U , f ∗ IdU = 0; then we are done if we deduce f = 0. Since U = B+ B− , it suffices to deduce f (ab) = 0 for all a ∈ B+ and b ∈ B− . Indeed, we then have: f (ab) =
X
f ((ab)(1) )ε((ab)(2) )
(ab)
=
X
f (a(1) b(1) )ε(a(2) )ε(b(2) )
(a)(b)
=
X
f (a(1) b(1) )a(2) ε(b(2) )S + (a(3) )
(a)(b)
=
X
f (a(1) b(1) )a(2) b(2) S − (b(3) )S + (a(3) )
(a)(b)
=
X
(f ∗ IdU )(a(1) b(1) )S − (b(2) )S + (a(2) ) = 0.
(a)(b)
3. Finite-dimensional Hopf algebras Uh|i are quasitriangular The purpose of this section, is the proof that finite-dimensional Hopf algebras of the form U = Uh|i are always quasitriangular. In order to prove this, we will show that the sub-Hopf algebras B+ and B− of U are in fact dual, compute the Drinfeld double constructed from these dual ‘‘quantum Borel subalgebras’’, and then show it has a Hopf algebra epimorphism onto U. As was discussed in the introduction, quasitriangularity of U does not follow just by dualizing the braiding structure of A, since U does not equal A0 . Yet the ideas along this section are very natural. Unfortunately, checking the details requires technical tedious proofs and we apologize the reader for that. Proposition 3.1. Assume U is finite-dimensional over K. Then: 1. A K-bilinear non-degenerate form h|iD : B+ ⊗ B− → K is well-defined by the specification:
hλ+ (a)|ρ + (b)iD := ha|bi for all a, b ∈ A. 2. For all p, p0 ∈ B+ and m, m0 ∈ B− ,
hp|mm0 iD =
X hp(1) |miD hp(2) |m0 iD ,
and hp|1B− iD = εB+ (p)
(3.1)
and h1B+ |miD = εB− (m).
(3.2)
(p)
hold, as do
hpp0 |miD =
X hp|m(2) iD hp0 |m(1) iD , (m)
3. Define θ0 to be the K-linear map
θ0 : B− → (B+ )∗ ,
m 7→ h−|miD
induced by h|iD ; then θ0 induces a bialgebra isomorphism ≈
θ : B− → ((B+ )0 )cop . 4. For all m ∈ B+ and p ∈ B− ,
hS + (p)|S − (m)iD = hp|miD . (−1)
5. The form h|iD is (convolution) invertible, with two-sided inverse h|iD
well-defined by
1) hp|mi(− = h(S + )−1 (p)|miD = hp|S − (m)iD D
+
−
for all p ∈ B and m ∈ B . Proof. 1. h|iD is well defined since if ρ + (b) = ρ + (b0 ) and λ+ (a) = λ+ (a0 ) then
ha|bi = λ+ (a)(b) = λ+ (a0 )(b) = ha0 |bi = ρ + (b)(a0 ) = ρ + (b0 )(a0 ) = ha0 |b0 i.
(3.3)
J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
777
We show now that h|iD is right non-degenerate, i.e. has right radical = 0. Indeed, if m = ρ + (b) is in the right radical of h|iD , then for all a in A, 0 = hλ+ (a)|ρ + (b)iD = ha|bi = ρ + (b)(a) whence ρ + (b) = m = 0. Similarly, we see that h|iD is left non-degenerate. 2. We will first verify (3.2), leaving to the reader the precisely similar verification of (3.1). In proving (3.2) we may assume p = λ+ (a),
p0 = λ+ (a0 ), m = ρ + (b) with a, a0 , b ∈ A
in which case the two assertions to be proved become
X
hλ+ (a)λ+ (a0 )|ρ + (b)iD =
hλ+ (a)|(ρ + (b))(2) iD hλ+ (a0 )|(ρ + (b))(1) iD
(ρ + (m))
and
hλ+ (1)|ρ + (b)iD = ε(b). This follows now from the braiding axioms (Definition 1.1) and since (by Proposition 1.2) λ+ is an algebra antihomomorphism while ρ + is a coalgebra anti-homomorphism. 3. Observe that θ being a coalgebra homomorphism is precisely the assertion (3.2) while θ being an algebra homomorphism follows from (3.1). Since B+ and B− are finite-dimensional, it follows that their dimensions are equal, and that the K-linear transformation θ0 is a K-isomorphism (whence θ is an isomorphism of K-bialgebras.). 4. We first recall the standard fact that, given B+ is a finite-dimensional Hopf algebra with antipode S + , it follows that S + is invertible, and that ((B+ )0 )cop is a Hopf algebra with antipode ((S + )∗ )−1 . It follows from the uniqueness of the antipode, ≈
that the bialgebra isomorphism θ : B− → ((B+ )0 )cop takes S − to ((S + )∗ )−1 , which is precisely the assertion to be proved. 5. That h(S + )−1 (p)|miD = hp|S − (m)iD follows from Proposition 3.1.4. Since both S + and S − are bijective, it follows that 1) is well-defined via (3.3). h|i(− D Thus, to complete the proof, two things must be verified: First, that for all p ∈ B+ and m ∈ B− ,
X
hp(1) |m(1) iD hp(2) |S − (m(2) )iD = ε(pm)
(p),(m)
— indeed, the left side, by (3.1), equals
X hp|m(1) S − (m(2) )iD = hp|ε(m)1B− i = ε(m)ε(p) = ε(pm). (m)
Second, that for all p ∈ B+ and m ∈ B− ,
X
hp(1) |S − m(1) iD hp(2) |m(2) iD = ε(pm)
(p),(m)
— the proof is similar.
Lemma 3.2. For all a and b in A,
hλ+ a|λ− biD = ha|bi(−1) . Proof. We need to show that for all a and b in A,
X
ha(1) |b(1) ihλ+ a(2) |λ− b(2) iD = ε(ab).
(a),(b)
Indeed, using (3.1) and the fact that λ+ is a coalgebra map the left side equals
X
hλ+ (a(1) )|ρ + (b(1) )iD hλ+ (a(2) )|λ− (b(2) )iD =
(a),(b)
X
hλ+ (a)|ρ + (b(1) )λ− (b(2) )iD .
(b)
Since, by (1.2), ρ + ∗ λ− = 1L in L = HomK (A, U ), it follows that
X
ρ + (b(1) )λ− (b(2) ) = ε(b)1L = ε(b)1B− ;
(b)
inserting this into the preceding expression, we obtain
hλ+ (a)|ε(b)1B− iD = ε(λ+ (a))ε(b) = ε(a)ε(b), which completes the proof.
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J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
Let us next recall the construction of the Drinfeld double D(B+ ) [12] for the finite-dimensional Hopf algebra B+ . We follow notations in [13]. This construction utilizes the following two module-coalgebra actions: (1) The left module-coalgebra action of B+ on X := ((B+ ))0 cop given (for all p, p0 in B+ and x in X ) by
``
(p
x)(p0 ) =
``
X
x(S −1 (p(2) )p0 p(1) ).
(2) The right module-coalgebra action p
´´
(3.4)
(p)
x=
X
´´
of X on B+ given (for all p in B+ and x in X ) by
x(S −1 (p(3) )p(1) )p(2) .
(3.5)
(p)
The Drinfeld double D(B+ ) = X FG B+ has X ⊗ B+ as its underlying vector space; its coalgebra structure is the usual tensor product of coalgebras. Multiplication is given by:
X
(x FG p)(x0 FG p0 ) =
[x(p1
(p),(x0 )
``
x02 )] FG [(p2
´´
x01 )p0 ]
for all p, p0 ∈ B+ , x, x0 ∈ X . Equivalently, the same multiplication on D(B+ ) is given by
X
(x FG p)(x0 FG p0 ) =
x03 (p1 )x01 (S −1 (p3 ))xx02 FG p2 p0 .
