Heisenberg double and Drinfeld double of the quantum superplane
HHeisenberg double and Drinfeld double of the quantumsuperplane
Nezhla Aghaei a,b
Michal Pawelkiewicz c a Max Planck Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. b Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern,Sidlerstrasse 5, Bern, ch-3012, Switzerland. c Institut de Physique Theorique, CEA Saclay, 91191 Gif Sur Yvette, France.
E-mail: [email protected], [email protected]
Abstract:
We study infinite dimensional generalisations of the Heisenberg doubles of the Borelhalf of U q ( sl (2)) and of U q ( osp (1 | U q ( sl (2)) and of U q ( osp (1 | R -matrix for the former and derivea novel R -matrix for the latter representation. a r X i v : . [ m a t h . QA ] S e p ontents U q ( sl (2))
104 Countinuous Heisenberg double of the Borel half of U q ( osp (1 |
145 Drinfeld double 206 Outlook 25A Special functions 27
A.1 Faddeev’s quantum dilogarithm 27A.2 Supersymmetric non-compact quantum dilogarithm 29A.3 Binomial and q-binomial identites 32
B Toy model: continuous monomial algebra 32
The methods comming from the representation theory of quantum groups have found a widerange of applications to the mathematical and theoretical physics. Quantum groups are relevantin conformal field theory [1], where the algebras of screening charges and vertex operators satisfythe relations of q-deformed Lie algebras, which are themselves a well studied family of quantumgroups. The fusion matrices of the conformal field theories were realised as a 6j symbols forrepresentations of the associated quantum groups. Moreover, in the context of quantum integrablesystems a systematic method of obtaining scatterring matrices [2] has been developed. Thesesystems satisfied the so-called
Yang-Baxter equation [3, 4] R R R = R R R , (1.1)There exist a systematic procedure to obtain solutions to the Yang-Baxter equation, which isbased on the so-called Drinfeld double construction [5–8]. It allows to associate a new, quasi-triangular Hopf algebra, i.e. a Hopf algebra that admits a universal R -matrix, which satisfies theYang-Baxter equation, to an arbitrary Hopf algebra and its dual.Another existing double construction, called the Heisenberg double construction [9, 10], ad-mits a canonical element that satisfies not the Yang-Baxter equation, but rather a pentagonequation [11] S S S = S S . (1.2)– 1 –sing Heisenberg doubles one can design the representations of Drinfeld doubles, as one canembed the elements of the Drinfeld double into a tensor square of Heisenberg doubles [9, 10].In mathematical physics, Heisenberg doubles appeared in particular in the context of thequantum Teichm¨uller theory of Riemann surfaces [12–17]. The Teichm¨uller theory is the theoryof the deformations of the complex structures on Riemann surfaces. On the space of complexstructures one can define the local coordinates using the triangulations of Riemann surfaces —however, descriptions given by equivalent triangulations should be related by similarity trans-formations. The transfromation that realises this, i.e. the so-called flip move, relates differenttriangulations of a quadrilateral and is one of the generators of the Ptolemy grupoid. The canon-ical element of the Heisenberg double of the quantum plane (that is, of a Borel half of U q ( sl (2)))evaluated on a particular family of infinite dimensional representations realises this flip move[13]. Moreover, the Fock coordinates associated to the edges of a quadrilateral correspond to theelements of the aformentioned Heisenberg double.From the Heisenberg double of the quantum plane Kashaev obtained a class of representa-tions of the U q ( sl (2)) quantum group, as well as its associated R -matrix. Those representationswere identified to be the class of infinite dimensional representations P α studied [18, 19] in con-nection with the Liouville field theory, which constitutes a prototypical non-trivial example ofthe noncompact conformal field theory [20, 21]. Using the means of harmonic analysis, Ponsotand Teschner investigated their properties. They have shown that the relation between the fusioncategory of the conformal field theory and the representation category of quantum group holdsin the case of the Liouville field theory and U q ( sl (2)) quantum group. Moreover, the consistencyof the bootstrap for the Liouville theory, i.e. the fact that the crossing-symmetry equation issatisfied by the three point function, was verified [18, 19, 22].The canonical element S in the Teichm¨uller theory context was expressed in terms of aparticular hyperbolic special function called the Faddeev’s quantum dilogarithm [23, 24], e b ( z ) = exp (cid:18)(cid:90) C e − izw sinh( w b) sinh( w/ b) d w w (cid:19) , (1.3)and it can be regarded as a quantisation of the Roger’s dilogarithm. In fact, the quantumdilogarithm has many elegant properties. In particular, the pentagon equation that it satisfies e b ( p ) e b ( x ) = e b ( x ) e b ( p + x ) e b ( p ) , (1.4)for non-commutative variables x , p such that [ p , x ] = πi , is a generalisation of the five-term rela-tion for Roger’s five-term identity. The quantum dilogarithm also found applications in conformalfield theory, topological field theory and hyperbolic geometry.In the context of super Teichm¨uller theory [25], one can define the supersymmetric analoguesof the Faddeev’s quantum dilogarithm function e R ( x ) = e b (cid:18) x i − b − ) (cid:19) e b (cid:18) x − i − b − ) (cid:19) ,e NS ( x ) = e b (cid:18) x i − ) (cid:19) e b (cid:18) x − i − ) (cid:19) . (1.5)– 2 –or self-adjoint operators p , x such that [ p , x ] = πi those supersymmetric quantum dilogarithmshave been shown to satisfy four pentagon relations f + ( p ) f + ( x ) = f + ( x ) f + ( x + p ) f + ( p ) − if − ( x ) f − ( x + p ) f − ( p ) , (1.6a) f + ( p ) f − ( x ) = − if + ( x ) f − ( x + p ) f − ( p ) + f − ( x ) f + ( x + p ) f + ( p ) , (1.6b) f − ( p ) f + ( x ) = f + ( x ) f + ( x + p ) f − ( p ) − if − ( x ) f − ( x + p ) f + ( p ) , (1.6c) f − ( p ) f − ( x ) = if + ( x ) f − ( x + p ) f + ( p ) − f − ( x ) f + ( x + p ) f − ( p ) , (1.6d)where f ± ( x ) = e R ( x ) ± e NS ( x ). As a consequence, the supersymmetric analogue of the flip oper-ator satisfies the (graded) pentagon equation.In this paper, we aim at constructing the canonical elements which satisfy pentagon equa-tion in terms of the elements of continuous Heisenberg doubles of the Borel half of U q ( sl (2)) and U q ( osp (1 | R -matrices. Theexpression for the U q ( osp (1 | R -matrix is a new one.The paper is organised as follows. In the section 2 we present a general theory of ( Z -graded) Heisenberg doubles and consider some discrete algebras as demonstrative examples. Inthe section 3 we define the infinite dimensional, continuous version of the Borel half of U q ( sl (2))and construct its Heisenberg double along with the canonical element corresponding to the oneobtained previously by Kashaev [10]. We also consider an infinite dimensional representation ofthis algebra, which was found to be relevant for Teichm¨uller theory. In the section 4 we considerthe continuous analogue of the Borel half of U q ( osp (1 | U q ( osp (1 | Z -graded case, and consider therepresentations of the continuous versions of the Drinfeld doubles which stem from the Heisenbergdouble representations. We derive also previously obtained R -matrix in the U q ( sl (2)) case and anew R -matrix for the U q ( osp (1 | In this chapter we will shortly describe the basic notions about Z -graded Heisenberg doubles.We will sketch how the Heisenberg double construction works. The exposition is structured in away that is similar to [10], it is however generalised to work in the Z -graded setting.We start from a short description of Z -graded Hopf algebras, and using the Hopf action wewill define a smash product of a Hopf algebra with its dual. At the end of this chapter we will– 3 –llustrate the Heisenberg double construction on the examples of discrete versions of U q ( sl (2)) and U q ( osp (1 | U q ( sl (2)) and U q ( osp (1 | Z -graded Hopf-algebra ( A , m, η, ∆ , (cid:15), γ ), where A is a Z -graded vectorspace equipped with the multiplication m : A ⊗ A → A , an unit η : C → A , a co-multiplication∆ : A → A ⊗ A , a co-unit (cid:15) : A → C and an