Yetter--Drinfeld structures on Heisenberg doubles and chains
aa r X i v : . [ m a t h . QA ] O c t YETTER–DRINFELD STRUCTURES ON HEISENBERG DOUBLES ANDCHAINS
A.M. SEMIKHATOVA
BSTRACT . For a Hopf algebra B with bijective antipode, we show that the Heisenbergdouble H p B (cid:6)q is a braided commutative Yetter–Drinfeld module algebra over the Drin-feld double D p B q . The braiding structure allows generalizing H p B (cid:6)q (cid:21) B (cid:6) cop ' B to“Heisenberg n -tuples” and “chains” . . . ' B (cid:6) cop ' B ' B (cid:6) cop ' B ' . . . , all of which areYetter–Drinfeld D p B q -module algebras. For B a particular Taft Hopf algebra at a 2 p throot of unity, the construction is adapted to yield Yetter–Drinfeld module algebras overthe 2 p -dimensional quantum group U q s ℓ p q . I NTRODUCTION
We establish the properties of H p B (cid:6)q — the Heisenberg double of a (dual) Hopf alge-bra — relating it to two popular structures: Yetter–Drinfeld modules (over the Drinfelddouble D p B q ) and braiding. In fact, we construct new examples of Yetter–Drinfeld mod-ule algebras , some of which are in addition braided commutative.Heisenberg doubles [1, 2, 3, 4] have been the subject of some attention, notably inrelation to Hopf algebroid constructions [5, 6, 7] (the basic observation being that H p B (cid:6)q is a Hopf algebroid over B (cid:6) [5]) and also from various other standpoints [8, 9, 10, 11]. We show that they are a rich source of Yetter–Drinfeld D p B q -module algebras: H p B (cid:6)q isa Yetter–Drinfeld module algebra over the Drinfeld double D p B q ; it is, moreover, braidedcommutative. Reinterpreting the construction of H p B (cid:6)q in terms of the braiding in theYetter–Drinfeld category then allows generalizing Heisenberg doubles to “ n-tuples ,” or“Heisenberg chains” (cf. [18]), which are all Yetter–Drinfeld D p B q -module algebras.In Sec. 2, we establish that H p B (cid:6)q is a Yetter–Drinfeld D p B q -module algebra, and inSec. 3 that it is braided ( D p B q -) commutative [19]; there, B denotes a Hopf algebra withbijective antipode. In Sec. 4, where we construct Yetter–Drinfeld module algebras for aquantum s ℓ p q at an even root of unity [20, 21, 22, 12, 13], B becomes a particular TaftHopf algebra.For the left and right regular actions of a Hopf algebra B on B (cid:6) , we use the respectivenotation b á b (cid:16) x b , b y b and b à b (cid:16) x b , b y b , where b P B (cid:6) and b P B (and x , y is The “true,” underlying motivation (deriving from [12, 13, 14, 15, 16, 17]) of our interest in H p B (cid:6)q isentirely left out here. A slight mockery of the statistical-mechanics meaning of a “Heisenberg chain” may give way to agenuine, and deep, relation in the context of the previous footnote.
SEMIKHATOV the evaluation). The left and right regular actions of B (cid:6) on B are b á b (cid:16) x b , b b and b à b (cid:16) x b , b b . We assume the precedence ab à b (cid:16) p ab qà b , ab á a (cid:16) p ab qá a ,and so on. For a Hopf algebra H and a left H -comodule U , we write the coaction d : U Ñ H b U as d p u q (cid:16) u p(cid:1) q b u p q ; then x e , u p(cid:1) qy u p q (cid:16) u and u q b u q b u p q (cid:16) u p(cid:1) q b u p qp(cid:1) q b u p qp q . H p B (cid:6)q AS A Y ETTER –D RINFELD D p B q - MODULE ALGEBRA
The purpose of this section is to show that H p B (cid:6)q is a Yetter–Drinfeld D p B q -modulealgebra. The key ingredients are the D p B q -comodule algebra structure from [4], whichwe recall in , and the D p B q -module algebra structure from [17], which we recallin . The claim then follows by direct computation. H p B (cid:6)q . The Heisenberg double H p B (cid:6)q is the smash prod-uct B (cid:6) B with respect to the left regular action of B on B (cid:6) , which means that the com-position in H p B (cid:6)q is given by(2.1) p a a qp b b q (cid:16) a p a b q a b , a , b P B (cid:6) , a , b P B . We recall from [4] that H p B (cid:6)q can also be obtained by twisting the product onthe Drinfeld double D p B q (see Appendix A) as follows. Let h : D p B q b D p B q Ñ k be given by h p m b m , n b n q (cid:16) x m , y x e , n y x n , m y and let (cid:4) h : D p B q b D p B q Ñ D p B q be defined as M (cid:4) h N (cid:16) M N h p M , N , M , N P D p B q . A simple calculation shows that (cid:4) h coincides with the product in (2.1): p m b m q (cid:4) h p n b n q (cid:16) m p m n q b m n , m , n P B (cid:6) , m , n P B . From this construction of H p B (cid:6)q , it readily follows [4] that the coproduct of D p B q ,viewed as a map d : H p B (cid:6)q Ñ D p B q b H p B (cid:6)q b b ÞÑ p b b