(3.6)
(p)(x0 )
The algebra thus defined on D(B+ ) is unital, with unity element 1D(B+ ) = 1X FG 1B+ . By [12], D(B+ ) is quasitriangular with its universal R-matrix R0 constructed as follows: Choose any basis {ei }i∈I for B+ over K, and let {ei }i∈I be the dual basis for (B+ )∗ ; let iB+ : B+ → D(B+ );
iX : X → D(B+ )
denote the canonical Hopf algebra monomorphisms; then R0 =
X
iB+ (ei ) ⊗ iX (ei ) ∈ D(B+ ) ⊗ D(B+ ).
(3.7)
i∈I
For the computations in the next subsection, it will be more convenient to replace the preceding construction of D(B+ ) by the following slight variation: Let us replace the pair (B+ , X ) = ((B+ )0 )cop in the preceding construction, by the pair (B+ , B− ), using the Hopf algebra isomorphism ≈
θ : B− → ((B+ )0 )cop of Proposition 3.1(4). It is readily verified that with this replacement, (3.4) and (3.5), respectively, go over into the two following module-coalgebra actions: (1a) The left module-coalgebra action of B+ on B− uniquely specified by requiring that, for all p, p0 ∈ B+ and all `` − m∈B ,
hp0 |p
``
miD =
X
θ (m)(S −1 (p(2) )p0 p(1) ) =
(p)
and, (2a) The right module-coalgebra action p
´´
m=
X
(3.8)
(p)
´´
θ (m)(S −1 (p(3) )p(1) )p(2)
(p)
X hS −1 (p(2) )p0 p(1) |miD
of B− on B+ uniquely specified by requiring that for all p ∈ B+ , m ∈ B− ,
X = h(S −1 (p(3) )p(1) )|miD p(2) .
(3.9)
(p)
In these terms, the preceding construction of the Drinfeld double D(B+ ), obviously goes over into the naturally isomorphic quasitriangular Hopf algebra B− FG B+ constructed as follows: B− FG B+ has B− ⊗ B+ as its underlying vector space and the coalgebra structure of the usual tensor product of coalgebras — i.e. for all p ∈ B+ and m ∈ B− we have
∆(m FG p) =
X
(m(1) FG p(1) ) ⊗ (m(2) FG p(2) )
(m),(p)
and
ε(m FG p) = ε(m)ε(p).
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779
Multiplication is now induced from the product on D(B+ ) by θ ⊗ Id and is thus given (for all p, p0 ∈ B+ and m, m0 ∈ B− ) by:
X
(m FG p)(m0 FG p0 ) =
(θ (m0(3) ))(p(1) )(θ (m0(1) ))(S −1 (p(3) ))[mm0(2) ] FG [p(2) p0 ].
(m0 ),(p)
Since θ is an anti-coalgebra map from B− to B∗ , we have (for all p, p0 ∈ B+ and m, m0 ∈ B− ):
X
(m FG p)(m0 FG p0 ) =
hp(1) |m0(1) iD hS −1 (p(3) )|m0(3) iD [mm0(2) ] FG [p(2) p0 ].
(3.10)
(m0 ),(p)
The resulting K-algebra structure on B− FG B+ is clearly unital, with unity element 1B− FG 1B+ . Finally, the bialgebra isomorphism θ −1 ⊗ Id : D(B+ ) → B− ⊗ B+ applied to (3.7) induces as follows a universal R-matrix on B− FG B+ : Choose any basis {ei }i∈I for B+ over K, and let {f i }i∈I be the dual basis for B− with respect to h|iD so that
hei |f j iD = δij for all i and j in I . Let iB+ : B+ → B− FG B+ ;
iB− : B− → B− FG B+
denote the canonical Hopf algebra monomorphisms; then R=
X
iB+ (ei ) ⊗ iB− (f i ) ∈ [B− FG B+ ] ⊗ [B− FG B+ ]
(3.11)
i∈I
furnishes the desired quasitriangular structure on B− FG B+ . We are now finally ready to prove: Theorem 3.3. Let (A, h|i) be a bialgebra with a braiding structure h|i. If Uh|i is finite-dimensional then it is a Hopf algebra quotient of a Drinfeld double and thus a quasitriangular Hopf algebra. Proof. Since U = Uh|i is finite-dimensional, it follows from Theorem 2.4 that it is a Hopf algebra. We wish to show that U is a Hopf-algebra quotient of the quasitriangular Hopf algebra B− FG B+ . Consider the K-linear map
Ψ : B− FG B+ → U ,
m ⊗ p = m FG p 7→ mp.
This is surjective by Proposition 2.3.3, and since ∆ is multiplicative it follows that Ψ is a coalgebra map. Thus, to prove Theorem 3.3, it suffices to show that Ψ is an algebra map, i.e. that, for all p, p0 in B+ and m, m0 in B− ,
Ψ ((m FG p)(m0 FG p0 )) = mpm0 p0 . Using (3.10), it thus remains to show that, for all p, p0 in B+ and m, m0 in B−
X
hp(1) |m01 iD hS −1 (p(3) )|m0(3) iD [mm0(2) p(2) p0 ] = mpm0 p0 .
(3.12)
(m0 ),(p)
And one final obvious reduction: to complete the proof of the present theorem, it thus suffices to show that, for all p ∈ B+ and m ∈ B− ,
X
hp(1) |m(1) iD hS −1 (p(3) )|m(3) iD m(2) p(2) = pm.
(3.13)
(p),(m)
Proof of (3.13). Since S + and S − are bijective (by Proposition 2.2), there exist a and b in A such that p = S + (λ+ (a)) = ρ − (a) and m = S − (λ− (b)) = ρ + (b). Then Proposition 2.3.1 and (3.3) imply: 1) hp|miD(−1) = hS + λ+ (a)|ρ + (b)i(− = hλ+ (a)|ρ + (b)iD = ha|bi. D
(3.14)
Similarly, Lemma 3.2 implies
hp|miD = hS + λ+ (a)|ρ + (b)iD = ha|bi(−1) . Since ρ
= S ◦ λ and ρ are anti-coalgebra maps (by Proposition 1.2), we then have: X LHS of (3.13) = h(S + )−1 (p(3) )|(m(3) )iD hp(1) |m(1) iD m(2) p(2) −
+
+
+
(p),(m)
=
1) hp(3) |(m(3) )i(− hp(1) |m(1) iD m(2) p(2) (by (3.3)) D
X (p),(m)
=
X
1) + hS + (λ+ (a(1) ))|ρ + (b(1) )i(− hS (λ+ (a(3) ))|ρ + (b(3) )iD ρ + (b(2) )ρ − (a(2) ) D
(a),(b)
(3.15)
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J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
which using (3.14) and (3.15) is seen to equal
X
ha(1) |b(1) iha(3) |b(3) i(−1) ρ + (b(2) )ρ − (a(2) ).
(a),(b)
Thus, to prove (3.13), it suffices to show this last expression equals the RHS of (3.13), i.e., equals pm = ρ − (a)ρ + (b)—but this equality is precisely the content of (2.2). This completes the proof of (3.13), and so the proof of Theorem 3.3. Remark 3.4. If we now again drop the hypothesis that U is finite-dimensional, two open questions suggest themselves: (I) If h|i is a braiding product on the bialgebra R, is Uh|i a Hopf algebra, and is it in some sense quasitriangular, or at least a quotient of some Drinfeld double? (II) In particular, if R is a Yang–Baxter operator, is this the case for the Faddeev–Reshetikhin–Takhtajan quantum enveloping algebra UFRT (R)? Affirmative answers in the very special case that R = RQ is the Sudbery Yang–Baxter operator associated with the Artin–Schelter–Tate multiparameter deformation of GL(N ), may be found in the [7,10]. 4. The special case of U (RQ ) and n = 2 In this section we characterize all finite-dimensional Hopf algebras of the form U = U (RQ ) where RQ is the Yang–Baxter operator supplied in [5,6] for the case n = 2. These Hopf algebras are quasitriangular by the previous section. We start with a brief summary; the reader is referred to [4] for the full details. n Let Q = {r 6= 1, pij }1≤i
(
pij
κ = r i j
qji = r /pji
i < j, i = j, i > j.