antipode γ : A → A . A decomposes into a directsum of two sub-spaces A = A ⊕ A , which are called even and odd respectively. We denote thedegree of an element x ∈ A i by | x | = i , and we will call an element x even if | x | = 0 and oddotherwise. The graded tensor product of two algebras A and B is then defined by the followingequation for a , a ∈ A , b , b ∈ B ,( a ⊗ b ) · ( a ⊗ b ) = ( − | b || a | a a ⊗ b b . (2.1)The maps m and η are subjected to the associativity and unitality relations m ◦ ( m ⊗ id ) = m ◦ ( id ⊗ m ) , (2.2) m ◦ ( η ⊗ id ) = id = m ◦ ( id ⊗ η ) , (2.3)while maps ∆ and (cid:15) have to satisfy the co-associativity and co-unitality relations(∆ ⊗ id ) ◦ ∆ = ∆ ◦ ( id ⊗ ∆) , (2.4)( (cid:15) ⊗ id ) ◦ ∆ = id = ( id ⊗ (cid:15) ) ◦ ∆ . (2.5)Moreover, the co-product ∆ and co-unit (cid:15) are algebra homomorphisms, and the antipode γ is agraded algebra anti-homomorphism and a graded co-algebra anti-homomorphism which satisfiesthe relations m ◦ ( id ⊗ γ ) ◦ ∆ = m ◦ ( γ ⊗ id ) ◦ ∆ = η ◦ (cid:15). (2.6)All the above maps are grade preserving.Moreover, we consider a Hopf algebra ( A ∗ , ˆ m, ˆ η, ˆ∆ , ˆ (cid:15), ˆ γ ) which is dual to A . The Hopf algebras A and A ∗ are dual in a sense that the vector spaces A and A ∗ are dual as vector spaces, andthere exists a non-degenerate duality pairing (also called a Hopf pairing ) ( , ) : A × A ∗ → C , forwhich the following relations are satisfied( x, f g ) = (∆( x ) , f ⊗ g ) , ( xy, f ) = ( x ⊗ y, ˆ∆( f )) , (2.7)between multiplications and co-multiplications,( η (1) , f ) = ˆ (cid:15) ( f ) , (cid:15) ( x ) = ( x, ˆ η (1)) , (2.8)between unit and co-unit maps, ( γ ( x ) , f ) = ( x, ˆ γ ( f )) , (2.9)– 4 –nd between antipodes, where( x ⊗ y, f ⊗ g ) = ( − | y || f | ( x, f )( y, g ) , for x, y ∈ A , f, g ∈ A ∗ .The ordinary tensor product A ∗ ⊗ A has a straight-forward product given by (1 ⊗ x )( f ⊗
1) = x ⊗ f . However, in order to construct a Heisenberg double, we are interested in equipping thespace A ∗ ⊗ A with a non-trivial algebra structure between the elements which belong to thesubalgebras A and A ∗ . In order to achieve that, we will use the Hopf pairing ( , ) to define a leftaction of A on A ∗ and consequently a smash product algebra A ∗ (cid:111) A .Using the duality pairing ( , ) we can define a left action (cid:46) of a Hopf-algebra A on A ∗ givenby x ⊗ f (cid:55)→ (cid:88) ( f ) ( − | f (1) || f (2) | ( x, f (2) ) f (1) =: x (cid:46) f, (2.10)where x ∈ A , f ∈ A ∗ and where we denote the coproduct ˆ∆( f ) = (cid:80) ( f ) f (1) ⊗ f (2) using usualSweedler notation [26]. The action (2.10) makes A ∗ into a module algebra over the Hopf algebra A , i.e. the action is compatible with the multiplication in A ∗ in the sense that x (cid:46) ( f g ) = (cid:88) ( x ) ( − | f || x (2) | ( x (1) (cid:46) f ) ( x (2) (cid:46) g ) , (2.11)where x ∈ A and f, g ∈ A ∗ . Using a left action (cid:46) one can construct a smash product algebra H ( A ) = A ∗ (cid:111) A by defining the multiplication( f ⊗ x )( g ⊗ y ) = (cid:88) ( x ) ( − | g || x (2) | f ( x (1) (cid:46) g ) ⊗ x (2) y, (2.12)where x, y ∈ A , f, g ∈ A ∗ . Definition 2.1 A Heisenberg double of a Hopf algebra A is the smash product algebra H ( A ) = A ∗ (cid:111) A with the multiplication given by the equation (2.12) . The Heisenberg double has A and A ∗ as subalgebras through canonical embeddings A ∗ (cid:51) f (cid:55)→ f ⊗ ∈ H ( A ) and A (cid:51) x (cid:55)→ ⊗ x ∈ H ( A ).It will be convienient to recast the definition above in a basis dependent way. In order todo that, we first choose a basis of A . The basis will be given by a collection of vectors { e α } α ∈ I ,where I is a (possibly infinite) set. Then, the multiplication and co-multiplication of the basiselements is given by e α e β = (cid:88) γ ∈ I m γαβ e γ , ∆( e α ) = (cid:88) β,γ ∈ I µ βγα e β ⊗ e γ , (2.13)– 5 –here m γαβ and µ βγα are respectively multiplication and comultiplication coefficients. With thechoice of a basis { e α } α ∈ I of A ∗ dual to { e α } α ∈ I in the sense( e α , e β ) = δ βα , (2.14)the multiplication and co-multiplication on A ∗ are as follows e α e β = (cid:88) γ ∈ I ( − | α || β | µ αβγ e γ , ˆ∆( e α ) = (cid:88) β,γ ∈ I ( − | β || γ | m αβγ e β ⊗ e γ . (2.15)The Heisenberg double H ( A ) is thus spanned by basis elements { e α ⊗ e β } α,β ∈ I , written in termsof the basis elements of A and A ∗ . The multiplication (2.12) on basis elements e α ⊗ e β has thefollowing form( e α ⊗ e β )( e γ ⊗ e δ ) = (cid:88) (cid:15),π,ρ,σ,τ ∈ I ( − | β || γ | + | π || (cid:15) | + | π || α | + | (cid:15) | m γπ(cid:15) m τρδ µ (cid:15)ρβ µ απσ e σ ⊗ e τ . (2.16)It is important to note that the Heisenberg double H ( A ) is not a Hopf algebra. The algebrastructure given by (2.12) is not compatible with the co-products on ∆, ˆ∆ defined on the initialHopf algebras A and A ∗ . In this it differs from the Drinfeld double algebra (which will be discussedin section 5), which is a (quasi-triangular) Hopf algebra, and not only an algebra. Definition 2.2
For Heisenberg algebra of interest to us is a canonical element S ∈ H ( A ) ⊗ H ( A ) S = (cid:88) α ∈ I ( − | α | (1 ⊗ e α ) ⊗ ( e α ⊗ , (2.17) Proposition 2.1
The canonical element S satisfies the graded pentagon relation S S S = S S , (2.18)where we use a notation for which S = S ⊗ (1 ⊗ S = (1 ⊗ ⊗ S and S = (cid:80) α ∈ I ( − | α | e α ⊗ (1 ⊗ ⊗ e α .Although, as we have mentioned previously, the Heisenberg double is not a Hopf algebra, thecanonical element S does encode the co-products on the initial Hopf algebras A and A ∗ in thefollowing way ∆( e α ) = S − (cid:0) (1 ⊗ ⊗ (1 ⊗ e α ) (cid:1) S, ˆ∆( e α ) = S (cid:0) ( e α ⊗ ⊗ (1 ⊗ (cid:1) S − , (2.19)where ∆( e α ) and ˆ∆( e α ) should be understood as elements of the Heisenberg double obtained byembedding the co-products under the canonical embeddings A (cid:44) → H ( A ) and A ∗ (cid:44) → H ( A ). Remark 2.1
To keep the notation compact, from now on we will denote the elements ⊗ e α and e α ⊗ of the Heisenberg double H ( A ) simply as e α and e α respectively. – 6 – xample 2.1 Heisenberg double of the Borel half of U q ( sl (2))As an instructive example, lets us consider a Heisenberg double of the Borel half of U q ( sl (2)),which should be considered as a discrete prototype of the continuous algebra considered in chap-ter 3. The Borel half algebra A = B ( U q ( sl (2))) = { H m E n } ∞ m,n =0 is generated by elements H, E with a commutation relation [
H, E ] = − ibE, (2.20)and a coproduct as follows∆( H ) = H ⊗ ⊗ H, ∆( E ) = E ⊗ e πbH + 1 ⊗ E, (2.21)where q = e πib is the deformation parameter. In addition, the antipode is γ ( H ) = − H, γ ( E ) = qEe − πbH . (2.22)We can choose the basis elements of A in the following way e m,n = q n m !( q ) n ( ib − H ) m ( iE ) n , (2.23)where q-numbers ( q ) n are defined as ( q ) n = (1 − q ) . . . (1 − q n ) and n, m ∈ N .Using the properties of the generators H, E and the binomial and q-binomial formulae onecan find the multiplication and co-multiplication of the basis elements e m,n e k,l = k (cid:88) j =0 (cid:18) m + jj (cid:19)(cid:18) n + ll (cid:19) q ( − n ) k − j ( k − j )! e m + j,n + l , ∆( e m,n ) = m (cid:88) k =0 n (cid:88) l =0 ∞ (cid:88) p =0 (cid:18) k + pk (cid:19) ( n − l ) p ( − πib ) p e m − k,n − l ⊗ e k + p,l . where (cid:0) nk (cid:1) is an ordinary and (cid:0) nk (cid:1) q = ( q ) n ( q ) k ( q ) n − k is q-deformed binomial coefficient.The dual Borel half algebra A ∗ = B ( U q ( sl (2))) = { ˆ H m F n } ∞ m,n =0 on the other hand is gener-ated by elements ˆ H, F which satisfy a commutation relation[ ˆ
H, F ] = + ibF, (2.24)and have a coproduct given byˆ∆( ˆ H ) = ˆ H ⊗ ⊗ ˆ H, ˆ∆( F ) = F ⊗ e − πb ˆ H + 1 ⊗ F. (2.25)The antipode is ˆ γ ( ˆ H ) = − ˆ H, ˆ γ ( F ) = qF e πb ˆ H . (2.26)The basis elements for A ∗ are given by e m,n = (2 πb ˆ H ) m ( iF ) n , (2.27)– 7 –nd their multiplication and co-multiplication is as follows e m,n e k,l = k (cid:88) j =0 (cid:18) kj (cid:19) ( n ) k − j ( − πib ) k − j e m + j,n + l , ˆ∆( e m,n ) = m (cid:88) k =0 n (cid:88) l =0 ∞ (cid:88) p =0 (cid:18) mk (cid:19)(cid:18) nl (cid:19) q ( − n + l ) p p ! e m − k,n − l ⊗ e k + p,l . By inspection the multiplication and co-multiplication coefficients are equal to m r,sm,n ; k,l = (cid:18) rr − m (cid:19)(cid:18) n + ll (cid:19) q ( − n ) k − r + m ( k − r + m )! Θ( r − m )Θ( k − r + m ) δ s,n + l ,µ m,n ; k,lr,s = (cid:18) kr − m (cid:19) ( n ) k − r + m ( − πib ) k − r + m Θ( r − m )Θ( k − r + m ) δ s,n + l , and that A and A ∗ are dual as Hopf algebras. The Heisenberg double H ( A ) is then given by thegenerators H, ˆ H, E, F , which satisfy the commutation relations (2.12), which are as follows[