1q b p b b , (2.2)makes H p B (cid:6)q into a left D p B q -comodule algebra (i.e., d is an algebra morphism). Simultaneously, H p B (cid:6)q is a D p B q -module algebra , i.e.,(2.3) M ⊲ p AC q (cid:16) p M ⊲ A qp M ⊲ C q for all M P D p B q and A , C P H p B (cid:6)q , under the D p B q action defined in [17]: ETTER–DRINFELD STRUCTURES ON HEISENBERG DOUBLES AND CHAINS 3 (2.4) p m b m q ⊲ p a a q (cid:16) m m a q S (cid:6)(cid:1) p m
2q m aS p m S (cid:6)(cid:1) p m , m b m P D p B q , a a P H p B (cid:6)q . Evidently, the right-hand side here factors into the actions of B (cid:6) cop and B : p m b m q ⊲ p a a q (cid:16) p m b q ⊲ (cid:0)p e b m q ⊲ p a a q(cid:8) , where p e b m q ⊲ p a a q (cid:16) p m a q m aS p m and p m b q ⊲ p a a q (cid:16) m a S (cid:6)(cid:1) p m
2q a à S (cid:6)(cid:1) p m . This allows verifying that (2.4) is indeed an action of D p B q independently of the argumentin [17]: it suffices to show that the actions of B (cid:6) cop and B taken in the “reverse” ordercombine in accordance with the Drinfeld double multiplication, i.e., to show that p e b m q ⊲ (cid:0)p m b q ⊲ p a a q(cid:8) (cid:16) (cid:0)p e b m qp m b q(cid:8) ⊲ p a a q (2.5) (cid:16) (cid:0)p m m à S (cid:1) p m m ⊲ p a a q . We do this in
B.1 .The D p B q -module algebra property was shown in [17], but a somewhat less bulkyproof can be given by considering the actions of m b e b m separately. The routinecalculations are in B.2 . H p B (cid:6)q is a ( left–left ) Yetter–Drinfeld D p B q -module algebra. By this we mean a left module algebra and a left comodule algebra with the Yetter–Drinfeld compatibility condition(2.6) p M ⊲ A qp(cid:1) q M M ⊲ A qp q (cid:16) M A p(cid:1) q b p M ⊲ A p qq for all M P D p B q and A P H p B (cid:6)q . (For Yetter–Drinfeld modules, see [23, 24, 25, 26, 27,19].) Condition (2.6) has to be shown for the D p B q action and coaction in (2.4) and (2.2). To simplify the calculation leading to (2.6), we again use that theaction of m b m P D p B q factors through the actions of m b e b m .First, for M (cid:16) e b m , we evaluate the left-hand side of (2.6) as (cid:0)p e b m ⊲ p a a q(cid:8)p(cid:1) qp e b m
2q b pp e b m ⊲ p a a qqp q(cid:16) (cid:0)p m p q á a q2 b p m p q aS p m p qqq1p e b m p qq(cid:8) b (cid:0)p m p q á a q1 m p q aS p m p qqq2(cid:8)(cid:16) (cid:0)p m p q á a
2q b p m p q aS p m p qqq1 m p q(cid:8) b (cid:0) a m p q aS p m p qqq2(cid:8)(cid:16) (cid:0)p m p q á a
2q b m p q a S p m p qq m p q(cid:8) b (cid:0) a m p q a S p m p qq(cid:8)(cid:16) (cid:0)p m p q á a
2q b m p q a a m p q a S p m p qq(cid:8) SEMIKHATOV but the right-hand side is given by (cid:0)p e b m a a e b m ⊲ p a a m p q á a S (cid:1) p m p qqq b m p q a m p q á a
1q m p q a S p m p qq(cid:8)(cid:16) (cid:0)p m p q á a
2q b m p q a m p q S (cid:1) p m p qqá a
1q m p q a S p m p qq(cid:8) (because a a m q (cid:16) p m á a
1q b a ), which is the same as the left-hand side.Second, for M (cid:16) m b
1, using the D p B q -identity(2.7) (cid:0) e b p a à S (cid:6)(cid:1) p m m q (cid:16) m S (cid:6)(cid:1) p m a q , we evaluate the left-hand side of (2.6) as (cid:0)p m q ⊲ p a a q(cid:8)p(cid:1) qp m q b (cid:0)p m q ⊲ p a a q(cid:8)p q(cid:16) (cid:1)(cid:0) m p q a S (cid:6)(cid:1) p m p qq(cid:8)2 b (cid:0) a à S (cid:6)(cid:1) p m p qq(cid:8)1p m p q b q(cid:9)b (cid:1)(cid:0) m p q a S (cid:6)(cid:1) p m p qq(cid:8)1 a à S (cid:6)(cid:1) p m p qq(cid:8)2(cid:9)(cid:16) (cid:1)(cid:0) m p q a S (cid:6)(cid:1) p m p qq b p a S (cid:6)(cid:1) p m p qqq(cid:8)p m p q b q(cid:9) b (cid:0) m p q a S (cid:6)(cid:1) p m p qq a (2.7) (cid:16) (cid:0) m p q a S (cid:6)(cid:1) p m p qq m p q b p S (cid:6)(cid:1) p m p qqá a m p q a S (cid:6)(cid:1) p m p qq a m p q a S (cid:6)(cid:1) p m p qqá a m p q a S (cid:6)(cid:1) p m p qq a m p q a a m p q a S (cid:6)(cid:1) p m p qq a S (cid:6)(cid:1) p m p qqq(cid:8)(cid:16) (cid:0)p m qp a a m q ⊲ p a a , which is the right-hand side. H p B (cid:6)q AS A BRAIDED COMMUTATIVE ALGEBRA