(4.1)
Let A(RQ ) be the bialgebra obtained by the FRT-construction [2] and described in [5]. Then A(RQ ) is generated as an j
j
j
k algebra by {Ti }1≤i,j≤n so that ∆(Ti ) = k Ti ⊗ Tk . 0 Let U = U (RQ ) ⊂ (A(RQ )) be the Hopf algebra obtained by the second FRT-construction and equivalently by Definition 1.3. These Hopf algebras were discussed in detail in [4,7,10]. U has generators of the form 1 {Kt , Lt , Eji , Fij }1≤t ≤n, 1≤i
P
Theorem 4.1. Let U be given as above, then: 1. U is finite-dimensional if and only if r and all pij are roots of unity. 2. If two such Hopf algebras have the same group of grouplike elements then they can be obtained from each other by twisting the comultiplication. The grouplike elements Kj , Lj ∈ U are the algebra homomorphisms defined on the generators Til of A(RQ ) by j
Kj (Til ) = δil κi
Lj (Til ) = δil (κji )−1 .
(4.2)
From Theorem 4.1.2 it follows that different (up to twisting) Hopf algebras of the form U (RQ ) are characterized via their grouplike elements. In what follows we describe all possible groups that may occur as groups of grouplike elements of such Hopf algebras where n = 2. Let T denote the set of all ordered triples (a, b, m), m > 0 of integers, such that gcd(a, b, m) = 1. We shall often (abusively) also regard a, b as defined modulo m. We wish to compute G(UQa,b,m ) where Qa,b,m is the associated multi-parameter given by p1,2 = e2π ia/m ,
q1,2 = e2π ib/m ,
r = p1,2 q1,2 .
Lemma 4.2. If (p, q, m) ε T, then G(UQa,b,m ) is naturally isomorphic to the subgroup Ga,b,m of Z/(m) × Z/(m) generated by the 4 elements
α1 = (a + b, a),
α2 = (b, a + b),
β1 = (a + b, b),
β2 = (a, a + b).
J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
781
Proof. The map
α1 7→ K1 ,
α2 7→ K2 ,
1 β1 → 7 L1 −1 , β2 7→ L− 2 is extended to an isomorphism Ga,b,m ∼ = G(UQa,b,m ).
Lemma 4.3. Let (p, q, m) ε T; then: (1) (a, b) ∈ Ga,b,m ⇒ (b, a) ∈ Ga,b,m (2) Ga,b,m = Gb,a,m (3) If (u, m) = 1 then Ga,b,m = Gup,uq,m and (a, b) ∈ Ga,b,m ⇒ (ua, ub) ∈ Ga,b,m . Proof. These facts are all immediate consequences of Lemma 4.2.
Observe that if m = m1 m2 · · · ms with the mi pairwise prime then there is a natural isomorphism (arising from the Chinese remainder theorem)
Z/(m) × Z/(m) ∼ =
s Y
(Z/(mi ) × Z/(mi ))
i =1
imply that Ga,b,m ∼ =
s Y
Ga,b,mi .
(4.3)
i=1
Proposition 4.4. Let (p, q, m) ε T, and let m = ps , s > 0, p prime. (1) If p 6= 3 then Ga,b,m = Z/(m) × Z/(m). (2) If p = 3 then (a) If a − b is not divisible by 3 then Ga,b,m = Z/(3s ) × Z/(3s ) (b) If a ≡ b (mod 3), then Ga,b,m coincides with G0 (m) := {(x, y) ∈ Z/(3s ) × Z/(3s ) : x ≡ −y (mod 3)} ∼ = Z/(3s ) × Z/(3s−1 ). Proof. Since p does not divide both a and b we may assume that p - b, thus p is invertible in Z/(m). Consider the 4 × 2 matrix
b a a + b a+b
a+b a + b . a b
By elementary operations over Z/(m) one can check that the rows do not generate Z/(m) × Z/(m) if and only if p | (b − a)(b + a) p | ((a + b)2 − ab) and
p | (a2 + 2ab).
First, p | (b2 − a2 ) implies that p | (a + b) or p | (a − b). Assume p | (a + b). Then p - ((a + b)2 − ab). Otherwise, if p | ((a + b)2 − ab) then p|ab and thus p|a or p|b implying that p | a and p | b, a contradiction. Assume now that p|(a − b) and p | ((a + b)2 − ab). Since (a + b)2 − ab = (a − b)2 + 3ab it follows that p | 3ab. If p 6= 3 then p | ab which yields a contradiction again. If p = 3 and a = b(mod 3) then a2 + 2ab = 3a2 (mod 3) and we have also 3 | (a2 + 2ab). It follows that this is the only case where Ga,b,m is not equal Z/(m) × Z/(m). We wish to compute G(a, b, 3s ) when a = b(mod 3). Let a − b = 3t u with u prime to 3. Choose v so uv = 1(mod m). We may assume (replacing if necessary a, b by v a, v b) that a − b = 3t . Hence Ga,b,m contains the following elements in Z/(m)× Z/(m): β2 −α2 = (a − b, 0) = (3t , 0), (0, 3t ), β2 = (a, a + b) = (b + 3t , 2b + 3t ), (b, 2b) Since b is invertible it contains also
(1, 2), (2, 1), 2(2, 1) − (1, 2) = (3, 0), (0, 3), (2, 1) − (1, 2) = (1, −1). Since G0 (m) is generated by (3, 0), (0, 3), (1, −1) considered as elements in Z/(3s ) × Z/(3s )) it follows that Ga,b,m contains G0 (m), while the converse inclusions is verified by noting that G0 (m) contains the four generators α1 , α2 , β1 , β2 of Ga,b,m . We claim that G0 (m) ∼ = Z/(3s ) × Z/(3s−1 ). The isomorphism is defined by
x+y (x, y) 7→ x, (mod 3s−1 ) , 3
(x, y) ∈ G0 (m).
This map is clearly monic. Given (¯x, y¯ ) ∈ Z/(3s ) × Z/(3s−1 ) take x = x¯ , y = (3y¯ − x)(mod 3s ) then (x, y) ∈ G0 (m) and its image is (¯x, y¯ ).
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By (4.3) and Proposition 4.4 we have: Theorem 4.5. Let (p, q, m) ε T then: (1) If either m or a − b is not divisible by 3 then Ga,b,m = Z/(m) × Z/(m). (2) If m = 3s and a − b is divisible by 3 then Ga,b,m coincides with G0 (m) := {(a, b) ∈ Z/(m) × Z/(m) : a ≡ −b(mod 3)} = Z/(m/3) × Z/(m).
Acknowledgements Both authors wish to thank the University of Illinois at Chicago, where we visited while this paper was written, for extending its hospitality. We are thankful to Loretta Allen, for helping us TeX this paper. We wish also to acknowledge our gratitude to Miriam Cohen, David Radford, Bhama Shrinivasan and L. Takhtajan for helpful discussions. References [1] R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of ‘‘quantum group’’ and ‘‘quantum Lie algebra’’, Comm. Algebra 19 (1991) 3295–3345. [2] L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantization of Lie groups and Lie Algebras, in: Algebraic Analysis, vol. I, Academic Press, Boston, MA, 1988, pp. 129–139. [3] L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97–110. [4] J. Towber, Multiparameter quantum forms of the enveloping algebra UglN related to the Faddeev–Reshetikhin–Takhtajan U (R) constructions, J. Knot Theory Ramifications 4 (2) (1995) 263–317. [5] M. Artin, W. Schelter, J. Tate, Quantum deformations of GLn , Comm. Pure Appl. Math. 44 (8–9) (1991) 879–895. [6] A. Sudbery, Consistent multiparameter quantisation of GL(n), J. Phys. A 23 (15) (1990) 697–704. [7] J. Towber, Some new finite-dimensional quasitriangular deformations of U (glN ), preprint. [8] N. Andruskiewitsch, S. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. (to appear), math.QA/0502157. [9] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1) (1990) 257–296. [10] S. Westreich, Hopf algebras of type An , twistings and the FRT-construction, Algebr. Represent. Theory 11 (1) (2008) 63–82. [11] M.E. Sweedler, Hopf Algebras, W.A. Benjamin, Inc., New York, 1969. [12] V.G. Drinfeld, Quantum groups, in: A.M. Gleason (Ed.), Proceedings of the International Congress of Mathematicians, Berkeley, Amer. Math. Soc., Providence, RI, 1986, pp. 798–820. [13] C. Kassel, Quantum Groups, in: Graduate Texts in Mathematics, vol. 155, Springer-Verlag, 1995.