H, E ] = − ibE, [ H, F ] = + ibF, [ ˆ
H, E ] = 0 , [ ˆ H, F ] = + ibF, [ H, ˆ H ] = 12 πi , [ E, F ] = ( q − q − ) e πbH . (2.28)The canonical element S is given in terms of generators as S = exp(2 πiH ⊗ ˆ H )( − qE ⊗ F ; q ) − ∞ , (2.29)where the special function ( x ; q ) ∞ , known under a name of a quantum dilogarithm , is defined as( x ; q ) − ∞ = ∞ (cid:89) k =0 − xq k = ∞ (cid:88) k =0 x k ( q ) k . (2.30)Using the properties of the quantum dilogarithm function one can check explicitly that the pen-tagon equation is satisfied. In particular, it reduces to the identity( V ; q ) ∞ ( U ; q ) ∞ = ( U ; q ) ∞ ((1 − q ) − [ U, V ]; q ) ∞ ( V ; q ) ∞ , (2.31)for U = − q ⊗ E ⊗ F, V = − qE ⊗ F ⊗
1, which was shown to be satisfied by the quantumdilogarithm ( x ; q ) ∞ . The square brackets denote the commutator, and operators U and V satisfythe following algebraic relations [10] W = U V − q V U, [ U, W ] = [
V, W ] = 0 . (2.32)– 8 – xample 2.2 Heisenberg double of the Borel half of U q ( osp (1 | U q ( osp (1 | A = B ( osp (1 | { H m v (+) n } ∞ m,n =0 is generatedby an even graded element H and an odd graded element v (+) with a commutation relation[ H, v (+) ] = − ibv (+) , (2.33)a coproduct ∆( H ) = H ⊗ ⊗ H, ∆( v (+) ) = v (+) ⊗ e πbH + 1 ⊗ v (+) , (2.34)and the antipode as follows γ ( H ) = − H, γ ( v (+) ) = q v (+) e − πbH , (2.35)where q = e iπb is the deformation parameter. We can choose the basis elements of A in thefollowing way e m,n = ( − n ( n − / q n/ m !( − q ) n ( ib − H ) m ( iv (+) ) n . (2.36)Using the properties of the generators H, v (+) and the binomial and q-binomial formulae one canfind the multiplication and co-multiplication of the basis elements e m,n e k,l = k (cid:88) j =0 (cid:18) m + jj (cid:19)(cid:18) n + ll (cid:19) − q ( − n ) k − j ( k − j )! e m + j,n + l , (2.37)∆( e m,n ) = m (cid:88) k =0 n (cid:88) l =0 ∞ (cid:88) p =0 (cid:18) k + pk (cid:19) ( n − l ) p ( − πib ) p e m − k,n − l ⊗ e k + p,l . (2.38)The dual Borel half algebra A ∗ = B ( U q ( osp (1 | { ˆ H m v ( − ) n } ∞ m,n =0 on the other hand isgenerated by elements ˆ H, v ( − ) which satisfy a commutation relation[ ˆ H, v ( − ) ] = + ibv ( − ) , (2.39)and have a coproduct given byˆ∆( ˆ H ) = ˆ H ⊗ ⊗ ˆ H, ˆ∆( v ( − ) ) = v ( − ) ⊗ e − πb ˆ H + 1 ⊗ v ( − ) , (2.40)with the antipode ˆ γ ( ˆ H ) = − ˆ H, ˆ γ ( v ( − ) ) = q − v ( − ) e πb ˆ H . (2.41)The basis elements for A ∗ are given by e m,n = ( πb ˆ H ) m ( iv ( − ) ) n , (2.42)– 9 –nd their multiplication and co-multiplication are as follows e m,n e k,l = k (cid:88) j =0 (cid:18) kj (cid:19) ( n ) k − j ( − πib ) k − j e m + j,n + l , (2.43)ˆ∆( e m,n ) = m (cid:88) k =0 n (cid:88) l =0 ∞ (cid:88) p =0 (cid:18) mk (cid:19)(cid:18) nl (cid:19) − q ( − n + l ) p p ! e m − k,n − l ⊗ e k + p,l . (2.44)By inspection the multiplication and co-multiplication coefficients are equal to m r,sm,n ; k,l = (cid:18) rr − m (cid:19)(cid:18) n + ll (cid:19) − q ( − n ) k − r + m ( k − r + m )! Θ( r − m )Θ( k − r + m ) δ s,n + l ,µ m,n ; k,lr,s = (cid:18) kr − m (cid:19) ( n ) k − r + m ( − πib ) k − r + m Θ( r − m )Θ( k − r + m ) δ s,n + l , and that A and A ∗ are dual as Hopf algebras. The discrete Heisenberg double can be defined asan algebra generated by the even elements H and ˆ H and the odd elements v (+) and v ( − ) satisfying(anti-)commutation relations[ H, ˆ H ] = 1 πi , { v (+) , v ( − ) } = − e πbH ( q + q − ) , [ H, v (+) ] = − ibv (+) , [ H, v ( − ) ] = ibv ( − ) , [ ˆ H, v (+) ] = 0 , [ ˆ H, v ( − ) ] = + ibv ( − ) . (2.45)The canonical element S is given in terms of generators as S = exp( πiH ⊗ ˆ H )( − q v (+) ⊗ v ( − ) ; − q ) − ∞ . (2.46)Using the properties of the quantum dilogarithm function one can check explicitly that the pen-tagon equation is satisfied. In particular, it reduces to the identity (2.31) for U = − q ⊗ v (+) ⊗ v ( − ) , V = − q v (+) ⊗ v ( − ) ⊗ q → − q . The operators U and V satisfythe following algebraic relations W = U V + qV U, [ U, W ] = [
V, W ] = 0 . (2.47) U q ( sl (2)) In this section we will provide a discussion of a continuous version of the Heisenberg double ofthe Borel half of U q ( sl (2)). We will describe the multiplication and co-multiplication structuresof the continuous Borel half, which follow from the commutation relations and the co-product forthe generators of the discrete algebra and the continuous version of the binomial and q-binomialformulae. Afterwards, we construct the canonical element S satisfying the pentagon equation,which is expressed using the Faddeev’s quantum dilogarithm. At the end of the section we recalla particular representation of continuous Heisenberg double which was introduced by Kashaev [14].As already described in the section 2, the discrete version of the Heisenberg double H ( A )of the Borel half A = B ( U q ( sl (2))) can be defined as an algebra spanned by the elements– 10 – ˆ H m F n H k E l } ∞ m,n,k,l =0 subjected to the commutation relations (2.28), where q = e iπb for aparameter b such that b ∈ R / Q .In the case of the continuous version of the algebra, instead of integer powers of the generators,we are interested in the generators taken to the pure imaginary powers — i.e. instead of H i E i for i j ∈ Z ≥ we would like to consider H iα E iα for α j ∈ R . This modification would not involveany subtleties if all the generators were positive (or at least non-negative) definite. The situationwould be similar to taking a complex power of a positive real number, which does not requirespecifting the branch of the logarithm — however, taking a complex power of a negative realnumber does.Unfortunately, for the algebra A not all generators are be positive. We assume that thegenerator E will be realised as positive operator while the generator H will not be. This positivityissue will be resolved by using principal value prescription for those generators which belong tothe Cartan subalgebra of the Borel half and its dual. The principal value is given by( (cid:15)y ) ispv = | y | is Θ( (cid:15)y ) + e − πs | y | is Θ( − (cid:15)y ) , (3.1)where y > (cid:15) = ± y ) is a Heaviside theta function. The Hopf algebra com-posed only of the Cartan subalgebra constitutes an instructive toy model, which because of itssimplicity clarifies the construction for the full B ( U q ( sl (2))). We refer to the appendix B for itsdiscussion.Starting from the equations (2.20), (2.34) one can derive the multiplication and co-multiplicationrelations for the elements of the form ( ± H ) ispv E ib − t using the continuous binomial formulae (A.39),the continuous version of the q-binomial formula (A.41) as well as the Mellin transform of theexponential function (B.7). The result of that calculation is presented below as the relationssatisfied by the basis elements.Let us start with the continuous version of the Borel half A of U q ( sl (2)). We define it as analgebra A spanned by the elements { e ( s, (cid:15), t ) } s,t ∈ R ,(cid:15) = ± , which satisfy the following multiplication e ( s, (cid:15), t ) e ( s (cid:48) , (cid:15), t (cid:48) ) = (cid:90) d σ (2 π ) (cid:18) − i ( s + s (cid:48) − σ ) − is (cid:19) Γ Γ( − iσ ) (cid:18) i ( t + t (cid:48) ) it (cid:19) b e − πitt (cid:48) ×× | πt | iσ [ e πσ/ Θ( − (cid:15)t ) + Θ( (cid:15)t ) e − πσ/ ] e ( s + s (cid:48) − σ, (cid:15), t + t (cid:48) ) ,e ( s, (cid:15), t ) e ( s (cid:48) , − (cid:15), t (cid:48) ) = (cid:90) d σ (2 π ) (cid:18) − i ( s + s (cid:48) − σ ) − is (cid:19) Γ Γ( − iσ ) (cid:18) i ( t + t (cid:48) ) it (cid:19) b e − πitt (cid:48) ×× | πt | iσ [ e πσ/ Θ( (cid:15)t ) + e − πσ/ Θ( − (cid:15)t )] (cid:34) e − π ( s (cid:48) − σ ) (1 − e − πs )1 − e − π ( s (cid:48) + s − σ ) e ( s + s (cid:48) − σ, (cid:15), t + t (cid:48) )++ e − πs (1 − e − π ( s (cid:48) − σ ) )1 − e − π ( s (cid:48) + s − σ ) e ( s + s (cid:48) − σ, − (cid:15), t + t (cid:48) ) (cid:35) , (3.2) – 11 –nd co-multiplication relations ∆( e ( s, (cid:15), t )) = (cid:90) d σ d˜ σ π d τ (cid:18) − i ( s − σ + ˜ σ ) − i ˜ σ (cid:19) Γ | τ | i ˜ σ e ( σ, (cid:15), τ ) ⊗⊗ (cid:26)(cid:18) Θ( (cid:15)τ ) + e − π ˜ σ (1 − e − π ( s − σ ) )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) (cid:19) e ( s − σ + ˜ σ, (cid:15), t − τ )++ e − π ( s − σ ) (1 − e − π ˜ σ )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) e ( s − σ + ˜ σ, − (cid:15), t − τ ) (cid:27) , (3.3) where (cid:0) sσ (cid:1) Γ = Γ( σ )Γ( s − σ )Γ( s ) is a continuous version of the binomial coefficent and (cid:0) tτ (cid:1) b = G b ( − τ ) G b ( − t + τ ) G b ( − t ) is a continuous version of the q-binomial coefficient, defined using a special function G b which isrelated to the Fadeev’s quantum dilogarithm (c.f. appendix A) e b ( x ) = exp[ − iπ − iπ ( b + b − )] G b ( Q − ix ) . (3.4)The elements e ( s, (cid:15), t ) admit a presentation in terms of the generators H, E (that is the elementswhich generated the discrete version of the algebra in section 2) in the following way e ( s, (cid:15), t ) = f (cid:15) ( s, t )( (cid:15) πH ) ispv E ib − t , (3.5)where f (cid:15) ( s, t ) = 12 π Γ( − is ) G − ( Q + it ) e − πtQ e πs/ . One can reproduce the discrete multiplication and co-multiplication relations by analytically con-tinueing the relations (3.2) and (3.3) to the values s = − im , t = − ibn , where n, m ∈ Z ≥ .After describing the continuous version of the algebra A , we focus on the algebra A ∗ dualto it. Let us define it as being spanned by the elements { ˆ e ( s, (cid:15), t ) } s,t ∈ R ,(cid:15) = ± , with the followingproduct ˆ e ( s, (cid:15), t )ˆ e ( s (cid:48) , (cid:15), t (cid:48) ) = (cid:90) d σ π (cid:18) − is (cid:48) − iσ (cid:19) Γ | t | iσ (cid:34) Θ( − (cid:15)t ) e − πσ (1 − e − π ( s (cid:48) − σ ) )1 − e − πs (cid:48) + Θ( (cid:15)t ) (cid:35) ×× ˆ e ( s + s (cid:48) − σ, (cid:15), t + t (cid:48) ) , ˆ e ( s, (cid:15), t )ˆ e ( s (cid:48) , − (cid:15), t (cid:48) ) = (cid:90) d σ π (cid:18) − is (cid:48) − iσ (cid:19) Γ | t | iσ Θ( − (cid:15)t ) e − π ( s (cid:48) − σ ) (1 − e − πσ )1 − e − πs (cid:48) ˆ e ( s + s (cid:48) − σ, (cid:15), t + t (cid:48) ) , (3.6) and coproduct ∆(ˆ e ( s, (cid:15), t )) = (cid:90) d σ π d˜ σ π d τ Γ( − i ˜ σ ) (cid:18) − is − iσ (cid:19) Γ (cid:18) itiτ (cid:19) b e − πiτ ( t − τ ) | πτ | i ˜ σ ×× (cid:110) ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, (cid:15), τ ) ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ )++ e − π ( s − σ ) (1 − e − πσ )1 − e − πs ( e π ˜ σ/ Θ( (cid:15)τ ) + e − π ˜ σ/ Θ( − (cid:15)τ ))ˆ e ( σ, (cid:15), τ ) ⊗ ˆ e ( s − σ + ˜ σ, − (cid:15), t − τ )++ e − πσ (1 − e − π ( s − σ ) )1 − e − πs ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, − (cid:15), τ ) ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ ) (cid:27) . (3.7) – 12 –he elements ˆ e ( s, (cid:15), t ) admit a presentation in terms of generators ˆ H, F satisfying equations(2.24)-(2.25) ˆ e ( s, (cid:15), t ) = | ˆ H | is Θ( (cid:15) ˆ H ) F ib − t . (3.8)As in the case of the initial Hopf algebra A , one can reproduce the discrete multiplication andco-multiplication relations by analytical continuation to the values s = − im , t = − ibn of thepowers, where n, m ∈ Z ≥ .The algebras A and A ∗ are dual to each other in the sense of relation (2.14) with respect toa duality pairing defined ( e ( s, (cid:15), t ) , ˆ e ( s (cid:48) , (cid:15) (cid:48) , t (cid:48) )) = δ ( s − s (cid:48) ) δ ( t − t (cid:48) ) δ (cid:15),(cid:15) (cid:48) . (3.9)Alternatively, one can see that the multiplication coefficients e ( s, (cid:15), t ) e ( s (cid:48) , (cid:15) (cid:48) , t (cid:48) ) = (cid:88) (cid:15) (cid:48)(cid:48) = ± (cid:90) d σ d τ m σ,(cid:15) (cid:48)(cid:48) ,τs,(cid:15),t,s (cid:48) ,(cid:15) (cid:48) ,t (cid:48) e ( σ, (cid:15) (cid:48)(cid:48) , τ ) , ˆ e ( s, (cid:15), t )ˆ e ( s (cid:48) , (cid:15) (cid:48) , t (cid:48) ) = (cid:88) (cid:15) (cid:48)(cid:48) = ± (cid:90) d σ d τ ˆ m s,(cid:15),t,s (cid:48) ,(cid:15) (cid:48) ,t (cid:48) σ,(cid:15) (cid:48)(cid:48) ,τ ˆ e ( σ, (cid:15) (cid:48)(cid:48) , τ ) , (3.10)and the co-multiplication coefficients∆( e ( s, (cid:15), t )) = (cid:88) (cid:15) (cid:48) ,(cid:15) (cid:48)(cid:48) = ± (cid:90) d σ (cid:48) d σ (cid:48)(cid:48) d τ (cid:48) d τ (cid:48)(cid:48) µ σ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) s,(cid:15),t e ( σ (cid:48) , (cid:15) (cid:48) , τ (cid:48) ) ⊗ e ( σ (cid:48)(cid:48) , (cid:15) (cid:48)(cid:48) , τ (cid:48)(cid:48) ) , ˆ∆(ˆ e ( s, (cid:15), t )) = (cid:88) (cid:15) (cid:48) ,(cid:15) (cid:48)(cid:48) = ± (cid:90) d σ (cid:48) d σ (cid:48)(cid:48) d τ (cid:48) d τ (cid:48)(cid:48) ˆ µ s,(cid:15),tσ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) ˆ e ( σ (cid:48) , (cid:15) (cid:48) , τ (cid:48) ) ⊗ ˆ e ( σ (cid:48)(cid:48) , (cid:15) (cid:48)(cid:48) , τ (cid:48)(cid:48) ) , (3.11)defined as above do satisfy the following equalities m s,(cid:15),tσ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) = ˆ µ s,(cid:15),tσ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) , ˆ m σ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) s,(cid:15),t = µ σ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) s,(cid:15),t . (3.12)Given the above, one uses equation (2.12) to define the commutation relations for the Heisenbergdouble of A .We want to stress that, as before for A and A ∗ separately, one can reproduce the relations(2.28) using the analytical continuation to the appropriate values of σ, τ . When one analyticallycontinues, the poles of gamma and Fadeev’s quantum dilogarithm functions present in the inte-grand are pinching the contours of integration. The residues of those poles then contribute as theterms of the sums of the product of the discrete H ( A ).By applying the definition (2.17) we obtain the following expression for the canonical element S in terms of the basis elements of H ( A ) S = (cid:88) (cid:15) = ± (cid:90) R d s d t e ( s, (cid:15), t ) ⊗ ˆ e ( s, (cid:15), t ) , (3.13)– 13 –hich can be written in terms of the generators H, ˆ H, E, F by using the explicit presentation ofbasis elements in equations (3.5) and (3.8), S = e πiH ⊗ ˆ H g − ( E ⊗ F ) , (3.14)where e b ( x ) = g b ( e πbx ). We see that quantum dilogarithm ( x, q ) ∞ , which was present in thediscrete version of the algebra, has been replaced by its continuous analogue g b ( x ).Moreover, the pentagon equation (2.18) for S is manifestly satisfied as it can be reduced tothe pentagon relation for Faddeev’s quantum dilogarithm (equation (A.25) in appendix A). Representations of the Heisenberg double of the Borel half of U q ( sl (2))In this part we will consider a representation theory of the continuous Heisenberg double H ( A )described above. This representations was first considered by Kashaev [13] in the context of ap-plications to the Teichm¨uller theory of Riemann surfaces. The Heisenberg double evaluated onthose representations have a direct interpretation as the operators in the quantum Teichm¨uller the-ory [13, 14].Following [13], we introduce the representations π : H ( A ) → Hom ( L ( R )) of the Heisenbergdouble H ( A ) on L ( R ) representation space using the following realisation of the generators H, ˆ H, E, F π ( H ) = p , π ( E ) = e πb x ,π ( ˆ H ) = x , π ( F ) = e πb ( p − x ) , (3.15)where p , x are self-adjoint operators on L ( R ) and [ p , x ] = πi . One can show that these generatorssatisfy the commutation relations (2.28).The canonical element S (3.14) evaluated on those representations can be written as follows( π ⊗ π )( S ) = e πi p x e − ( x + p − x ) . (3.16)This representation of the canonical element has been considered in the context of Teichm¨ullertheory as a realisation of the flip operator [13, 14]. U q ( osp (1 | This section is devoted to the study of the continuous Heisenberg double of the Borel half of U q ( osp (1 | U q ( osp (1 | Z -graded. Afterwards, we consider aninfinite dimensional representations of the Heisenberg double on L ( R ) ⊗ C | with the focus oncanonical element S .The discrete Heisenberg double H ( A ) of A = B ( U q ( osp (1 | B ( U q ( sl (2))). This is caused bythe fact that one cannot take a complex power of elements which have an odd degree and producea homogenous elements. In particular, we cannot simply take the imaginary powers of the oddelements v ( ± ) . In order to resolve this issue, we can consider a decomposition of those particularelements v (+) = Eκ, v ( − ) = F ˆ κ, (4.1)into the even elements E, F which satisfy non-trivial commutation relations with generators H, ˆ H [ H, E ] = − ibE, [ ˆ H, F ] = + ibF. (4.2)The odd elements κ, ˆ κ commute trivially with the even ones[ H, κ ] = [