The category of Yetter–Drinfeld modules is well known to be braided, with the braiding c U , V : U b V Ñ V b U given by c U , V : u b v ÞÑ p u p(cid:1) q ⊲ v q b u p q . The inverse is c (cid:1) U , V : v b u ÞÑ u p q b S (cid:1) p u p(cid:1) qq ⊲ v . A left H -module and left H -comodule algebra X is said to be braidedcommutative [7] (or H -commutative [19, 28]) if(3.1) yx (cid:16) p y p(cid:1) q ⊲ x q y p q for all x , y P X . H p B (cid:6)q is braided commutative with respect to the braiding associatedwith the Yetter–Drinfeld module structure. ETTER–DRINFELD STRUCTURES ON HEISENBERG DOUBLES AND CHAINS 5 (1) The braided { H -commutativity property may be compared with “quantum com-mutativity” [29]. We recall that for a quasitriangular Hopf algebra H , its modulealgebra X is called quantum commutative if(3.2) yx (cid:16) p R p q ⊲ x qp R p q ⊲ y q (cid:17) (cid:4) p R ⊲ p x b y qq , x , y P X , where R (cid:16) R p q b R p q P H b H is the universal R -matrix (and the dot denotes themultiplication in X ). A minor source of confusion is that this useful property(see, e.g., [29, 5, 6]) is sometimes also referred to as H -commutativity [29]. For aYetter–Drinfeld module algebra X over a quasitriangular H , the properties in (3.1)and (3.2) are different (for example, a “quantum commutative” analogue of The-orem does not hold for H p B (cid:6)q ). We therefore consistently speak of (3.1) asof “braided commutativity” (this term is also used in [30] in related contexts, al-though in more than one).(2) The two properties, Eqs. (3.1) and (3.2), are “morally” similar, however. To seethis, recall that a Yetter–Drinfeld H -module is the same thing as a D p H q -module,the D p H q action on a left–left Yetter–Drinfeld module X being defined as p p b h q ⊲ x (cid:16) x S (cid:6)(cid:1) p p q , p h ⊲ x qp(cid:1) qy p h ⊲ x qp q , p P H (cid:6) , h P H , x P X . Let then R (cid:16) ¸ A p e b e A q b p e A b q P D p H q b D p H q be the universal R -matrix for the double. It follows that (cid:4) p R (cid:1) ⊲ p x b y qq (cid:16) (cid:0)p e b S p e A qq ⊲ x (cid:8)(cid:0)p e A b q ⊲ y (cid:8)(cid:16) x e A , S (cid:1) p y p(cid:1) qqy (cid:0) S p e A q ⊲ x (cid:8) y p q (cid:16) p y p(cid:1) q ⊲ x q y p q for all x , y P X , and therefore the braided commutativity property can be equiva-lently stated in the form yx (cid:16) (cid:4) p R (cid:1) ⊲ p x b y qq similar to Eq. (3.2) (the occurrence of R (cid:1) instead of R may be attributed to ourchoice of left–left Yetter–Drinfeld modules). We evaluate the right-hand side of (3.1) for X (cid:16) H p B (cid:6)q as (cid:0)p b b qp(cid:1) q ⊲ p a a q(cid:8)p b b qp q(cid:16) pp b b ⊲ p a a qqp b b b p qp b p q á a q S (cid:6)(cid:1) p b p qq b p q aS p b p qqà S (cid:6)(cid:1) p b p qq(cid:8)(cid:9)p b p q b p qq(cid:16) (cid:0) b p qp b p q á a q S (cid:6)(cid:1) p b p qq(cid:0) b p q aS p b p qqà S (cid:6)(cid:1) p b p qq(cid:8)1 á b p q(cid:8) b p q aS p b p qqà S (cid:6)(cid:1) p b p qq(cid:8)2 b p q SEMIKHATOV (cid:16) (cid:0) b p qp b p q á a q S (cid:6)(cid:1) p b p qq(cid:0)p b p q aS p b p qqq1 à S (cid:6)(cid:1) p b p qq(cid:8)á b p q(cid:8) b p q aS p b p qqq2 b p q X (cid:16) b p qp b p q á a q S (cid:6)(cid:1) p b p qq b p qx S (cid:6)(cid:1) p b p qq b p q , p b p q aS p b p qqq1y b p q aS p b p qqq2 b p q(cid:16) b p qp b p q á a q S (cid:6)(cid:1) p b p qq b p q b p q aS p b p qq b p q(cid:16) b p b p q á a q b p q a (cid:16) p b b qp a a q , where in X (cid:16) we used that p a à a qá b (cid:16) b ab , a y . We now somewhat generalize the observation leading to .We first recall the definition of a braided product, then see when braided commutativityis hereditary under taking a braided product, and verify the corresponding condition for B (cid:6) cop and B ; their braided product, which is therefore a braided commutative Yetter–Drin-feld module algebra, actually coincides with H p B (cid:6)q . If H is a Hopf algebra and X and Y two (left–left) Yetter–Drinfeld module alge-bras, their braided product X ' Y is defined as the tensor product with the composition(3.3) p x ' y qp v ' u q (cid:16) x p y p(cid:1) q ⊲ v q ' y p q u , x , v P X , y , u P Y . This is a Yetter–Drinfeld module algebra. (Indeed, the associativity of (3.3) is ensuredby Y being a comodule algebra and X being a module algebra. As a tensor product ofYetter–Drinfeld modules, X ' Y is a Yetter–Drinfeld module under the diagonal action(via iterated coproduct) and codiagonal coaction of H . By the Yetter–Drinfeld axiom for Y and the module algebra properties of X and Y , moreover, X ' Y is a module algebra; theroutine verification is given in B.3 for completeness. That X ' Y is a comodule algebrafollows from the comodule algebra properties of X and Y and the Yetter–Drinfeld axiomfor Y ; this is also recalled in B.3 .) We say that two Yetter–Drinfeld modules X and Y are braided symmetric if c Y , X (cid:16) c (cid:1) X , Y (note that both sides here are maps Y b X Ñ X b Y ), that is, p y p(cid:1) q ⊲ x q b y p q (cid:16) x p q b (cid:0) S (cid:1) p x p(cid:1) q q ⊲ y (cid:8) . Let X and Y be braided symmetric Yetter–Drinfeld modules, each of whichis a braided commutative Yetter–Drinfeld module algebra. Then their braided productX ' Y is also braided commutative.