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa
Hopf algebras constructed by the FRT-construction Jacob Towber a , Sara Westreich b,∗ a
Department of Mathematics, De Paul University, Chicago, IL, United States
b
Interdisciplinary Department of the Social Sciences, Bar-Ilan University, Ramat-Gan, Israel
article
info
Article history: Received 10 January 2008 Received in revised form 31 August 2008 Available online 13 November 2008 Communicated by C. Kassel This article is dedicated to the memory of Jack Towber, a good friend and colleague MSC: Primary: 16W30 17B37
a b s t r a c t Given any bialgebra A and a braiding product h|i on A, a bialgebra Uh|i was constructed in [R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of ‘‘quantum group’’ and ‘‘quantum Lie algebra’’, Comm. Algebra 19 (1991) 3295–3345], contained in the finite dual of A. This construction generalizes a (not very well known) construction of Fadeev, Reshetikhin and Takhtajan [L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97–110]. In the present paper it is proved that when Uh|i is finite-dimensional (even if A is not), then it is a quasitriangular Hopf algebra. © 2008 Elsevier B.V. All rights reserved.
0. Introduction One of the most fundamental constructions in the theory of Lie groups, is that which associates to a Lie or algebraic group G, its Lie algebra g. Expressing this in Hopf-algebraic terms, we have a construction
Φ : A(G) 7→ U (g) which inputs the bialgebra A(G) of representative functions on an affine algebraic group G (over the ground field K) and constructs from it the Hopf K-algebra U (g). It is remarkable that in the last 20 or so years, when so much attention has been devoted to generalizing results from Lie groups and Lie algebras to quantum groups, there seems to have been no proposed generalization of Φ to quantum groups. One purpose of this paper is to investigate such a proposed generalization. This paper completes a train of thought begun 15 years ago in [1], where a construction Φ , closely related to the remarkable constructions of Faddeev, Reshetikhin and Takhtajan [2,3] was given.1 Let us recall some facts about the constructions in [2,3] and the related LT-construction. Given an invertible Yang–Baxter operator R, Faddeev, Reshetikhin and Takhtajan associate to it two bialgebras, which we shall here denote by AFRT (R) and UFRT (R). It was proved in [1, Th. 5.1] that AFRT (R) has a braiding structure h|iR depending only on R. Unlike the AFRT (R)-construction which is quite known, the second one, UFRT (R), seems to have been largely forgotten. Going further it was proved in [1, Section 6] that for any bialgebra A with a braiding structure h|i, there exists a general Φ : (A, h|i) → Uh|i , where Uh|i is a bialgebra contained in the finite dual of (A, h|i). When (A, h|i) = AFRT (R) then the bialgebra Uh|i is naturally isomorphic to UFRT (R) and thus the LT-construction is a generalization of the UFRT (R)-construction. All the constructions mentioned above are of bialgebras. However, known specific cases of UFRT (R) are all Hopf algebras. Such are for example the multiparameter deformations of U (sln ) described in [4]. Thus, a natural question to ask is whether Uh|i is always a Hopf algebra. In this paper we give a partial positive answer to this question. We prove:
∗
Corresponding author. E-mail addresses: [email protected] (J. Towber), [email protected] (S. Westreich).
1 In this connection, L. Takhtajan was kind enough to supply some very valuable help, which guided us into the related literature. 0022-4049/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2008.09.012
J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
773
Theorem 2.4. Let (A, h|i) be a bialgebra with a braiding structure h|i. If Uh|i is finite-dimensional then it is a Hopf algebra. Let it be emphasized that Uh|i can be finite-dimensional even when (A, h|i) is not. The Uh|i -construction utilizes the braiding product in a simple but not quite straightforward way; it is NOT merely a question of taking the finite dual of A. As a result, neither the dimension nor quasitriangularity of Uh|i follow just by dualizing the structure of (A, h|i) (Example 1.4). We also prove: Theorem 3.3. Let (A, h|i) be a bialgebra with a braiding structure h|i. If Uh|i is finite-dimensional then it is a Hopf algebra quotient of a Drinfeld double and thus a quasitriangular Hopf algebra. An important class of Hopf algebras obtained by the Uh|i -construction is the family of Hopf algebras denoted by U (RQ ), where RQ is the Yang–Baxter operator associated with the multiparameter deformation of GLn supplied in [5,6]. These Hopf algebras were described and discussed in detail in [4,7]. In this paper we characterize all finite-dimensional Hopf algebras (thus necessarily quasitriangular) obtained by this construction for the case n = 2. Though finite-dimensional pointed Hopf algebras were fully characterized by [8], the characterization does not tell quasitriangular Hopf algebras from the others. Aside from the one-parameter deformations of Jimbo, Drinfeld, Lusztig et al., evaluated at roots of unity [9], (which are not always quasitriangular), there have it seems hitherto been no general examples of finite-dimensional quasitriangular Hopf algebras. Thus, it may be of interest that the Uh|i -construction produces infinitely many new such examples, non-isomorphic and unrelated by a cocycle twist [10]. 1. Preliminaries Throughout this paper (except where otherwise specified), all bialgebras considered will be taken over a fixed groundfield K. ⊗ will then mean ⊗K . Let A be a bialgebra over K. We denote by P (A) the set of all K-linear maps
h|i : A ⊗ A → K,
a ⊗ b 7→ ha|bi.
and note the natural associative product ∗A on P (A), defined for all h|i1 and h|i2 in P (A) and a, b ∈ A by
(h|i1 ∗A h|i2 )(a ⊗ b) =
X
ha(1) | b(1) i1 ha(2) | b(2) i2 .
(a),(b)
Note that the 2-sided identity element 1(∗A ) with respect to ∗A is given by 1(∗A )(a ⊗ b) := ε(ab) . We say that h|i ∈ P (A) is (convolution) invertible if it has a 2-sided inverse with respect to ∗A (which we shall denote by h|i(−1) ), i.e. a product which satisfies
h|i ∗A h|i(−1) = h|i(−1) ∗A h|i = 1(∗A ). If h|i is (convolution) invertible we shall also denote by h|i∗ the element of P (A) defined (for all a, b ∈ A) by
ha|bi∗ := hb|ai(−1) .
(1.1)
Definition 1.1. An element h|i in P (A) will be called a left braiding product for A, precisely when it satisfies the following four conditions, for all a, b and c in A: (B1) h|i P is (convolution) invertible. P (B2) (a),(b) ha(1) |b(1) ib(2) a(2) = (a),(b) ha(2) |b(2) ia(1) b(1) . P (B3) ha|bc i = h a(1) |biha(2) |c i. ( a ) P (B4) hab|c i = (c ) hb|c(1) iha|c(2) i. We define the braiding maps on A, to be the following four maps
λ+ : A → A0 , λ− : A → A0 ,
a 7→ ha|−i ∗
a 7→ ha|−i
ρ + : A → A0 , a 7→ h−|ai; ρ − : A → A0 , a → 7 h−|ai∗ .
Note that the definition of h|i∗ also implies immediately that
λ+ ∗ ρ − = 1L = ρ − ∗ λ+ and
λ− ∗ ρ + = 1L = ρ + ∗ λ− .
(1.2)
0
It is known that these four maps land in the finite dual A . Moreover, Proposition 1.2. λ± are algebra anti-homomorphisms and coalgebra homomorphisms, while ρ ± are algebra homomorphisms and coalgebra anti-homomorphisms. Retaining the above assumptions, we are ready to define the following bialgebra: Definition 1.3. Define Uh|i to be the subalgebra of A0 generated by the set of all λ+ (a), λ− (a), ρ + (a) and ρ − (a) with a ∈ A.