E, κ ] = 0 , [ ˆ H, ˆ κ ] = [ F, ˆ κ ] = 0 , (4.3)and they satisfy the following identites κ = − , ˆ κ = − . (4.4)The decomposition (4.1) informs one how one needs to modify the definition of the Hopf algebra A in the continuous case. It allows to straightforwardly take the imaginary powers of the evenpart of the decomposition, while constraining the powers of the odd part to integers only.Let us start with the Borel half A of U q ( osp (1 | { e ( s, (cid:15), t, n ) } s,t ∈ R ,(cid:15) = ± ,n =0 , , where the basis elements are given in terms of the generators by e ( s, (cid:15), t, n ) = f (cid:15),n ( s, t )( (cid:15)πH ) ispv E ib − t κ n , (4.5)where f (cid:15), ( s, t ) = 14 π ζ Γ( − is ) e − πtQ/ e πs/ G − ( Q + it ) ,f (cid:15), ( s, t ) = i π ζ Γ( − is ) e − πtQ/ e πs/ G − ( Q + it ) , (4.6)where the special functions G R , G NS are related to the supersymmetric analogues of Faddeev’squantum dilogarithm functions e − ( r ) + e − ( r ) = ζ (cid:90) d te πitr e − πtQ G NS ( Q + it ) ,e − ( r ) − e − ( r ) = ζ (cid:90) d te πitr e − πtQ G R ( Q + it ) , (4.7)that are described in more details in the appendix A, with ζ = exp( − iπ ( b + b − ) / e ( s, (cid:15), t, n ) e ( s (cid:48) , (cid:15), t (cid:48) , n (cid:48) ) = (cid:90) d σ π (cid:18) − is (cid:48) − iσ (cid:19) Γ | πt | iσ f (cid:15),n ( s, t ) f (cid:15),n (cid:48) ( s (cid:48) , t (cid:48) ) f (cid:15),n + n (cid:48) ( s + s (cid:48) − σ, t + t (cid:48) ) ×× [Θ( − (cid:15)t ) + Θ( (cid:15)t ) e − πσ ] e ( s + s (cid:48) − σ, (cid:15), t + t (cid:48) , n + n (cid:48) ) ,e ( s, (cid:15), t, n ) e ( s (cid:48) , − (cid:15), t (cid:48) , n (cid:48) ) = (cid:90) d σ π (cid:18) − is (cid:48) − iσ (cid:19) Γ | πt | iσ [Θ( (cid:15)t ) + Θ( − (cid:15)t ) e − πσ ] ×× (cid:34) e − π ( s (cid:48) − σ ) (1 − e − πs )1 − e − π ( s (cid:48) + s − σ ) f (cid:15),n ( s, t ) f − (cid:15),n (cid:48) ( s (cid:48) , t (cid:48) ) f (cid:15),n + n (cid:48) ( s + s (cid:48) − σ, t + t (cid:48) ) e ( s + s (cid:48) − σ, (cid:15), t + t (cid:48) , n + n (cid:48) )++ e − πs (1 − e − π ( s (cid:48) − σ ) )1 − e − π ( s (cid:48) + s − σ ) f (cid:15),n ( s, t ) f − (cid:15),n (cid:48) ( s (cid:48) , t (cid:48) ) f − (cid:15),n + n (cid:48) ( s + s (cid:48) − σ, t + t (cid:48) ) e ( s + s (cid:48) − σ, − (cid:15), t + t (cid:48) , n + n (cid:48) ) (cid:35) , (4.8)while the co-product is given by ∆( e ( s, (cid:15), t, ζ (cid:90) d σ d˜ σ (2 π ) d τ Γ( − i ˜ σ ) (cid:32) − is − iσ (cid:33) Γ e π ˜ σ/ | τ | i ˜ σ (cid:20) G R ( Q + it ) G NS ( Q + iτ ) G R ( − iτ + Q + it ) ×× e ( σ, (cid:15), τ, ⊗ (cid:40) (cid:18) Θ( (cid:15)τ ) + e − π ˜ σ (1 − e − π ( s − σ ) )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) (cid:19) f (cid:15), ( s, t ) f (cid:15), ( σ, τ ) f (cid:15), ( s − σ + ˜ σ, t − τ ) ×× e ( s − σ + ˜ σ, (cid:15), t − τ,
1) + e − π ( s − σ ) (1 − e − π ˜ σ )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) f (cid:15), ( s, t ) f (cid:15), ( σ, τ ) f − (cid:15), ( s − σ + ˜ σ, t − τ ) ×× e ( s − σ + ˜ σ, − (cid:15), t − τ, (cid:41) + G R ( Q + it ) G R ( Q + iτ ) G NS ( − iτ + Q + it ) e ( σ, (cid:15), τ, ⊗⊗ (cid:26)(cid:18) Θ( (cid:15)τ ) + e − π ˜ σ (1 − e − π ( s − σ ) )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) (cid:19) f (cid:15), ( s, t ) f (cid:15), ( σ, τ ) f (cid:15), ( s − σ + ˜ σ, t − τ ) e ( s − σ + ˜ σ, (cid:15), t − τ, e − π ( s − σ ) (1 − e − π ˜ σ )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) f (cid:15), ( s, t ) f (cid:15), ( σ, τ ) f − (cid:15), ( s − σ + ˜ σ, t − τ ) e ( s − σ + ˜ σ, − (cid:15), t − τ, (cid:27)(cid:21) , (4.9) ∆( e ( s, (cid:15), t, ζ (cid:90) d σ d˜ σ (2 π ) d τ Γ( − i ˜ σ ) (cid:32) − is − iσ (cid:33) Γ e π ˜ σ/ | τ | i ˜ σ (cid:20) G NS ( Q + it ) G NS ( Q + iτ ) G NS ( − iτ + Q + it ) ×× e ( σ, (cid:15), τ, ⊗ (cid:40) (cid:18) Θ( (cid:15)τ ) + e − π ˜ σ (1 − e − π ( s − σ ) )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) (cid:19) f (cid:15), ( s, t ) f (cid:15), ( σ, τ ) f (cid:15), ( s − σ + ˜ σ, t − τ ) ×× e ( s − σ + ˜ σ, (cid:15), t − τ,
0) + e − π ( s − σ ) (1 − e − π ˜ σ )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) f (cid:15), ( s, t ) f (cid:15), ( σ, τ ) f − (cid:15), ( s − σ + ˜ σ, t − τ ) ×× e ( s − σ + ˜ σ, − (cid:15), t − τ, (cid:41) + G NS ( Q + it ) G R ( Q + iτ ) G R ( − iτ + Q + it ) e ( σ, (cid:15), τ, ⊗⊗ (cid:26)(cid:18) Θ( (cid:15)τ ) + e − π ˜ σ (1 − e − π ( s − σ ) )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) (cid:19) f (cid:15), ( s, t ) f (cid:15), ( σ, τ ) f (cid:15), ( s − σ + ˜ σ, t − τ ) e ( s − σ + ˜ σ, (cid:15), t − τ, e − π ( s − σ ) (1 − e − π ˜ σ )1 − e − π ( s − σ +˜ σ ) Θ( − (cid:15)τ ) f (cid:15), ( s, t ) f (cid:15), ( σ, τ ) f − (cid:15), ( s − σ + ˜ σ, t − τ ) e ( s − σ + ˜ σ, − (cid:15), t − τ, (cid:27)(cid:21) . (4.10)By analytically continuing the values of s, t in the equations (4.8)-(4.10) one can recover thecommutation (2.37) and co-product relations (2.38) for the discrete algebra elements generatedby H, v (+) considered in section 2. The values corresponding to the discrete algebra basis elementare s = − im , t = − ibn , n = 1 for m ∈ Z ≥ , n ∈ Z ≥ + 1 and s = − im , t = − ibn , n = 0 for– 16 – ∈ Z ≥ , n ∈ Z ≥ .After describing the Hopf algebra A , we consider the dual Hopf algebra A ∗ . This Hopf algebrais spanned by the elements { ˆ e ( s, (cid:15), t, n ) } s,t ∈ R ,(cid:15) = ± ,n =0 , . The dual basis can be expressed in termsof the generators satisfying the relations (4.2)-(4.4)ˆ e ( s, (cid:15), t, n ) = | ˆ H | is Θ( (cid:15) ˆ H ) F ib − t ˆ κ n . (4.11)The multiplication relations for those elements are as follows ˆ e ( s, (cid:15), t, n )ˆ e ( s (cid:48) , (cid:15), t (cid:48) , n (cid:48) ) = (cid:90) d σ π (cid:18) − is (cid:48) − iσ (cid:19) Γ | t | iσ (cid:34) Θ( − (cid:15)t ) e − πσ (1 − e − π ( s (cid:48) − σ ) )1 − e − πs (cid:48) + Θ( (cid:15)t ) (cid:35) ×× ˆ e ( s + s (cid:48) − σ, (cid:15), t + t (cid:48) , n + n (cid:48) ) , ˆ e ( s, (cid:15), t, n )ˆ e ( s (cid:48) , − (cid:15), t (cid:48) , n (cid:48) ) = (cid:90) d σ π (cid:18) − is (cid:48) − iσ (cid:19) Γ | t | iσ Θ( − (cid:15)t ) e − π ( s (cid:48) − σ ) (1 − e − πσ )1 − e − πs (cid:48) ×× ˆ e ( s + s (cid:48) − σ, (cid:15), t + t (cid:48) , n + n (cid:48) ) , (4.12) while the co-multiplication has the following form ∆(ˆ e ( s, (cid:15), t, ζ (cid:90) d σ π d˜ σ π d τ Γ( − i ˜ σ ) (cid:32) − is − iσ (cid:33) Γ | πτ | i ˜ σ ×× (cid:20) G NS ( Q + it ) G NS ( Q + iτ ) G NS ( − iτ + Q + it ) (cid:110) ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ, e − π ( s − σ ) (1 − e − πσ )1 − e − πs ( e π ˜ σ/ Θ( (cid:15)τ ) + e − π ˜ σ/ Θ( − (cid:15)τ ))ˆ e ( σ, (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, − (cid:15), t − τ, e − πσ (1 − e − π ( s − σ ) )1 − e − πs ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, − (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ, (cid:27) ++ G NS ( Q + it ) G R ( Q + iτ ) G R ( − iτ + Q + it ) (cid:110) ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ, e − π ( s − σ ) (1 − e − πσ )1 − e − πs ( e π ˜ σ/ Θ( (cid:15)τ ) + e − π ˜ σ/ Θ( − (cid:15)τ ))ˆ e ( σ, (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, − (cid:15), t − τ, e − πσ (1 − e − π ( s − σ ) )1 − e − πs ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, − (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ, (cid:27)(cid:21) , (4.13) ∆(ˆ e ( s, (cid:15), t, ζ (cid:90) d σ π d˜ σ π d τ Γ( − i ˜ σ ) (cid:32) − is − iσ (cid:33) Γ | πτ | i ˜ σ ×× (cid:20) G R ( Q + it ) G R ( Q + iτ ) G NS ( − iτ + Q + it ) (cid:110) ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ, e − π ( s − σ ) (1 − e − πσ )1 − e − πs ( e π ˜ σ/ Θ( (cid:15)τ ) + e − π ˜ σ/ Θ( − (cid:15)τ ))ˆ e ( σ, (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, − (cid:15), t − τ, e − πσ (1 − e − π ( s − σ ) )1 − e − πs ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, − (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ, (cid:27) ++ G R ( Q + it ) G NS ( Q + iτ ) G R ( − iτ + Q + it ) (cid:110) ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ, e − π ( s − σ ) (1 − e − πσ )1 − e − πs ( e π ˜ σ/ Θ( (cid:15)τ ) + e − π ˜ σ/ Θ( − (cid:15)τ ))ˆ e ( σ, (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, − (cid:15), t − τ, e − πσ (1 − e − π ( s − σ ) )1 − e − πs ( e π ˜ σ/ Θ( − (cid:15)τ ) + e − π ˜ σ/ Θ( (cid:15)τ ))ˆ e ( σ, − (cid:15), τ, ⊗ ˆ e ( s − σ + ˜ σ, (cid:15), t − τ, (cid:27)(cid:21) . (4.14)– 17 –s in the case of A , the multiplication and co-multiplication relations above reduce to the productand co-product (2.43)-(2.44) of the discrete dual Hopf algebra from section 2 by the means ofappropriate analytic continuation.One can see that the multiplications and co-multiplications of A and A ∗ are dual to eachother in the sense of (2.14) with respect to a duality bracked defined( e ( s, (cid:15), t, n ) , ˆ e ( s (cid:48) , (cid:15) (cid:48) , t (cid:48) , n (cid:48) )) = δ ( s − s (cid:48) ) δ ( t − t (cid:48) ) δ (cid:15),(cid:15) (cid:48) δ n,n (cid:48) . (4.15)Alternatively, one can see that with the multiplication and co-multiplication coefficients definedin the following way e ( s, (cid:15), t, n ) e ( s (cid:48) , (cid:15) (cid:48) , t (cid:48) , n (cid:48) ) = (cid:88) (cid:15) (cid:48)(cid:48) = ± (cid:88) n (cid:48)(cid:48) =0 (cid:90) d σ d τ m σ,(cid:15) (cid:48)(cid:48) ,τ,n (cid:48)(cid:48) s,(cid:15),t,n,s (cid:48) ,(cid:15) (cid:48) ,t (cid:48) ,n (cid:48) e ( σ, (cid:15) (cid:48)(cid:48) , τ, n (cid:48)(cid:48) ) , ˆ e ( s, (cid:15), t, n )ˆ e ( s (cid:48) , (cid:15) (cid:48) , t (cid:48) , n (cid:48) ) = (cid:88) (cid:15) (cid:48)(cid:48) = ± (cid:88) n (cid:48)(cid:48) =0 (cid:90) d σ d τ ˆ m s,(cid:15),t,n,s (cid:48) ,(cid:15) (cid:48) ,t (cid:48) ,n (cid:48) σ,(cid:15) (cid:48)(cid:48) ,τ,n (cid:48)(cid:48) ˆ e ( σ, (cid:15) (cid:48)(cid:48) , τ, n (cid:48)(cid:48) ) , (4.16) and ∆( e ( s, (cid:15), t, n )) = (cid:88) (cid:15) (cid:48) ,(cid:15) (cid:48)(cid:48) = ± ,n (cid:48) ,n (cid:48)(cid:48) =0 , (cid:90) d σ (cid:48) d σ (cid:48)(cid:48) d τ (cid:48) d τ (cid:48)(cid:48) µ σ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,n (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) ,n (cid:48)(cid:48) s,(cid:15),t,n e ( σ (cid:48) , (cid:15) (cid:48) , τ (cid:48) , n (cid:48) ) ⊗ e ( σ (cid:48)(cid:48) , (cid:15) (cid:48)(cid:48) , τ (cid:48)(cid:48) , n (cid:48)(cid:48) ) , ˆ∆(ˆ e ( s, (cid:15), t, n )) = (cid:88) (cid:15) (cid:48) ,(cid:15) (cid:48)(cid:48) = ± ,n (cid:48) ,n (cid:48)(cid:48) =0 , (cid:90) d σ (cid:48) d σ (cid:48)(cid:48) d τ (cid:48) d τ (cid:48)(cid:48) ˆ µ s,(cid:15),t,nσ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,n (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) ,n (cid:48)(cid:48) ˆ e ( σ (cid:48) , (cid:15) (cid:48) , τ (cid:48) , n (cid:48) ) ⊗ ˆ e ( σ (cid:48)(cid:48) , (cid:15) (cid:48)(cid:48) , τ (cid:48)(cid:48) , n (cid:48)(cid:48) ) , (4.17) satisfy the equality m s,(cid:15),t,nσ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,n (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) ,n (cid:48)(cid:48) = ( − n (cid:48) n (cid:48)(cid:48) ˆ µ s,(cid:15),t,nσ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,n (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) ,n (cid:48)(cid:48) , ˆ m σ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,n (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) ,n (cid:48)(cid:48) s,(cid:15),t,n = ( − n (cid:48) n (cid:48)(cid:48) µ σ (cid:48) ,(cid:15) (cid:48) ,τ (cid:48) ,n (cid:48) ,σ (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ,τ (cid:48)(cid:48) ,n (cid:48)(cid:48) s,(cid:15),t,n . (4.18)Then, one uses (2.12) to find the exchange relations for the Heisenberg double H ( A ). By analyticcontinuation, one can produce the (anti-)commutation relations for the generators H, ˆ H, E, F, κ, ˆ κ which are as follows: the even generators satisfy[ E, F ] = e πbH ( q − q − ) , [ H, E ] = − ibE, [ H, F ] = + ibF, [ ˆ
H, E ] = 0 , [ ˆ H, F ] = + ibF, (4.19)while κ, ˆ κ commute trivially with all even generators[ H, κ ] = [ ˆ
H, κ ] = [
E, κ ] = [
F, κ ] = 0 , [ H, ˆ κ ] = [ ˆ H, ˆ κ ] = [ E, ˆ κ ] = [ F, ˆ κ ] = 0 , (4.20)and they satisfy the following identites between each other κ = ˆ κ = − , κ = ˆ κ. (4.21)– 18 –t is worthwile to note that, taking into account the decomposition (4.1) from which we startedthis section, one can recover the (anti-)commutation relations of the discrete Heisenberg double(2.45) using the exchange relations that come from (2.12).Let us now take a look at the canonical element S of this Heisenberg double. Using thedefinition (2.17) we obtain the relation for S in terms of the basis elements (4.5) and (4.11) S = (cid:88) (cid:15) = ± (cid:88) n =0 (cid:90) d s d t ( − n e ( s, (cid:15), t, n ) ⊗ ˆ e ( s, (cid:15), t, n ) , (4.22)which written in terms of the generators has the form S = 12 e iπH ⊗ ˆ H (cid:2)(cid:0) g − ( E ⊗ F ) + g − ( E ⊗ F ) (cid:1) ⊗ i (cid:0) g − ( E ⊗ F ) − g − ( E ⊗ F ) (cid:1) κ ⊗ ˆ κ (cid:3) , (4.23)where we used equation (4.7) and the relations g R ( x ) = e R ( e πbx ) , g NS ( x ) = e NS ( e πbx ) . (4.24)The fact that the canonical element S , equation (4.23), satisfies the graded pentagon equationfollows directly from the supersymmetric pentagon identities (1.6) and has been checked explicitlyin [25]. Representations of the Heisenberg double of the Borel half of U q ( osp (1 | π : H ( A ) → Hom ( L ( R ) ⊗ C | ) of theHeisenberg double H ( A ) of the quantum superplane A that is a supersymmetric analogue of therepresentation (3.15). The generators are represented as the following operators π ( H ) = p (cid:32) (cid:33) , π ( E ) = e πb x (cid:32) (cid:33) , π ( κ ) = i (cid:32) (cid:33) ,π ( ˆ H ) = x (cid:32) (cid:33) , π ( F ) = e πb ( p − x ) (cid:32) (cid:33) , π (ˆ κ ) = i (cid:32) (cid:33) , (4.25)where [ p , x ] = πi are operators on L ( R ). The canonical element S (4.23) evaluated on therepresentation (4.25) has the form( π ⊗ π )( S ) = 12 e πi p x (cid:40)(cid:104) e − ( x + p − x ) + e − ( x + p − x ) (cid:105) (cid:32) (cid:33) ⊗ (cid:32) (cid:33) + − i (cid:104) e − ( x + p − x ) − e − ( x + p − x ) (cid:105) (cid:32) (cid:33) ⊗ (cid:32) (cid:33) (cid:41) . (4.26)This representation of the canonical element has been considered in the context of superTeichm¨uller theory as a realisation of the supersymmetric flip operator [25].– 19 – Drinfeld double
In this section, we will present the definition of the Z -graded Drinfeld double D ( A ) of a Hopfalgebra A , given in terms of the basis elements, and remind ourselves some facts about the univer-sal element R satisfying the Yang-Baxter equation. Then, we will describe an algebra morphismbetween H ( A ) ⊗ H ( A ∗ ) and D ( A ), which constitutes a Z -graded generalisation of a morphismdescribed in [10]. Furthermore, we state the relation between the universal elements of the Heisen-berg doubles and the universal R -matrix of the Drinfeld double. We present the U q ( sl (2)) and U q ( osp (1 | A , m, η, ∆ , (cid:15), γ ), subjected to the axioms (2.2)-(2.6).With the choice of a basis { e α } α ∈ I which algebraically spans A we can describe the multiplicationand co-multiplication e α e β = (cid:88) γ ∈ I m γαβ e γ , ∆( e α ) = (cid:88) β,γ ∈ I µ βγα e β ⊗ e γ , (5.1)and an antipode γ ( e α ) = (cid:88) β ∈ I γ βα e β . (5.2)In addition to the algebra A we can consider the algebra A ∗ , that is a Hopf algebra dual to A .In terms of a basis { e α } α ∈ I of A ∗ the multiplication and co-multiplication relations of this Hopfalgebra are as follows e α e β = (cid:88) γ ∈ I ( − | α || β | µ αβγ e γ , ˆ∆ op ( e α ) = (cid:88) β,γ ∈ I ( − | β || γ | m αβγ e β ⊗ e γ , (5.3)with an antipode ˆ γ ( e α ) = (cid:88) β ∈ I γ αβ e β . (5.4)They are dual to each other with respect to a duality bracket ( , ) satisfying the relations (2.7)-(2.9),and given explicitly on the basis by (2.14).Given those two Hopf algebras, it is possible to define a quasi-triangular Hopf algebra asa double cross product of Hopf algebras [8]. Explicitly, we can define a Hopf algebra D ( A ) = { x ⊗ f | x ∈ A , f ∈ A ∗ } equipped with the product( x ⊗ ( − | f | f )( y ⊗ ( − | g | g ) = (cid:88) ( y ) , ( f ) ( − | y (1) || f | + | y (2) | ( | f (1) | + | f (2) | )+ | y (3) || f (1) | + | f (2) || f (3) | ×× ( − | f (1) | ( | f (2) | + | f (3) | )+ | x | ( | y (1) | + | f (3) | )+ | g | ( | f (1) | + | x (3) | ) ×× ( y (1) , S − ( f (3) ))( y (3) , f (1) ) xy (2) ⊗ ( − | f (2) | + | g | f (2) g, (5.5)the co-product ∆( x ⊗ f ) = (cid:88) ( x ) , ( f ) ( − | f (1) || f (2) | + | x (2) || f (2) | x (1) ⊗ f (2) ⊗ x (2) ⊗ f (1) , (5.6)– 20 –nd the antipode γ ( x ⊗ f ) = ( − | x || f | (1 ⊗ ˆ γ cop ( f ))( γ ( x ) ⊗ , (5.7)where ˆ γ cop = ˆ γ − is the antipode of ( A ∗ ) cop , i.e. the antipode of the Hopf algebra dual to A equipped with the opposite coproduct. We also use the following notation for the coproduct(∆ ⊗ id )∆( x ) = (cid:88) ( x ) x (1) ⊗ x (2) ⊗ x (3) , ( ˆ∆ ⊗ id ) ˆ∆( f ) = (cid:88) ( f ) f (1) ⊗ f (2) ⊗ f (3) . Definition 5.1 A Drinfeld double of a Hopf algebra A is a quasi-triangular Hopf algebra D ( A ) with the multiplication, co-multiplication and antipode given by the equations (5.5) - (5.7) . The Hopf algebra D ( A ) has A and ( A ∗ ) cop as subalgebras through canonical embeddings( A ∗ ) cop (cid:51) f (cid:55)→ ⊗ f ∈ D ( A ) and A (cid:51) x (cid:55)→ x ⊗ ∈ D ( A ).In terms of a basis, the Drinfeld double D ( A ) is spanned by a collection of elements { e α ⊗ e β } α,β ∈ I which satisfy the following multiplication relations( e α ⊗ ⊗ e β ) = e α ⊗ e β , ( e α ⊗ e β ⊗
1) = m γαβ ( e γ ⊗ , (1 ⊗ e α )(1 ⊗ e β ) = ( − | α || β | µ αβγ (1 ⊗ e γ ) , (1 ⊗ e α )( e β ⊗
1) = ( − | µ | ( | σ | + | δ | ) m µνγ m αµδ µ (cid:15)ρβ µ σνρ ( γ − ) δ(cid:15) e σ ⊗ e γ , (5.8)and co-multiplication relations∆( e α ⊗
1) = (cid:88) β,γ ∈ I µ βγα ( e β ⊗ ⊗ ( e γ ⊗ , ∆(1 ⊗ e α ) = (cid:88) β,γ ∈ I m αγβ (1 ⊗ e β ) ⊗ (1 ⊗ e γ ) , (5.9)and is equipped with the antipode γ ( e α ⊗
1) = (cid:88) β ∈ I γ βα e β ⊗ , γ (1 ⊗ e α ) = (cid:88) β ∈ I ( γ cop ) βα ⊗ e β , (5.10)where (ˆ γ cop ) αβ = ( γ − ) αβ . Equivalently, instead of the 4th exchange relation in (5.8) one can usethe crossing relation (cid:88) γ,ρ,σ ∈ I ( − | β || σ | µ σγα m βγρ e σ ⊗ e ρ = (cid:88) γ,ρ,σ ∈ I ( − | ρ || γ | m βργ µ γσα (1 ⊗ e ρ ) ⊗ ( e σ ⊗ , (5.11)which indeed defines the same algebra. Definition 5.2
In the case of Drinfeld double one has a canonical element R ∈ D ( A ) ⊗ D ( A ) called the universal R -matrix R = (cid:88) α ∈ I ( e α ⊗ ⊗ (1 ⊗ e α ) . (5.12)– 21 – roposition 5.1 The universal R -matrix satisfies the Yang-Baxter equation R R R = R R R . (5.13) where we use a notation for which R = R ⊗ (1 ⊗ , R = (1 ⊗ ⊗ R and R = (cid:80) α ∈ I e α ⊗ (1 ⊗ ⊗ e α . The Drinfeld double D ( A ) can be related to the Heisenberg algebras in terms of an algebramorphism. Lets consider again the Heisenberg double H ( A ), as described in section 2. Let usrecall that one can regard it as an algebra of elements { e β ⊗ e α } α,β ∈ I subjected to the set ofrelations( e α ⊗ e β )( e γ ⊗ e δ ) = (cid:88) (cid:15),π,ρ,σ,τ ∈ I ( − | β || γ | + | π || (cid:15) | + | π || α | + | (cid:15) | m γπ(cid:15) m τρδ µ (cid:15)ρβ µ απσ e σ ⊗ e τ . (5.14)Moreover, we can also construct an additional Heisenberg double H ( A ∗ ) = { ˜ e α ⊗ ˜ e β } α,β ∈ I startingfrom the dual algebra A ∗ . It has the following relations(˜ e α ⊗ ˜ e β )(˜ e γ ⊗ ˜ e δ ) = (cid:88) (cid:15),π,ρ,σ,τ ∈ I ( − | ρ || π | + | ρ || (cid:15) | + | π || δ | µ ρ(cid:15)γ µ πδτ m β(cid:15)π m σαρ ˜ e σ ⊗ ˜ e τ . (5.15)We will denote the flipped (i.e. the one with the tensor factors reversed) canonical element of thisHeisenberg double as ˜ S = ˜ e α ⊗ ˜ e α . It satisfies a “reversed” pentagon equation of the form˜ S ˜ S = ˜ S ˜ S ˜ S . (5.16)We can relate those 2 algebras by the means of the following proposition: Proposition 5.2
There exists an algebra anti-isomorphism ξ : H ( A ∗ ) → H ( A ) given by ξ (˜ e α ) = ( − c | α | γ βα e β , ξ (˜ e α ) = ( − ( c +1) | α | ( γ − ) αβ e β , (5.17) where c = 0 , . The anti-isomorphism can be implemented on representation spaces in terms of super-transposition(i.e. the graded analogue of ordinary transposition). The super-transposition for square even( n | m )-matrices, i.e. linear transformations belonging to the space of Hom ( C ( n | m ) , C ( n | m ) ), isgiven by (cid:32) A BC D (cid:33) st = (cid:32) A t C t − B t D t (cid:33) , (5.18)where A ∈ Hom ( C n , C n ) , D ∈ Hom ( C m , C m ) are even, and B ∈ Hom ( C m , C n ) , C ∈ Hom ( C n , C m )are odd, and t denotes ordinary, not graded matrix transposition. Moreover, the transpositionon L ( R ) is implemented by the following action on the momentum and position operators: p t = − p, q t = q .Then, we claim that the following proposition is true:– 22 – roposition 5.3 A map η : D ( A ) → H ( A ) ⊗ H ( A ∗ ) defined as follows η ( a,b ) ( e α ⊗
1) = ( − a | β | + b | γ | µ βγα e β ⊗ ˜ e γ , η ( a,b ) (1 ⊗ e α ) = ( − a (cid:48) | β | + b (cid:48) | γ | m αγβ e β ⊗ ˜ e γ , (5.19) for the choice of the parameters a, a (cid:48) , b, b (cid:48) ∈ Z ≥ such that ( − a + a (cid:48) = − , ( − b + b (cid:48) = 1 , (5.20) is an algebra homomorphism. This fact can be checked by a direct calculation.Using the morphism η from the proposition 5.3 in addition to the canonical element S forthe Heisenberg double H ( A ) and the (flipped) canonical element ˜ S for H ( A ∗ ), one can define thefollowing two elements S (cid:48) = ( − ( a + b (cid:48) +1) | α | ˜ e α ⊗ e α and S (cid:48)(cid:48) = ( − ( a + b (cid:48) ) | α | e α ⊗ ˜ e α . It can be shownthat they satisfy a set of 6 pentagon-like equations S (cid:48) S (cid:48) S = S S (cid:48) , ˜ S S (cid:48) = S (cid:48) S (cid:48) ˜ S ,S S (cid:48)(cid:48) S (cid:48)(cid:48) = S (cid:48)(cid:48) S , S (cid:48)(cid:48) ˜ S = ˜ S S (cid:48)(cid:48) S (cid:48)(cid:48) , (5.21) S (cid:48) ˜ S S (cid:48)(cid:48) = S (cid:48)(cid:48) S (cid:48) , S (cid:48)(cid:48) S (cid:48) = S (cid:48) S S (cid:48)(cid:48) . Then, we claim that one can construct the R-matrix of the Drinfeld double D ( A ) as follows Proposition 5.4
Under an algebra map η one has the following relation ( η ( a,b ) ⊗ η ( a,b ) ) R = S (cid:48)(cid:48) S ˜ S S (cid:48) . (5.22) Remark 5.1
To keep the notation compact, from now on we will denote the elements ⊗ e α and e α ⊗ of the Drinfeld double D ( A ) simply as e α and e α respectively. Example 5.1
Drinfeld double of the Borel half of U q ( sl (2))Using the definitions above, one can obtain the Drinfeld double D ( A ) commutation relationsfor A = B ( U q ( sl (2)))[ H, E ] = − ibE, [ H, F ] = + ibF, [ ˆ
H, E ] = − ibE, [ ˆ H, F ] = + ibF, [ H, ˆ H ] = 0 , [ E, F ] = ( q − q − )( e πbH − e − πb ˆ H ) , (5.23)with the coproduct ∆( H ) = H ⊗ ⊗ H, ∆( E ) = E ⊗ e πbH + 1 ⊗ E, ∆( ˆ H ) = ˆ H ⊗ ⊗ ˆ H, ∆( F ) = F ⊗ e − πb ˆ H + 1 ⊗ F, (5.24)and the antipode γ ( H ) = − H, γ ( E ) = qEe − πbH ,γ ( ˆ H ) = − ˆ H, γ ( F ) = qF e πb ˆ H . (5.25)– 23 –sing the map η as well as the algebra anti-homomorphism ξ , one can obtain a representation π D ( A ) : D ( A ) → L ( R ) ⊗ from the representation (3.15) π D ( A ) ( H ) = p + p , π D ( A ) ( E ) = e πb ( p + x ) + e πb ( x + p ) ,π D ( A ) ( ˆ H ) = x − x , π D ( A ) ( F ) = e − πb ( x + p ) + e πb ( p − x ) , (5.26)where p , x are the momentum and position operators satisfying [ p , x ] = πi . Moreover, the uni-versal R -matrix realised using this representation is given by( π D ( A ) ⊗ π D ( A ) ) R = e πi ( p + p )( x − x ) g − ( e πb ( x + p − x − p ) ) g − ( e πb ( x + p + p − x ) ) ×× g − ( e πb ( p + x − x − p ) ) g − ( e πb ( p + x + p − x ) ) . (5.27) Example 5.2