We must show that(3.4) (cid:0)p x ' y qp(cid:1) q ⊲ p v ' u q(cid:8)p x ' y qp q (cid:16) p x ' y qp v ' u q ETTER–DRINFELD STRUCTURES ON HEISENBERG DOUBLES AND CHAINS 7 for all x , v P X and y , u P Y . For this, we write the condition c X , Y (cid:16) c (cid:1) Y , X as p x p(cid:1) q ⊲ y q b x p q (cid:16) y p q b (cid:0) S (cid:1) p y p(cid:1) qq ⊲ x (cid:8) and use it to establish an auxiliary identity,(3.5) (cid:0)p x p(cid:1) q ⊲ y qp(cid:1) q ⊲ x p q (cid:8) b p x p(cid:1) q ⊲ y qp q (cid:16) (cid:1) y p qp(cid:1) q ⊲ (cid:0) S (cid:1) p y p(cid:1) qq ⊲ x (cid:8)(cid:9) b y p qp q(cid:16) (cid:0) y q S (cid:1) p y qq ⊲ x (cid:8) b y p q (cid:16) x b y . The left-hand side of (3.4) is (cid:0)p x ' y qp(cid:1) q ⊲ p v ' u q(cid:8)p x ' y qp q(cid:16) (cid:0) x p(cid:1) q y p(cid:1) q ⊲ p v ' u q(cid:8)p x p q ' y p q q(cid:16) (cid:0)p x q y q ⊲ v q ' p x q y q ⊲ u q(cid:8)p x p q ' y p qq(cid:16) p x q y q ⊲ v q(cid:0)p x q y q ⊲ u qp(cid:1) q ⊲ x p q (cid:8) ' p x q y q ⊲ u qp q y p q(cid:16) p x p(cid:1) q y q ⊲ v q(cid:0)p x p qp(cid:1) q ⊲ p y q ⊲ u qqp(cid:1) q ⊲ x p q p q(cid:8) ' p x p q p(cid:1) q ⊲ p y q ⊲ u qqp q y p q(cid:16) p x p(cid:1) q y q ⊲ v q x p q ' p y q ⊲ u q y p q , just because of (3.5) in the last line. But the right-hand side of (3.4) is p x ' y qp v ' u q (cid:16) x p y p(cid:1) q ⊲ v q ' y p q u (cid:16) p x p(cid:1) q y p(cid:1) q ⊲ v q x p q ' p y p q p(cid:1) q ⊲ u q y p qp q because X and Y are both braided commutative. The two expressions coincide. Because the braided symmetry condition is symmetric with respect tothe two modules, we also have the braided symmetric Yetter–Drinfeld module algebra Y ' X , with the product p y ' x qp u ' v q (cid:16) y p x p(cid:1) q ⊲ u q ' x p q v . In addition to the multiplication inside Y and inside X , this formula expresses the relations xu (cid:16) p x p(cid:1) q ⊲ u q x p q satisfied in Y ' X by x P X and u P Y . Because c X , Y (cid:16) c (cid:1) Y , X , these arethe same relations ux (cid:16) p u p(cid:1) q ⊲ x q u p q that we have in X ' Y . Somewhat more formally,the isomorphism f : X ' Y Ñ Y ' X is given by f : x ' y ÞÑ p x p(cid:1) q ⊲ y q ' x p q . This is a module map by virtue of the Yetter–Drinfeld condition, and it is immediate to verify that d p f p x ' y qq (cid:16) p id b f qp d p x ' y qq .That f is an algebra map follows by calculating f p x ' y q f p v ' u q (cid:16) pp x p(cid:1) q ⊲ y q ' x p q qpp v p(cid:1) q ⊲ u q ' v p qq(cid:16) p x p(cid:1) q ⊲ y qp x p qp(cid:1) q v p(cid:1) q ⊲ u q ' x p q p q v p q(cid:16) p x q ⊲ y qp x q v p(cid:1) q ⊲ u q ' x p q v p q SEMIKHATOV (cid:16) x p(cid:1) q ⊲ (cid:0) y p v p(cid:1) q ⊲ u q(cid:8) ' x p q v p q and f pp x ' y qp v ' u qq (cid:16) f (cid:0) x p y p(cid:1) q ⊲ v q ' y p q u (cid:8)(cid:16) p x p(cid:1) qp y p(cid:1) q ⊲ v qp(cid:1) q ⊲ p y p q u qq ' x p q p y p(cid:1) q ⊲ v qp q X (cid:16) x p(cid:1) q ⊲ p y p q u qp q ' x p q(cid:0) S (cid:1) p y p qp(cid:1) q u p(cid:1) qq ⊲ p y p(cid:1) q ⊲ v q(cid:8)(cid:16) x p(cid:1) q ⊲ p y p q u p qq ' x p q(cid:0) S (cid:1) p y q u p(cid:1) qq y q ⊲ v (cid:8)(cid:16) x p(cid:1) q ⊲ p yu p qq ' x p q (cid:0) S (cid:1) p u p(cid:1) qq ⊲ v (cid:8) X (cid:16) x p(cid:1) q ⊲ (cid:0) y p v p(cid:1) q ⊲ u q(cid:8) ' x p q v p q , where the braided symmetry condition was used in each of the X (cid:16) equalities. Further examples of Yetter–Drinfeld module alge-bras are provided by multiple braided products X ' . . . ' X N (of Yetter–Drinfeld H -module algebras X i ), defined as the corresponding tensor products with the diagonal ac-tion and codiagonal coaction of H and with the relations(3.6) x r i s ' y r j s (cid:16) p x p(cid:1) q ⊲ y qr j s ' x p qr i s for all i ¡ j , where z r i s P X i . (The inverse relations are x r i s ' y r j s (cid:16) y p qr j s ' p S (cid:1) p y p(cid:1) qq ⊲ x qr i s , i j .)It readily follows from the Yetter–Drinfeld module algebra axioms for each of the X i that X ' . . . ' X N is an associative algebra and, in fact, a Yetter–Drinfeld H -module algebra.More specifically, let X and Y be braided symmetric Yetter–Drinfeld H -module alge-bras, as in , and consider the “alternating” products(3.7) X ' Y ' X ' Y ' . . . , with an arbitrary number of factors (or a similar product with the leftmost Y , or actuallytheir inductive limits). We let X r i s denote the i th copy of X , and similarly with Y r j s . Forarbitrary x r i s P X r i s and y r j s P Y r j s , we then have the relations(3.8) x r i (cid:0) s ' y r j s (cid:16) p x p(cid:1) q ⊲ y qr j s ' x p qr i (cid:0) s , which by (3.6) are satisfied for all i > j ; but by the braided symmetry condition, re-lations (3.8) hold for all i and j (replicating the relations between elements of X andelements of Y in X ' Y ). In (3.7), also,(3.9) x r i (cid:0) s ' v r j (cid:0) s (cid:16) p x p(cid:1) q ⊲ v qr j (cid:0) s ' x p q r i (cid:0) s , x , v P X , y r i s ' u r j s (cid:16) p y p(cid:1) q ⊲ u qr j s ' y p qr i s , y , u P Y , i ¡ j . (These formulas also hold for i (cid:16) j if X and Y are braided commutative.) ETTER–DRINFELD STRUCTURES ON HEISENBERG DOUBLES AND CHAINS 9 H p B (cid:6)q as a braided product. Theorem can be reinterpreted by saying that theHeisenberg double of B (cid:6) is a braided product, H p B (cid:6)q (cid:16) B (cid:6) cop ' B , with the braiding b b b ÞÑ p b p(cid:1) q ⊲ b q b b p q , b P B , b P B (cid:6) , where we abbreviate the action of B in to m ⊲ p b b q (cid:16) p m b q m bS p m , m P B , and further use ⊲ for the restriction to B (cid:6) , viz., m ⊲ b (cid:16) m á b . It is also understood that B (cid:6) cop and B are viewed as left D p B q -comodule algebras via d : b ÞÑ p b q b b , d : b ÞÑ p e b b