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To emphasize the fact that Uh|i is more then just taking the finite dual we give the following example. Example 1.4. Let M = {g i , i ≥ 0} be an abelian monoid and let A = kM be the monoid algebra. A has a bialgebra structure by letting ∆(g i ) = g i ⊗ g i and ε(g i ) = 1 for all i ≥ 0. Fix some q ∈ k. We put a braiding structure on A by defining
hg i |g j i := qi+j . It is straightforward that this is indeed a braiding structure on the bialgebra A and that h|i(−1) is given by:
hg i |g j i(−1) = q−(i+j) . Since the braiding is symmetric we have λ+ = µ+ and λ− = µ− . The construction of Uh|i implies now that λ+ (g i ) is the algebra character given by λ+ (g i )(g j ) = qi+j while λ− (g i ) is the algebra character given by λ− (g i )(g j ) = q−(i+j) . Thus, λ+ (g i ) and λ− (g i ) are group inverses of each other inside Uh|i , and so Uh|i , is the group algebra of the the group consisting of the algebra characters defined above. In particular Uh|i is a Hopf algebra. Observe also that if q is a root of unity then we obtain a finite set of characters. In this case Uh|i is a finite-dimensional Hopf algebra while A is an infinite-dimensional bialgebra. 2. Finite-dimensional Hopf algebras obtained as Uh|i The purpose of this section is the proof that finite-dimensional bialgebras of the form Uh|i are always Hopf algebras. In order to prove this, we shall define and utilize the ‘‘quantum Borel subalgebras’’ B+ and B− of Uh|i which will be shown to be Hopf algebras satisfying B+ B− = B− B+ when they are finite-dimensional. Proposition 2.1. Let Uh|i the sub-bialgebra of A0 generated by the images of the four maps λ+ , λ− , ρ + , ρ − as defined in Definition 1.3. Then
(Im ρ + )(Im ρ − ) = (Im ρ − )(Im ρ + ) and
(Im λ+ )(Im λ− ) = (Im λ− )(Im λ+ ) where (Im ρ + )(Im ρ − ) denotes the K-span of all products of the form ρ + (a)ρ − (a0 ), with a and a0 in A, and similarly for the other three products involved. Proof. It was proved [1, Prop. 2.8] that h|i∗ also satisfies the braiding axioms. Thus, for all a, b, c ∈ A,
(B∗ 2)
X
a(2) c(2) ha(1) |c(1) i(−1) =
X
c(1) a(1) ha(2) |c(2) i(−1)
(obtained from (B2) upon replacing h|i by h|i∗ , a by c and b by a). Also
(B∗ 4)
X X ha|bc i(−1) = ha(2) |bi(−1) ha(1) |c i(−1) .
Hence for all a, b, c ∈ A we have
(ρ + (a)ρ − (b))(c ) = = = = = =
X (ρ + (a))(c(1) )(ρ − (b))(c(2) ) X hc(1) |aihb|c(2) i(−1) X ε(a(2) b(2) )hc(1) |a(1) ihb(1) |c(2) i(−1) X hb(2) |a(2) i(−1) hb(3) |a(3) ihc(1) |a(1) ihb(1) |c(2) i(−1) X hc(1) |a(1) ihb(1) |a(2) c(2) i(−1) hb(2) |a(3) i by (B∗ 4) X hc(2) |a(2) ihb(1) |c(1) a(1) i(−1) hb(2) |a(3) i in (B2) replace a by c, b by a;
then apply hb(1) |−i(−1) to both sides X = hc(2) |a(2) ihb(2) |c(1) i(−1) hb(1) |a(1) i(−1) hb(3) |a(3) i X = hb(1) |a(1) i(−1) hb(3) |a(3) i(ρ − (b(2) )ρ + (a(2) ))(c ).
by (B∗ 4)
Hence
ρ + (a)ρ − (b) =
X
hb(1) |a(1) i(−1) hb(3) |a(3) iρ − (b(2) )ρ + (a(2) ).
(2.1)
Thus, (Im ρ + )(Im ρ − ) ⊆ (Im ρ − )(Im ρ + ). The converse inclusion follows similarly and we have:
ρ − (a)ρ + (b) =
X
ha(1) |b(1) iha(3) |b(3) i(−1) ρ + (b(2) )ρ − (a(2) )
(2.2)
J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
775
and so finally
(Im ρ + )(Im ρ − ) = (Im ρ − )(Im ρ + ). It is left to the reader to supply the precisely similar proof that
(Im λ+ )(Im λ− ) = (Im λ− )(Im λ+ ). We assume for the remainder of this paper (unless otherwise stated) that U := Uh|i is a finite-dimensional vector space over K. Set B+ := Im λ+
B− := Im λ− .
By Proposition 1.2, B+ and B− are K-sub-bialgebras of A0 , finite-dimensional as vector-spaces over K. Proposition 2.2. If U is finite-dimensional then B+ and B− are Hopf algebras with invertible antipodes. Proof. Let M denote the convolution algebra (HomK (B+ , B+ ), ∗). We shall first show that IdB+ is a left non-zero-divisor in M: Assume, P on the contrary, that IdB+ ∗ f = 0 for some non-zero f ∈ M. Thus, (x) x(1) f (x(2) ) = 0 for all x ∈ Im λ+ . Applying this with x = λ+ (a), we have (since λ+ is a coalgebra homomorphism, and using (1.2)) f (x) = f (λ+ (a)) =
X
X ε(a(1) )f (λ+ (a(2) )) = ρ − (a(1) )λ+ (a(2) )f (λ+ (a(3) )) X X = ρ − (a(1) ) (λ+ (a(2) )(1) f ((λ+ (a(2) )(2) ))) =
X
(a)
(λ+ (a(2) ))
ρ (a(1) )(I ∗ f )(λ+ (a(2) )) = 0. −
Hence f = 0, contradiction. Thus, IdB+ is a left non-zero divisor in M; a precisely symmetrical argument shows that it is a right non-zero divisor. Since M is finite-dimensional over K, it follows that IdB+ is a unit in M; i.e., that B+ is a Hopf algebra. A precisely similar argument shows that B− = Im λ− is a Hopf algebra. Finally, we apply to B+ and B− the theorem ([11, Cor.5.1.6]) that the antipode of a finite-dimensional Hopf algebra is invertible. Proposition 2.3. Let U be finite-dimensional over K, and let S + , S − denote the antipodes of B+ , B− , respectively. Then: 1. S + ◦ λ+ = ρ − and S − ◦ λ− = ρ + . 2. Im λ+ = Im ρ − and Im λ− = Im ρ + . Proof. 1. To prove the assertion that S + ◦ λ+ and ρ − are equal, we shall next show that each is a two-sided inverse to λ+ in the convolution algebra L := (HomK ((A, U ), ∗)). For all a in A, since λ+ is a coalgebra homomorphism,
((S + ◦ λ+ ) ∗ λ+ )(a) =
X
S + (λ+ (a(1) ))λ+ (a(2) )
(a)
=
X
S + ((λ+ (a))(1) )(λ+ (a)(2) )
(λ+ (a))
= ε(λ+ (a)) = ε(a) so that (using (1.2)),
(S + ◦ λ+ ) ∗ λ+ = 1L = ρ − ∗ λ+ . Similarly, we prove
λ+ ∗ (S + ◦ λ+ ) = 1L = λ+ ∗ ρ − . Hence S + ◦ λ+ and ρ − are equal. The proof that S − ◦ λ− = ρ + (since each is inverse to λ− ) is precisely similar. 2. From part 1 and the fact that S + : Im λ+ → Im λ+ is an invertible map, it follows immediately that Im λ+ = Im(S ◦ λ+ ) = Im ρ − . Similarly, Im ρ + = Im λ− . As a result we obtain Theorem 2.4. Let U = Uh|i the sub-bialgebra of A0 generated by the images of the four maps λ+ , λ− , ρ + , ρ − as defined in Definition 1.3. Assume U is finite-dimensional. Then U = B+ B− = B− B+ is a Hopf algebra.
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Proof. The fact that U = B+ B− = B− B+ follows from part 2 of Proposition 2.3 and from Proposition 2.1. To see that U is a Hopf algebra, it suffices as before to show that IdU is neither a left zero-divisor, nor a right zero-divisor in (hom(U , U ), ∗). The proofs of both parts are precisely similar; we shall here show that IdU is not a right zero-divisor in (hom(U , U ), ∗). Suppose f : U → U , f ∗ IdU = 0; then we are done if we deduce f = 0. Since U = B+ B− , it suffices to deduce f (ab) = 0 for all a ∈ B+ and b ∈ B− . Indeed, we then have: f (ab) =
X
f ((ab)(1) )ε((ab)(2) )
(ab)
=
X
f (a(1) b(1) )ε(a(2) )ε(b(2) )
(a)(b)
=
X
f (a(1) b(1) )a(2) ε(b(2) )S + (a(3) )
(a)(b)
=
X
f (a(1) b(1) )a(2) b(2) S − (b(3) )S + (a(3) )
(a)(b)
=
X
(f ∗ IdU )(a(1) b(1) )S − (b(2) )S + (a(2) ) = 0.