1q b b and left D p B q -module algebras via p m b m q ⊲ b (cid:16) m m á b q S (cid:6)(cid:1) p m , p m b m q ⊲ b (cid:16) p m bS p m S (cid:6)(cid:1) p m q . Both B (cid:6) cop and B are then Yetter–Drinfeld D p B q -module algebras, and each is braidedcommutative.Moreover, B (cid:6) cop and B are braided symmetric because c B (cid:6) cop , B (cid:16) c (cid:1) B , B (cid:6) cop , i.e., p b p(cid:1) q ⊲ b q b b p q (cid:16) b p q b p S (cid:1) D p b p(cid:1) qq ⊲ b q . The antipode here is that of D p B q , and therefore the right-hand side evaluates as b S (cid:6)p b ⊲ b q (cid:16) b b à S (cid:6)(cid:1) p S (cid:6)p b b b à b , which is immediately seen tocoincide with the left-hand side.Thus, the result that H p B (cid:6)q (cid:16) B (cid:6) cop ' B is a braided commutative Yetter–Drinfeld mod-ule algebra now follows from . (This offers a nice alternative to an unilluminatingbrute-force proof.) n -tuples { chains. It follows from that H p B (cid:6)q is also isomorphicto the braided commutative Yetter–Drinfeld module algebra B ' B (cid:6) cop , with the product p a ' a qp b ' b q (cid:16) a p b à S (cid:6)(cid:1) p a a b . We next consider “Heisenberg n -tuples { chains”— the alternating products H n (cid:16) B (cid:6) cop ' B ' B (cid:6) cop ' B ' . . . ' B , H n (cid:0) (cid:16) B (cid:6) cop ' B ' B (cid:6) cop ' B ' . . . ' B ' B (cid:6) cop . As we saw in , the relations are then given by b r i s b r j (cid:0) s (cid:16) p b b qr j (cid:0) s b i s for all i and j (where b P B and b P B (cid:6) , B (cid:6) cop Ñ B (cid:6) cop r j (cid:0) s and B Ñ B r i s are the morphisms onto the respective factors, and we omit ' for brevity) and a r i (cid:0) s b r j (cid:0) s (cid:16) p a b S (cid:6)(cid:1) p a j (cid:0) s a i (cid:0) s , a , b P B (cid:6) cop , a r i s b r j s (cid:16) p a bS p a j s a i s , a , b P B , i > j (Relations inverse to the last two are b r i (cid:0) s a r j (cid:0) s (cid:16) a j (cid:0) sp S (cid:6)p a ba i (cid:0) s and b r i s a r j s (cid:16) a j s p S (cid:1) p a ba i s for i j .)The chains with the leftmost B factor are defined entirely similarly. The obvious em-beddings allow defining (one-sided or two-sided) inductive limits of alternating chains.All the chains are Yetter–Drinfeld module algebras, but those with > Y ETTER –D RINFELD MODULE ALGEBRAS FOR U q s ℓ p q In this section, we construct Yetter–Drinfeld module algebras for U q s ℓ p q at an evenroot of unity q (cid:16) e i p p for an integer p > U q s ℓ p q is the 2 p -dimensional quantum group with generators E , K , and F and the relations KEK (cid:1) (cid:16) q E , KFK (cid:1) (cid:16) q (cid:1) F , r E , F s (cid:16) K (cid:1) K (cid:1) q (cid:1) q (cid:1) , E p (cid:16) F p (cid:16) , K p (cid:16) D p E q (cid:16) E b K (cid:0) b E , D p K q (cid:16) K b K , D p F q (cid:16) F b (cid:0) K (cid:1) b F , e p E q (cid:16) e p F q (cid:16) e p K q (cid:16) S p E q (cid:16) (cid:1) EK (cid:1) , S p K q (cid:16) K (cid:1) , S p F q (cid:16) (cid:1) KF . In [12, 13], U q s ℓ p q was arrived at as a subquotient of the Drinfeld double of a TaftHopf algebra (a trick also used, e.g., in [35] for a closely related quantum group). Itturns out that not only D p B q but also the pair p D p B q , H p B (cid:6)qq can be “truncated” to apair p U q s ℓ p q , H q s ℓ p qq of 2 p -dimensional algebras, with H q s ℓ p q being a braided com-mutative Yetter–Drinfeld U q s ℓ p q -module algebra. This is worked out in what follows. H q s ℓ p q — a “Heisenberg counterpart” of U q s ℓ p q — appears in . D p B q and H p B (cid:6)q for the Taft Hopf algebra B .4.1.1. The Taft Hopf algebra B . Let B (cid:16) Span p E m k n q , m p (cid:1) , n p (cid:1) , be the 4 p -dimensional Hopf algebra generated by E and k with the relations In an “applied” context (see, e.g., [14, 31, 32]), this quantum group first appeared in [12, 13]; sub-sequently, it gradually transpired (with the final picture having emerged from [33]) that that was just acontinuation of a series of previous (re)discoveries [20, 21, 22] (also see [34]). The ribbon and (somewhatstretching the definition) factorizable structures of U q s ℓ p q were worked out in [12]. ETTER–DRINFELD STRUCTURES ON HEISENBERG DOUBLES AND CHAINS 11 kE (cid:16) q Ek , E p (cid:16) , k p (cid:16) , (4.1)and with the comultiplication, counit, and antipode given by D p E q (cid:16) b E (cid:0) E b k , D p k q (cid:16) k b k , e p E q (cid:16) , e p k q (cid:16) , S p E q (cid:16) (cid:1) Ek (cid:1) , S p k q (cid:16) k (cid:1) . (4.2)Dual elements F , κ P B (cid:6) are introduced as x F , E m k n y (cid:16) d m , q (cid:1) n q (cid:1) q (cid:1) , x κ , E m k n y (cid:16) d m , q (cid:1) n { . Then [12] B (cid:6) (cid:16) Span p F a κ b q , a p (cid:1) , b p (cid:1) . D p B q . Straightforward calculation shows [12] that the Drin-feld double D p B q is the Hopf algebra generated by E , F , k , and κ with the relationsgiven byi) relations (4.1) in B ,ii) the relations κ F (cid:16) q F κ , F p (cid:16) , κ p (cid:16) B (cid:6) , andiii) the cross-relations k κ (cid:16) κ k , kFk (cid:1) (cid:16) q (cid:1) F , κ E κ (cid:1) (cid:16) q (cid:1) E , r E , F s (cid:16) k (cid:1) κ q (cid:1) q (cid:1) . The Hopf-algebra structure p D D , e D , S D q of D p B q is given by (4.2) and D D p F q (cid:16) κ b F (cid:0) F b , D D p κ q (cid:16) κ b κ , e D p F q (cid:16) , e D p κ q (cid:16) , S D p F q (cid:16) (cid:1) κ (cid:1) F , S D p κ q (cid:16) κ (cid:1) . H p B (cid:6)q . For the above B , H p B (cid:6)q is spanned by(4.3) F a κ b E c k d , a , c (cid:16) , . . . , p (cid:1) , b , d P Z {p p Z q , where κ p (cid:16) k p (cid:16) F p (cid:16)
0, and E p (cid:16)
0. Then the product in (2.1) becomes [17](4.4) p F r κ s E m k n qp F a κ b E c k d q(cid:16) ¸ u > q (cid:1) u p u (cid:1) q(cid:18) mu (cid:26)(cid:18) au (cid:26) r u s ! p q (cid:1) q (cid:1) q u q (cid:1) bn (cid:0) cn (cid:0) a p s (cid:1) n q(cid:0) u p c (cid:1) a (cid:1) b (cid:0) m (cid:1) s q(cid:2) F a (cid:0) r (cid:1) u κ b (cid:0) s E m (cid:0) c (cid:1) u k n (cid:0) d (cid:0) u . A convenient basis in H p B (cid:6)q can be chosen as p κ , z , l , Bq , where κ is understood as κ z (cid:16) (cid:1)p q (cid:1) q (cid:1) q e Ek (cid:1) , l (cid:16) κ k , B (cid:16) p q (cid:1) q (cid:1) q F . The relations in H p B (cid:6)q then become κ z (cid:16) q (cid:1) z κ , κ l (cid:16) q l κ , κ B (cid:16) q B κ , κ p (cid:16)
1, and l p (cid:16) , z p (cid:16) , B p (cid:16) , l z (cid:16) z l , l B (cid:16) B l , B z (cid:16) p q (cid:1) q (cid:1) q (cid:0) q (cid:1) z B . p U q s ℓ p q , H q s ℓ p qq pair.4.2.1. From D p B q to U q s ℓ p q . The “truncation” whereby D p B q yields U q s ℓ p q [12] con-sists of two steps: first, taking the quotient D p B q (cid:16) D p B q{p κ k (cid:1) q (4.5)by the Hopf ideal generated by the central element κ b k (cid:1) e b U q s ℓ p q as the subalgebra in D p B q spanned by F ℓ E m k n (tensor product omitted) with ℓ, m (cid:16) , . . . , p (cid:1) n (cid:16) , . . . , p (cid:1)
1. It then follows that U q s ℓ p q is a Hopf algebra —the one described at the beginning of this section, where K (cid:16) k . H p B (cid:6)q to H q s ℓ p q . In H p B (cid:6)q , dually, we take a subalgebra and then a quo-tient [17]. In the basis chosen above, the subalgebra (which is also a U q s ℓ p q submodule)is the one generated by z , B , and l . Its quotient by l p (cid:16) p -dimensionalalgebra H q s ℓ p q — the “Heisenberg counterpart” of U q s ℓ p q [17].As an associative algebra, H q s ℓ p q (cid:16) C q r z , Bs b p C r l s{p l p (cid:1) qq , with the p -dimensional algebra C q r z , Bs (cid:16) C r z , Bs{p z p , B p , B z (cid:1) p q (cid:1) q (cid:1) q (cid:1) q (cid:1) z Bq . The U q s ℓ p q action on H q s ℓ p q follows from (2.4) as E ⊲ l n (cid:16) q (cid:1) n r n s l n z , k ⊲ l n (cid:16) q (cid:1) n l , F ⊲ l n (cid:16) (cid:1) q n r n s l n B , E ⊲ z n (cid:16) (cid:1) q n r n s z n (cid:0) , k ⊲ z n (cid:16) q n z n , F ⊲ z n (cid:16) r n s q (cid:1) n z n (cid:1) , E ⊲ B n (cid:16) q (cid:1) n r n sB n (cid:1) , k ⊲ B n (cid:16) q (cid:1) n B n , F ⊲ B n (cid:16) (cid:1) q n r n sB n (cid:0) . The coaction d : H q s ℓ p q Ñ U q s ℓ p q b H q s ℓ p q follows from (2.2) as l ÞÑ b l , z m ÞÑ m ¸ s (cid:16) p(cid:1) q s q s p (cid:1) m qp q (cid:1) q (cid:1) q s (cid:18) ms (cid:26) E s k (cid:1) m b z m (cid:1) s , ETTER–DRINFELD STRUCTURES ON HEISENBERG DOUBLES AND CHAINS 13 B m ÞÑ m ¸ s (cid:16) q s p m (cid:1) s qp q (cid:1) q (cid:1) q s (cid:18) ms (cid:26) F s k (cid:1) p m (cid:1) s q b B m (cid:1) s . In particular, z ÞÑ k (cid:1) b z (cid:1) p q (cid:1) q (cid:1) q Ek (cid:1) b B ÞÑ k (cid:1) b B (cid:0) p q (cid:1) q (cid:1) q F b With the U q s ℓ p q action and coaction given above, H q s ℓ p q is a braided commu-tative Yetter–Drinfeld U q s ℓ p q -module algebra .Hence, in particular, C q r z , Bs is also a braided commutative Yetter–Drinfeld U q s ℓ p q -module algebra. The Heisenberg n -tuples { chains defined in can also be“truncated” similarly to how we passed from H p B (cid:6)q to H q s ℓ p q . An additional possibilityhere is to drop the coinvariant l altogether, which leaves us with the “ truly Heisenberg ”Yetter–Drinfeld U q s ℓ p q -module algebras HHH (cid:16) C (cid:6) p q rB s ' C p q r z s (cid:16) C q r z , B s , HHH n (cid:16) C (cid:6) p q rB s ' C p q r z s ' . . . ' C (cid:6) p q rB n (cid:1) s ' C p q r z n s , HHH n (cid:0) (cid:16) C (cid:6) p q rB s ' C p q r z s ' . . . ' C (cid:6) p q rB n (cid:1) s ' C p q r z n s ' C (cid:6) p q rB n (cid:0) s (or their infinite versions), where C (cid:6) p q rBs (cid:16) C rBs{B p and C p q r z s (cid:16) C r z s{ z p , with the braid-ing inherited from , which amounts to using the relations B i z j (cid:16) q (cid:1) q (cid:1) (cid:0) q (cid:1) z j B i for all (odd) i and (even) j , and z i z j (cid:16) q (cid:1) z j z i (cid:0) p (cid:1) q (cid:1) q z j , B i B j (cid:16) q B j B i (cid:0) p (cid:1) q qB j , i > j (and z pi (cid:16) B pi (cid:16)
0; our relations may be interestingly compared with those in para-Grassmann algebras in [36]).