(a)(b)
3. Finite-dimensional Hopf algebras Uh|i are quasitriangular The purpose of this section, is the proof that finite-dimensional Hopf algebras of the form U = Uh|i are always quasitriangular. In order to prove this, we will show that the sub-Hopf algebras B+ and B− of U are in fact dual, compute the Drinfeld double constructed from these dual ‘‘quantum Borel subalgebras’’, and then show it has a Hopf algebra epimorphism onto U. As was discussed in the introduction, quasitriangularity of U does not follow just by dualizing the braiding structure of A, since U does not equal A0 . Yet the ideas along this section are very natural. Unfortunately, checking the details requires technical tedious proofs and we apologize the reader for that. Proposition 3.1. Assume U is finite-dimensional over K. Then: 1. A K-bilinear non-degenerate form h|iD : B+ ⊗ B− → K is well-defined by the specification:
hλ+ (a)|ρ + (b)iD := ha|bi for all a, b ∈ A. 2. For all p, p0 ∈ B+ and m, m0 ∈ B− ,
hp|mm0 iD =
X hp(1) |miD hp(2) |m0 iD ,
and hp|1B− iD = εB+ (p)
(3.1)
and h1B+ |miD = εB− (m).
(3.2)
(p)
hold, as do
hpp0 |miD =
X hp|m(2) iD hp0 |m(1) iD , (m)
3. Define θ0 to be the K-linear map
θ0 : B− → (B+ )∗ ,
m 7→ h−|miD
induced by h|iD ; then θ0 induces a bialgebra isomorphism ≈
θ : B− → ((B+ )0 )cop . 4. For all m ∈ B+ and p ∈ B− ,
hS + (p)|S − (m)iD = hp|miD . (−1)
5. The form h|iD is (convolution) invertible, with two-sided inverse h|iD
well-defined by
1) hp|mi(− = h(S + )−1 (p)|miD = hp|S − (m)iD D
+
−
for all p ∈ B and m ∈ B . Proof. 1. h|iD is well defined since if ρ + (b) = ρ + (b0 ) and λ+ (a) = λ+ (a0 ) then
ha|bi = λ+ (a)(b) = λ+ (a0 )(b) = ha0 |bi = ρ + (b)(a0 ) = ρ + (b0 )(a0 ) = ha0 |b0 i.
(3.3)
J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
777
We show now that h|iD is right non-degenerate, i.e. has right radical = 0. Indeed, if m = ρ + (b) is in the right radical of h|iD , then for all a in A, 0 = hλ+ (a)|ρ + (b)iD = ha|bi = ρ + (b)(a) whence ρ + (b) = m = 0. Similarly, we see that h|iD is left non-degenerate. 2. We will first verify (3.2), leaving to the reader the precisely similar verification of (3.1). In proving (3.2) we may assume p = λ+ (a),
p0 = λ+ (a0 ), m = ρ + (b) with a, a0 , b ∈ A
in which case the two assertions to be proved become
X
hλ+ (a)λ+ (a0 )|ρ + (b)iD =
hλ+ (a)|(ρ + (b))(2) iD hλ+ (a0 )|(ρ + (b))(1) iD
(ρ + (m))
and
hλ+ (1)|ρ + (b)iD = ε(b). This follows now from the braiding axioms (Definition 1.1) and since (by Proposition 1.2) λ+ is an algebra antihomomorphism while ρ + is a coalgebra anti-homomorphism. 3. Observe that θ being a coalgebra homomorphism is precisely the assertion (3.2) while θ being an algebra homomorphism follows from (3.1). Since B+ and B− are finite-dimensional, it follows that their dimensions are equal, and that the K-linear transformation θ0 is a K-isomorphism (whence θ is an isomorphism of K-bialgebras.). 4. We first recall the standard fact that, given B+ is a finite-dimensional Hopf algebra with antipode S + , it follows that S + is invertible, and that ((B+ )0 )cop is a Hopf algebra with antipode ((S + )∗ )−1 . It follows from the uniqueness of the antipode, ≈
that the bialgebra isomorphism θ : B− → ((B+ )0 )cop takes S − to ((S + )∗ )−1 , which is precisely the assertion to be proved. 5. That h(S + )−1 (p)|miD = hp|S − (m)iD follows from Proposition 3.1.4. Since both S + and S − are bijective, it follows that 1) is well-defined via (3.3). h|i(− D Thus, to complete the proof, two things must be verified: First, that for all p ∈ B+ and m ∈ B− ,
X
hp(1) |m(1) iD hp(2) |S − (m(2) )iD = ε(pm)
(p),(m)
— indeed, the left side, by (3.1), equals
X hp|m(1) S − (m(2) )iD = hp|ε(m)1B− i = ε(m)ε(p) = ε(pm). (m)
Second, that for all p ∈ B+ and m ∈ B− ,
X
hp(1) |S − m(1) iD hp(2) |m(2) iD = ε(pm)
(p),(m)
— the proof is similar.
Lemma 3.2. For all a and b in A,
hλ+ a|λ− biD = ha|bi(−1) . Proof. We need to show that for all a and b in A,
X
ha(1) |b(1) ihλ+ a(2) |λ− b(2) iD = ε(ab).
(a),(b)
Indeed, using (3.1) and the fact that λ+ is a coalgebra map the left side equals
X
hλ+ (a(1) )|ρ + (b(1) )iD hλ+ (a(2) )|λ− (b(2) )iD =
(a),(b)
X
hλ+ (a)|ρ + (b(1) )λ− (b(2) )iD .
(b)
Since, by (1.2), ρ + ∗ λ− = 1L in L = HomK (A, U ), it follows that
X
ρ + (b(1) )λ− (b(2) ) = ε(b)1L = ε(b)1B− ;
(b)
inserting this into the preceding expression, we obtain
hλ+ (a)|ε(b)1B− iD = ε(λ+ (a))ε(b) = ε(a)ε(b), which completes the proof.
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J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
Let us next recall the construction of the Drinfeld double D(B+ ) [12] for the finite-dimensional Hopf algebra B+ . We follow notations in [13]. This construction utilizes the following two module-coalgebra actions: (1) The left module-coalgebra action of B+ on X := ((B+ ))0 cop given (for all p, p0 in B+ and x in X ) by
``
(p
x)(p0 ) =
``
X
x(S −1 (p(2) )p0 p(1) ).
(2) The right module-coalgebra action p
´´
(3.4)
(p)
x=
X
´´
of X on B+ given (for all p in B+ and x in X ) by
x(S −1 (p(3) )p(1) )p(2) .
(3.5)
(p)
The Drinfeld double D(B+ ) = X FG B+ has X ⊗ B+ as its underlying vector space; its coalgebra structure is the usual tensor product of coalgebras. Multiplication is given by:
X
(x FG p)(x0 FG p0 ) =
[x(p1
(p),(x0 )
``
x02 )] FG [(p2
´´
x01 )p0 ]
for all p, p0 ∈ B+ , x, x0 ∈ X . Equivalently, the same multiplication on D(B+ ) is given by
X
(x FG p)(x0 FG p0 ) =
x03 (p1 )x01 (S −1 (p3 ))xx02 FG p2 p0 .
(3.6)
(p)(x0 )
The algebra thus defined on D(B+ ) is unital, with unity element 1D(B+ ) = 1X FG 1B+ . By [12], D(B+ ) is quasitriangular with its universal R-matrix R0 constructed as follows: Choose any basis {ei }i∈I for B+ over K, and let {ei }i∈I be the dual basis for (B+ )∗ ; let iB+ : B+ → D(B+ );
iX : X → D(B+ )
denote the canonical Hopf algebra monomorphisms; then R0 =
X
iB+ (ei ) ⊗ iX (ei ) ∈ D(B+ ) ⊗ D(B+ ).