Acknowledgments.
I am grateful to A. Isaev for the useful comments. This work wassupported in part by the RFBR grant 07-01-00523, the RFBR–CNRS grant 09-01-93105,and the grant LSS-1615.2008.2.A
PPENDIX A. D RINFELD DOUBLE
We recall that the Drinfeld double of B , denoted by D p B q , is B (cid:6) b B as a vector space,endowed with the structure of a quasitriangular Hopf algebra given as follows. The co-algebra structure is that of B (cid:6) cop b B , the algebra structure is given by(A.1) p m b m qp n b n q (cid:16) m p m n à S (cid:1) p m m n We also recall that C q r z , Bs is in fact Mat p p C q [16]. for all m , n P B (cid:6) and m , n P B , the antipode is given by(A.2) S D p m b m q (cid:16) p e b S p m qqp S (cid:6)(cid:1) p m q b q (cid:16) p S p m S (cid:6)(cid:1) p m qà m
1q b S p m , and the universal R -matrix is(A.3) R (cid:16) ¸ I p e b e I q b p e I b q , where t e I u is a basis of B and t e I u its dual basis in B (cid:6) .A PPENDIX B. S TANDARD CALCULATIONS
B.1. Proof of the action in (2.4) . To show that (2.4) defines an action of D p B q , weverify (2.5) by first evaluating its right-hand side: (cid:0)p m m à S (cid:1) p m m ⊲ p a a q(cid:16) x m , S (cid:1) p m p qqy x m , m p qy p m q ⊲ (cid:0)p m p q á a q m p q aS p m p qq(cid:8)(cid:16) x m p q , S (cid:1) p m p qqy x m p q , m p qy m p qp m p qá a q S (cid:6)(cid:1) p m p qq m p q aS p m p qqà S (cid:6)(cid:1) p m p qqq(cid:16) x m p q , S (cid:1) p m p qqy x m p q , m p qy m p qp m p q á a q S (cid:6)(cid:1) p m p qq m p q a S p m p qq(cid:2) x m p q , m p q S (cid:1) p a S (cid:1) p m p qqy(cid:16) x m p q , S (cid:1) p a S (cid:1) p m p qqy x m p q , m p qy m p qp m p q á a q S (cid:6)(cid:1) p m p qq m p q a S p m p qq(cid:16) x S (cid:6)(cid:1) p m p qq , a
1y x S (cid:6)(cid:1) p m p qq , m p qy x m p q , m p qy x a , m p qy m p q a S (cid:6)(cid:1) p m p qq m p q a S p m p qq(cid:16) x S (cid:6)(cid:1) p m p qq , a
1y x m p q a S (cid:6)(cid:1) p m p qq , m p qy m p q a S (cid:6)(cid:1) p m p qq m p q a S p m p qq(cid:16) x S (cid:6)(cid:1) p m p qq , a
1y x(cid:0) m p q a S (cid:6)(cid:1) p m p qq(cid:8)2 , m p qy (cid:0) m p q a S (cid:6)(cid:1) p m p qq(cid:8)1 m p q a S p m p qq(cid:16) (cid:0) m p q áp m p q a S (cid:6)(cid:1) p m p qqq(cid:8) m p q(cid:0) a à S (cid:6)(cid:1) p m p qq(cid:8) S p m p qq , which is the same as the left-hand side: p e b m q ⊲ (cid:0)p m b q ⊲ p a a q(cid:8) (cid:16) p e b m q ⊲ (cid:0) m a S (cid:6)(cid:1) p m
2q a à S (cid:6)(cid:1) p m m m a S (cid:6)(cid:1) p m m a à S (cid:6)(cid:1) p m S p m . B.2. Proof of the D p B q -module algebra property for H p B (cid:6)q . To show (2.3) for theaction in (2.4), we do this for M (cid:16) e b m and M (cid:16) m b M (cid:16) e b m is (cid:0)p e b m ⊲ p a a q(cid:8)(cid:0)p e b m ⊲ p b b q(cid:8)(cid:16) (cid:0)p m p q á a q m p q aS p m p qq(cid:8)(cid:0)p m p q á b q m p q bS p m p qq(cid:8)(cid:16) p m p q á a q(cid:0)pp m p q aS p m p qqq1 m p qqá b (cid:8) m p q aS p m p qqq2 m p q bS p m p qq(cid:16) p m p q á a q(cid:0) m p q a b (cid:8) m p q a S p m p qq m p q bS p m p qq(cid:16) p m p q á a q(cid:0) m p q a b (cid:8) m p q a bS p m p qq , ETTER–DRINFELD STRUCTURES ON HEISENBERG DOUBLES AND CHAINS 15 which is the left-hand side p e b m q ⊲ (cid:0) a p a b q a b (cid:8) .Second, the left-hand side of (2.3) with M (cid:16) m b p m b q ⊲ (cid:0) a p a b q a b (cid:8)(cid:16) m a p a b q S (cid:6)(cid:1) p m
2q a b qà S (cid:6)(cid:1) p m m p q a p a b q S (cid:6)(cid:1) p m p qq a S (cid:6)(cid:1) p m p qqqp b à S (cid:6)(cid:1) p m p qqq . But the right-hand side of (2.3) evaluates the same: (cid:0)p m q ⊲ p a a q(cid:8)(cid:0)p m q ⊲ p b b q(cid:8)(cid:16) (cid:0) m p q a S (cid:6)(cid:1) p m p qq a à S (cid:6)(cid:1) p m p qqq(cid:8)(cid:0) m p q b S (cid:6)(cid:1) p m p qq b à S (cid:6)(cid:1) p m p qqq(cid:8)(cid:16) m p q a S (cid:6)(cid:1) p m p qq(cid:0)p a à S (cid:6)(cid:1) p m p qqq1 á m p q b S (cid:6)(cid:1) p m p qq(cid:8) a à S (cid:6)(cid:1) p m p qq(cid:8)2(cid:0) b à S (cid:6)(cid:1) p m p qq(cid:8)(cid:16) m p q a S (cid:6)(cid:1) p m p qq(cid:0)p a S (cid:6)(cid:1) p m p qqqáp m p q b S (cid:6)(cid:1) p m p qqq(cid:8) a b à S (cid:6)(cid:1) p m p qq(cid:8) (because D p a à m q (cid:16) p a m q b a ) (cid:16) m p q a S (cid:6)(cid:1) p m p qqx S (cid:6)(cid:1) p m p qqp m p q b S (cid:6)(cid:1) p m p qqq2 , a
1y p m p q b S (cid:6)(cid:1) p m p qqq1 a b à S (cid:6)(cid:1) p m p qq(cid:8) (simply because p a à a qá b (cid:16) b ab , a y ) (cid:16) x b S (cid:6)(cid:1) p m p qq , a m p q ab S (cid:6)(cid:1) p m p qq a b à S (cid:6)(cid:1) p m p qq(cid:8)(cid:16) x b , a
1y x S (cid:6)(cid:1) p m p qq , a m p q ab S (cid:6)(cid:1) p m p qq a b à S (cid:6)(cid:1) p m p qq(cid:8)(cid:16) m p q a p a b q S (cid:6)(cid:1) p m p qq a S (cid:6)(cid:1) p m p qqq(cid:0) b à S (cid:6)(cid:1) p m p qq(cid:8) . B.3. Standard checks for braided products.
Here, we give the standard calculationsestablishing the module algebra and comodule algebra properties for the product definedin (3.3).The module algebra property follows by calculating (cid:0) h ⊲ p x ' y q(cid:8)(cid:0) h ⊲ p v ' u q(cid:8) (cid:16) (cid:0)p h p q ⊲ x q ' p h p q ⊲ y q(cid:8)(cid:0)p h p q ⊲ v q ' p h p q ⊲ u q(cid:8)(cid:16) p h p q ⊲ x q(cid:0)p h p q ⊲ y qp(cid:1) q h p q ⊲ v (cid:8) ' p h p q ⊲ y qp qp h p q ⊲ u q(cid:16) p h p q ⊲ x qp h p q y p(cid:1) q ⊲ v q ' p h p q ⊲ y p qqp h p q ⊲ u q(cid:16) h ⊲ (cid:0) x p y p(cid:1) q ⊲ v q ' y p q u (cid:8) (cid:16) h ⊲ (cid:0)p x ' y qp v ' u q(cid:8) . To verify the comodule algebra property d (cid:0)p x ' y qp v ' u q(cid:8) (cid:16) d p x ' y q d p v ' u q , wecalculate the left-hand side using that X and Y are comodule algebras and that Y is Yetter–Drinfeld: d (cid:0)p x ' y qp v ' u q(cid:8) (cid:16) d (cid:0) x p y p(cid:1) q ⊲ v q ' y p q u (cid:8) (cid:16) (cid:0) x p y p(cid:1) q ⊲ v q(cid:8)p(cid:1) qp y p q u qp(cid:1) q b (cid:0) x p y p(cid:1) q ⊲ v q(cid:8)p q ' (cid:0) y p q u (cid:8)p q(cid:16) x p(cid:1) qp y p(cid:1) q ⊲ v qp(cid:1) q y p q p(cid:1) q u p(cid:1) q b x p qp y p(cid:1) q ⊲ v qp q ' y p qp q u p q(cid:16) x p(cid:1) qp y q ⊲ v qp(cid:1) q y q u p(cid:1) q b (cid:0) x p qp y q ⊲ v qp q ' y p q u p q(cid:8)(cid:16) x p(cid:1) q y q v p(cid:1) q u p(cid:1) q b (cid:0) x p q p y q ⊲ v p qq ' y p q u p q(cid:8) , which is the same as the right-hand side by another use of the comodule axiom for Y : d p x ' y q d p v ' u q (cid:16) (cid:0) x p(cid:1) q y p(cid:1) q b p x p q ' y p q q(cid:8)(cid:0) v p(cid:1) q u p(cid:1) q b p v p q ' u p qq(cid:8)(cid:16) p x p(cid:1) q y p(cid:1) q v p(cid:1) q u p(cid:1) qq b (cid:0) x p q p y p qp(cid:1) q ⊲ v p qq ' y p qp q u p q(cid:8)(cid:16) p x p(cid:1) q y q v p(cid:1) q u p(cid:1) qq b (cid:0) x p q p y q ⊲ v p q q ' y p q u p q(cid:8) . R EFERENCES [1] A.Yu. Alekseev and L.D. Faddeev, p T (cid:6) G q t : A toy model for conformal field theory , Commun. Math.Phys. 141 (1991) 413–422.[2] N.Yu. Reshetikhin, and M.A. Semenov-Tian-Shansky, Central extensions of quantum current groups ,Lett. Math. Phys. 19 (1990) 133–142.[3] M.A. Semenov-Tyan-Shanskii,
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