(3.7)
i∈I
For the computations in the next subsection, it will be more convenient to replace the preceding construction of D(B+ ) by the following slight variation: Let us replace the pair (B+ , X ) = ((B+ )0 )cop in the preceding construction, by the pair (B+ , B− ), using the Hopf algebra isomorphism ≈
θ : B− → ((B+ )0 )cop of Proposition 3.1(4). It is readily verified that with this replacement, (3.4) and (3.5), respectively, go over into the two following module-coalgebra actions: (1a) The left module-coalgebra action of B+ on B− uniquely specified by requiring that, for all p, p0 ∈ B+ and all `` − m∈B ,
hp0 |p
``
miD =
X
θ (m)(S −1 (p(2) )p0 p(1) ) =
(p)
and, (2a) The right module-coalgebra action p
´´
m=
X
(3.8)
(p)
´´
θ (m)(S −1 (p(3) )p(1) )p(2)
(p)
X hS −1 (p(2) )p0 p(1) |miD
of B− on B+ uniquely specified by requiring that for all p ∈ B+ , m ∈ B− ,
X = h(S −1 (p(3) )p(1) )|miD p(2) .
(3.9)
(p)
In these terms, the preceding construction of the Drinfeld double D(B+ ), obviously goes over into the naturally isomorphic quasitriangular Hopf algebra B− FG B+ constructed as follows: B− FG B+ has B− ⊗ B+ as its underlying vector space and the coalgebra structure of the usual tensor product of coalgebras — i.e. for all p ∈ B+ and m ∈ B− we have
∆(m FG p) =
X
(m(1) FG p(1) ) ⊗ (m(2) FG p(2) )
(m),(p)
and
ε(m FG p) = ε(m)ε(p).
J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
779
Multiplication is now induced from the product on D(B+ ) by θ ⊗ Id and is thus given (for all p, p0 ∈ B+ and m, m0 ∈ B− ) by:
X
(m FG p)(m0 FG p0 ) =
(θ (m0(3) ))(p(1) )(θ (m0(1) ))(S −1 (p(3) ))[mm0(2) ] FG [p(2) p0 ].
(m0 ),(p)
Since θ is an anti-coalgebra map from B− to B∗ , we have (for all p, p0 ∈ B+ and m, m0 ∈ B− ):
X
(m FG p)(m0 FG p0 ) =
hp(1) |m0(1) iD hS −1 (p(3) )|m0(3) iD [mm0(2) ] FG [p(2) p0 ].
(3.10)
(m0 ),(p)
The resulting K-algebra structure on B− FG B+ is clearly unital, with unity element 1B− FG 1B+ . Finally, the bialgebra isomorphism θ −1 ⊗ Id : D(B+ ) → B− ⊗ B+ applied to (3.7) induces as follows a universal R-matrix on B− FG B+ : Choose any basis {ei }i∈I for B+ over K, and let {f i }i∈I be the dual basis for B− with respect to h|iD so that
hei |f j iD = δij for all i and j in I . Let iB+ : B+ → B− FG B+ ;
iB− : B− → B− FG B+
denote the canonical Hopf algebra monomorphisms; then R=
X
iB+ (ei ) ⊗ iB− (f i ) ∈ [B− FG B+ ] ⊗ [B− FG B+ ]
(3.11)
i∈I
furnishes the desired quasitriangular structure on B− FG B+ . We are now finally ready to prove: Theorem 3.3. Let (A, h|i) be a bialgebra with a braiding structure h|i. If Uh|i is finite-dimensional then it is a Hopf algebra quotient of a Drinfeld double and thus a quasitriangular Hopf algebra. Proof. Since U = Uh|i is finite-dimensional, it follows from Theorem 2.4 that it is a Hopf algebra. We wish to show that U is a Hopf-algebra quotient of the quasitriangular Hopf algebra B− FG B+ . Consider the K-linear map
Ψ : B− FG B+ → U ,
m ⊗ p = m FG p 7→ mp.
This is surjective by Proposition 2.3.3, and since ∆ is multiplicative it follows that Ψ is a coalgebra map. Thus, to prove Theorem 3.3, it suffices to show that Ψ is an algebra map, i.e. that, for all p, p0 in B+ and m, m0 in B− ,
Ψ ((m FG p)(m0 FG p0 )) = mpm0 p0 . Using (3.10), it thus remains to show that, for all p, p0 in B+ and m, m0 in B−
X
hp(1) |m01 iD hS −1 (p(3) )|m0(3) iD [mm0(2) p(2) p0 ] = mpm0 p0 .
(3.12)
(m0 ),(p)
And one final obvious reduction: to complete the proof of the present theorem, it thus suffices to show that, for all p ∈ B+ and m ∈ B− ,
X
hp(1) |m(1) iD hS −1 (p(3) )|m(3) iD m(2) p(2) = pm.
(3.13)
(p),(m)
Proof of (3.13). Since S + and S − are bijective (by Proposition 2.2), there exist a and b in A such that p = S + (λ+ (a)) = ρ − (a) and m = S − (λ− (b)) = ρ + (b). Then Proposition 2.3.1 and (3.3) imply: 1) hp|miD(−1) = hS + λ+ (a)|ρ + (b)i(− = hλ+ (a)|ρ + (b)iD = ha|bi. D
(3.14)
Similarly, Lemma 3.2 implies
hp|miD = hS + λ+ (a)|ρ + (b)iD = ha|bi(−1) . Since ρ
= S ◦ λ and ρ are anti-coalgebra maps (by Proposition 1.2), we then have: X LHS of (3.13) = h(S + )−1 (p(3) )|(m(3) )iD hp(1) |m(1) iD m(2) p(2) −
+
+
+
(p),(m)
=
1) hp(3) |(m(3) )i(− hp(1) |m(1) iD m(2) p(2) (by (3.3)) D
X (p),(m)
=
X
1) + hS + (λ+ (a(1) ))|ρ + (b(1) )i(− hS (λ+ (a(3) ))|ρ + (b(3) )iD ρ + (b(2) )ρ − (a(2) ) D
(a),(b)
(3.15)
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J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
which using (3.14) and (3.15) is seen to equal
X
ha(1) |b(1) iha(3) |b(3) i(−1) ρ + (b(2) )ρ − (a(2) ).
(a),(b)
Thus, to prove (3.13), it suffices to show this last expression equals the RHS of (3.13), i.e., equals pm = ρ − (a)ρ + (b)—but this equality is precisely the content of (2.2). This completes the proof of (3.13), and so the proof of Theorem 3.3. Remark 3.4. If we now again drop the hypothesis that U is finite-dimensional, two open questions suggest themselves: (I) If h|i is a braiding product on the bialgebra R, is Uh|i a Hopf algebra, and is it in some sense quasitriangular, or at least a quotient of some Drinfeld double? (II) In particular, if R is a Yang–Baxter operator, is this the case for the Faddeev–Reshetikhin–Takhtajan quantum enveloping algebra UFRT (R)? Affirmative answers in the very special case that R = RQ is the Sudbery Yang–Baxter operator associated with the Artin–Schelter–Tate multiparameter deformation of GL(N ), may be found in the [7,10]. 4. The special case of U (RQ ) and n = 2 In this section we characterize all finite-dimensional Hopf algebras of the form U = U (RQ ) where RQ is the Yang–Baxter operator supplied in [5,6] for the case n = 2. These Hopf algebras are quasitriangular by the previous section. We start with a brief summary; the reader is referred to [4] for the full details. n Let Q = {r 6= 1, pij }1≤i
(
pij
κ = r i j
qji = r /pji
i < j, i = j, i > j.
(4.1)
Let A(RQ ) be the bialgebra obtained by the FRT-construction [2] and described in [5]. Then A(RQ ) is generated as an j
j
j
k algebra by {Ti }1≤i,j≤n so that ∆(Ti ) = k Ti ⊗ Tk . 0 Let U = U (RQ ) ⊂ (A(RQ )) be the Hopf algebra obtained by the second FRT-construction and equivalently by Definition 1.3. These Hopf algebras were discussed in detail in [4,7,10]. U has generators of the form 1 {Kt , Lt , Eji , Fij }1≤t ≤n, 1≤i
P
Theorem 4.1. Let U be given as above, then: 1. U is finite-dimensional if and only if r and all pij are roots of unity. 2. If two such Hopf algebras have the same group of grouplike elements then they can be obtained from each other by twisting the comultiplication. The grouplike elements Kj , Lj ∈ U are the algebra homomorphisms defined on the generators Til of A(RQ ) by j
Kj (Til ) = δil κi
Lj (Til ) = δil (κji )−1 .
(4.2)
From Theorem 4.1.2 it follows that different (up to twisting) Hopf algebras of the form U (RQ ) are characterized via their grouplike elements. In what follows we describe all possible groups that may occur as groups of grouplike elements of such Hopf algebras where n = 2. Let T denote the set of all ordered triples (a, b, m), m > 0 of integers, such that gcd(a, b, m) = 1. We shall often (abusively) also regard a, b as defined modulo m. We wish to compute G(UQa,b,m ) where Qa,b,m is the associated multi-parameter given by p1,2 = e2π ia/m ,
q1,2 = e2π ib/m ,
r = p1,2 q1,2 .
Lemma 4.2. If (p, q, m) ε T, then G(UQa,b,m ) is naturally isomorphic to the subgroup Ga,b,m of Z/(m) × Z/(m) generated by the 4 elements
α1 = (a + b, a),
α2 = (b, a + b),
β1 = (a + b, b),
β2 = (a, a + b).
J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
781
Proof. The map
α1 7→ K1 ,
α2 7→ K2 ,
1 β1 → 7 L1 −1 , β2 7→ L− 2 is extended to an isomorphism Ga,b,m ∼ = G(UQa,b,m ).
Lemma 4.3. Let (p, q, m) ε T; then: (1) (a, b) ∈ Ga,b,m ⇒ (b, a) ∈ Ga,b,m (2) Ga,b,m = Gb,a,m (3) If (u, m) = 1 then Ga,b,m = Gup,uq,m and (a, b) ∈ Ga,b,m ⇒ (ua, ub) ∈ Ga,b,m . Proof. These facts are all immediate consequences of Lemma 4.2.
Observe that if m = m1 m2 · · · ms with the mi pairwise prime then there is a natural isomorphism (arising from the Chinese remainder theorem)
Z/(m) × Z/(m) ∼ =
s Y
(Z/(mi ) × Z/(mi ))
i =1
imply that Ga,b,m ∼ =
s Y
Ga,b,mi .
(4.3)
i=1
Proposition 4.4. Let (p, q, m) ε T, and let m = ps , s > 0, p prime. (1) If p 6= 3 then Ga,b,m = Z/(m) × Z/(m). (2) If p = 3 then (a) If a − b is not divisible by 3 then Ga,b,m = Z/(3s ) × Z/(3s ) (b) If a ≡ b (mod 3), then Ga,b,m coincides with G0 (m) := {(x, y) ∈ Z/(3s ) × Z/(3s ) : x ≡ −y (mod 3)} ∼ = Z/(3s ) × Z/(3s−1 ). Proof. Since p does not divide both a and b we may assume that p - b, thus p is invertible in Z/(m). Consider the 4 × 2 matrix
b a a + b a+b
a+b a + b . a b
By elementary operations over Z/(m) one can check that the rows do not generate Z/(m) × Z/(m) if and only if p | (b − a)(b + a) p | ((a + b)2 − ab) and
p | (a2 + 2ab).
First, p | (b2 − a2 ) implies that p | (a + b) or p | (a − b). Assume p | (a + b). Then p - ((a + b)2 − ab). Otherwise, if p | ((a + b)2 − ab) then p|ab and thus p|a or p|b implying that p | a and p | b, a contradiction. Assume now that p|(a − b) and p | ((a + b)2 − ab). Since (a + b)2 − ab = (a − b)2 + 3ab it follows that p | 3ab. If p 6= 3 then p | ab which yields a contradiction again. If p = 3 and a = b(mod 3) then a2 + 2ab = 3a2 (mod 3) and we have also 3 | (a2 + 2ab). It follows that this is the only case where Ga,b,m is not equal Z/(m) × Z/(m). We wish to compute G(a, b, 3s ) when a = b(mod 3). Let a − b = 3t u with u prime to 3. Choose v so uv = 1(mod m). We may assume (replacing if necessary a, b by v a, v b) that a − b = 3t . Hence Ga,b,m contains the following elements in Z/(m)× Z/(m): β2 −α2 = (a − b, 0) = (3t , 0), (0, 3t ), β2 = (a, a + b) = (b + 3t , 2b + 3t ), (b, 2b) Since b is invertible it contains also
(1, 2), (2, 1), 2(2, 1) − (1, 2) = (3, 0), (0, 3), (2, 1) − (1, 2) = (1, −1). Since G0 (m) is generated by (3, 0), (0, 3), (1, −1) considered as elements in Z/(3s ) × Z/(3s )) it follows that Ga,b,m contains G0 (m), while the converse inclusions is verified by noting that G0 (m) contains the four generators α1 , α2 , β1 , β2 of Ga,b,m . We claim that G0 (m) ∼ = Z/(3s ) × Z/(3s−1 ). The isomorphism is defined by
x+y (x, y) 7→ x, (mod 3s−1 ) , 3
(x, y) ∈ G0 (m).
This map is clearly monic. Given (¯x, y¯ ) ∈ Z/(3s ) × Z/(3s−1 ) take x = x¯ , y = (3y¯ − x)(mod 3s ) then (x, y) ∈ G0 (m) and its image is (¯x, y¯ ).
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J. Towber, S. Westreich / Journal of Pure and Applied Algebra 213 (2009) 772–782
By (4.3) and Proposition 4.4 we have: Theorem 4.5. Let (p, q, m) ε T then: (1) If either m or a − b is not divisible by 3 then Ga,b,m = Z/(m) × Z/(m). (2) If m = 3s and a − b is divisible by 3 then Ga,b,m coincides with G0 (m) := {(a, b) ∈ Z/(m) × Z/(m) : a ≡ −b(mod 3)} = Z/(m/3) × Z/(m).
Acknowledgements Both authors wish to thank the University of Illinois at Chicago, where we visited while this paper was written, for extending its hospitality. We are thankful to Loretta Allen, for helping us TeX this paper. We wish also to acknowledge our gratitude to Miriam Cohen, David Radford, Bhama Shrinivasan and L. Takhtajan for helpful discussions. References [1] R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of ‘‘quantum group’’ and ‘‘quantum Lie algebra’’, Comm. Algebra 19 (1991) 3295–3345. [2] L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantization of Lie groups and Lie Algebras, in: Algebraic Analysis, vol. I, Academic Press, Boston, MA, 1988, pp. 129–139. [3] L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97–110. [4] J. Towber, Multiparameter quantum forms of the enveloping algebra UglN related to the Faddeev–Reshetikhin–Takhtajan U (R) constructions, J. Knot Theory Ramifications 4 (2) (1995) 263–317. [5] M. Artin, W. Schelter, J. Tate, Quantum deformations of GLn , Comm. Pure Appl. Math. 44 (8–9) (1991) 879–895. [6] A. Sudbery, Consistent multiparameter quantisation of GL(n), J. Phys. A 23 (15) (1990) 697–704. [7] J. Towber, Some new finite-dimensional quasitriangular deformations of U (glN ), preprint. [8] N. Andruskiewitsch, S. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. (to appear), math.QA/0502157. [9] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1) (1990) 257–296. [10] S. Westreich, Hopf algebras of type An , twistings and the FRT-construction, Algebr. Represent. Theory 11 (1) (2008) 63–82. [11] M.E. Sweedler, Hopf Algebras, W.A. Benjamin, Inc., New York, 1969. [12] V.G. Drinfeld, Quantum groups, in: A.M. Gleason (Ed.), Proceedings of the International Congress of Mathematicians, Berkeley, Amer. Math. Soc., Providence, RI, 1986, pp. 798–820. [13] C. Kassel, Quantum Groups, in: Graduate Texts in Mathematics, vol. 155, Springer-Verlag, 1995.