Drinfeld double of the Borel half of U q ( osp (1 | D ( A ) for A = B ( U q ( osp (1 | H, v (+) ] = − ibv (+) , [ H, v ( − ) ] = + ibv ( − ) , [ ˆ H, v (+) ] = − ibv (+) , [ ˆ H, v ( − ) ] = + ibv ( − ) , [ H, ˆ H ] = 0 , { v (+) , v ( − ) } = ( q + q − )( e πbH − e − πb ˆ H ) . (5.28)with the coproduct∆( H ) = H ⊗ ⊗ H, ∆( v (+) ) = v (+) ⊗ e πbH + 1 ⊗ v (+) , ∆( ˆ H ) = ˆ H ⊗ ⊗ ˆ H, ∆( v ( − ) ) = v ( − ) ⊗ e − πb ˆ H + 1 ⊗ v ( − ) , (5.29)and the antipode γ ( H ) = − H, γ ( v (+) ) = q v (+) e − πbH ,γ ( ˆ H ) = − ˆ H, γ ( v ( − ) ) = q − v ( − ) e πb ˆ H . (5.30)Using the map η as well as the algebra anti-homomorphism ξ , one can obtain a representation π D ( A ) : D ( A ) → L ( R ) ⊗ ⊗ ( C | ) ⊗ from the representation (4.25) π ( u,v ) D ( A ) ( H ) = ( p + p ) I ,π ( u,v ) D ( A ) ( v (+) ) = ( − u i (cid:34) e πb ( x + p ) (cid:32) (cid:33) ⊗ I + ( − v e πb ( p + x ) I ⊗ (cid:32) − (cid:33)(cid:35) ,π ( u,v ) D ( A ) ( ˆ H ) = ( x − x ) I ,π ( u,v ) D ( A ) ( v ( − ) ) = ( − u +1 i (cid:34) e πb ( p − x ) (cid:32) (cid:33) ⊗ I + ( − v e − πb ( x + p ) I ⊗ (cid:32) − (cid:33)(cid:35) , (5.31)where u, v = 0 , p , x are the momentum and position operators satisfying[ p , x ] = πi and the (1 | I = (cid:32) (cid:33) . Moreover, the– 24 –reviously not obtained universal R -matrix realised using this representation is given by (cid:16) π ( u,v ) D ( A ) ⊗ π ( u,v ) D ( A ) (cid:17) R = 116 e πi ( p + p )( x − x ) ×× (cid:20) h + ( x + p − x − p )) I ⊗ + ( − v ih − ( x + p − x − p ) (cid:18) (cid:19) ⊗ I ⊗ I ⊗ (cid:18) − (cid:19)(cid:21) ×× (cid:20) h + ( x + p + p − x )) I ⊗ − ih − ( x + p + p − x ) (cid:18) (cid:19) ⊗ I ⊗ (cid:18) (cid:19) ⊗ I (cid:21) ×× (cid:20) h + ( p + x − x − p )) I ⊗ + ih − ( p + x − x − p ) I ⊗ (cid:18) − (cid:19) ⊗ I ⊗ (cid:18) − (cid:19)(cid:21) ×× (cid:20) h + ( p + x + p − x )) I ⊗ + ( − v ih − ( p + x + p − x ) I ⊗ (cid:18) − (cid:19) ⊗ (cid:18) (cid:19) ⊗ I (cid:21) , (5.32) where h ± ( x ) = e − ( x ) ± e − ( x ). Let us conclude our paper by mentioning some interesting directions for future work suggestedby our results.As we mentioned in the introduction, a 1-parameter family of infinite dimensional represen-tations P α , α ∈ ( b + b − ) + i R of U q ( sl (2)) , q = e iπb has been studied by Bytsko, Ponsot, andTeschner [18, 19, 27] in connection to the Liouville theory. P α represents U q ( sl (2)) on a space ofanalytic function defined on a strip around { x ∈ C : | Im( x ) | < b } which possess a Fourier trans-form that is meromorphic on C with a specified set of allowed poles. The generators K, E, F ofthe quantum group are realised as positive, self-adjoint operators. The family P α is closed underthe the tensor product and the calculation of the 3j- and 6j-symbols allowed to make a connectionwith the fusion matrices of the Liouville theory. Moreover, on the same representation space actsa representation of U ˜ q ( sl (2)) for ˜ q = e iπb , and therefore P α constitute representations of themodular double of the quantum group introduced in a sense of Faddeev [28]. This modularityproperty, which ensures a self-duality of the exchange b → b , is crucial for the interpretation interms of Liouville theory, which exhibits the same symmetry. The representations P α were alsofound in during the study of the spectral problem of Dehn twists in quantum Teichm¨uller theoryutilising an algebra map similar to the one in proposition 5.3 — it differed form it however by atwisting of the co-product.This results for the class of infinite-dimensional representations P α of the quantum group U q ( sl ) has been generalised in [29, 30] for the case of the higher rank U q ( sl n +1 ) quantum groupsusing the cluster algebras methods. They describe the algebraic ingredients of a proof of theconjecture of Frenkel and Ip [31] that the category of the representations P λ of the quantumgroup U q ( sl n +1 ) is closed under tensor products.In the context of N = 1 supersymmetric Liouville theory, the attempt to find an U q ( osp (1 | P α resulted obtaining the 6j-symbols which reproduce only theNeveu-Schwarz sector of the theory [32, 33]. Given our results regarding the Heisenberg double ofthe Borel half of U q ( osp (1 | P α ,i.e. the one which will encode the entirety of the structure of N = 1 supersymmetric Liouville– 25 –heory, is a 1-parameter family of infinite-dimensional representations of U q ( osp (1 | L ( R ) ⊗ ( C | ) ⊗ and given by π ( u,v ) α ( K ) = e πb p I ⊗ I ,π ( u,v ) α ( v ( ± ) ) = ( − u iq ± (cid:34) e πb ( ∓ x + p ± α ) (cid:32) (cid:33) ⊗ I ± ( − v e πb ( ∓ x − p ∓ α ) I ⊗ (cid:32) (cid:33)(cid:35) , (6.1)where u, v = 0 , q = exp[ iπb ] and where the momentum and positionoperators p , x acting on L ( R ) satisfy [ p , x ] = πi . π α is a representation of an U q ( osp (1 | Kv ( ± ) = q ± v ( ± ) K, { v (+) , v ( − ) } = ( q + q − )( K − K − ) , (6.2)and the co-product ∆( K ) = K ⊗ K, ∆( v ( ± ) ) = v ( ± ) ⊗ K + K − ⊗ v ( ± ) . (6.3)The novel R -matrix for this family of representations is given by( π ( u,v ) α ⊗ π ( u,v ) β ) R = 116 F − e iπ ( p + α )( p − β ) S S S S F − , (6.4)where F = exp (cid:26) iπ p + α )( p − β ) − ( p − α )( p + β )] (cid:27) ,S = h + ( p + x + p + x )) I ⊗ − ( − v ih − ( p + x + p + x ) (cid:32) (cid:33) ⊗ I ⊗ I ⊗ (cid:32) (cid:33) ,S = h + ( p + x + p + β )) I ⊗ − ih − ( p + x + p + β ) (cid:32) (cid:33) ⊗ I ⊗ (cid:32) (cid:33) ⊗ I ,S = h + ( x + α + p + x )) I ⊗ + ih − ( x + α + p + x ) I ⊗ (cid:32) (cid:33) ⊗ I ⊗ (cid:32) (cid:33) ,S = h + ( x + α + p + β )) I ⊗ + ( − v ih − ( x + α + p + β ) I ⊗ (cid:32) (cid:33) ⊗ (cid:32) (cid:33) ⊗ I . It would be interesting to perform a harmonic analysis for this family of representations andcalculate the 3j- and 6j-symbols. In particular, of interest is whether 6j-symbols could reproducethe entire fusion matrix of the N = 1 supersymmetric Liouville theory.– 26 – cknowledgments We are very grateful to J¨org Teschner for explanations, suggestions and many helpful discussionsand comments. We also thank Rinat Kashaev for stimulating discussions.This project was initiated when NA and MP was supported by the German Science Founda-tion (DFG) within the Research Training Group 1670 ”Mathematics Inspired by String Theoryand QFT”. The work of N.A. was supported by Max Planck Institute of Mathematics (MPIM)inBonn. The work of M.P. was supported by the European Research Council (advanced grantNuQFT). We also thank and acknowledges the AEC centre at University of Bern and IHES fortheir hospitality during this project.
A Special functions
Quantum dilogarithm plays a key role in the constructions described in this paper. In thisappendix we review the Faddeev’s quantum dilogarithm and its most important properties. Wecollected the different definitions of related special functions which one may face in the references.
A.1 Faddeev’s quantum dilogarithm
The basic special function that appears in the context of the infinite dimensional representationsof the Heisenberg double of the quantum plane is Barnes’ double gamma function [34]. The doublegamma function is defined aslog Γ ( z | ω ) := ∂∂s (cid:88) m ,m ∈ Z ≥ ( z + m ω + m ω ) − s s =0 , and using which one can define Γ b ( x ) := Γ ( x | b , b − ) . For Re x > b ( x ) = (cid:90) ∞ d tt e − xt − e − Q t (1 − e − t b )(1 − e − t b ) − (cid:16) Q − x (cid:17) e t − Q − xt , (A.1)where Q = b + . One can analytically continue Γ b to a meromorphic function defined on theentire complex plane C . The most important property of Γ b is its behavior with respect to shiftsby b ± , Γ b ( x + b) = √ π b b x − Γ b ( bx ) Γ b ( x ) , Γ b ( x + b − ) = √ π b − b x + Γ b ( x b ) Γ b ( x ) . (A.2)These shift equation allows us to calculate residues of the poles of Γ b . When x →
0, for instance,one finds Γ b ( x ) = Γ b ( Q )2 πx + O (1) . (A.3)– 27 –rom Barnes’ Double Gamma function we can build other important special functions,Υ b ( x ) := 1Γ b ( x )Γ b ( Q − x ) , (A.4) S b ( x ) := Γ b ( x )Γ b ( Q − x ) , (A.5) G b ( x ) := e − iπ x ( Q − x ) S b ( x ) , (A.6) w b ( x ) := e πi ( Q + x ) G b ( Q − ix ) , (A.7) g b ( x ) := ζ b G b ( Q + πi b log x ) , (A.8) e b ( x ) := ζ b G b ( Q − ix ) , (A.9)where ζ b = exp[ − iπ − iπ (b + b − )] One refers to the function S b as double sine function. It isdefined by the following integral representationlog S b ( x ) = (cid:90) ∞ d tit (cid:18) sin 2 xt t sinh b − t − xt (cid:19) . (A.10)The S b function is meromorphic with poles and zeros in S b ( x ) = 0 ⇔ x = Q + n b + m b − , n, m ∈ Z ≥ ,S b ( x ) − = 0 ⇔ x = − n b − m b − , n, m ∈ Z ≥ . Other most important properties are as follows:Functional equation(Shift): S b ( x − i b /
2) = 2 cosh ( π b x ) S b ( x + i b /
2) (A.11)Self-duality: S b ( x ) = S / b ( x ) (A.12)Inversion relation(Reflection): S b ( x ) S b ( − x ) = 1 (A.13)Unitarity: S b ( x ) = 1 /S b ( x ) (A.14)Residue: res x = iQ/ S b ( x ) = e − iπ (1+ Q ) (2 πi ) − . (A.15)From the relation between the special functions and the shift property of Barnes’ double Gammafunction it is easy to derive the following shift and reflection properties of G b , G b ( x + b) = (1 − e πi b x ) G b ( x ) , (A.16) G b ( x ) G b ( Q − x ) = e πix ( x − Q ) . (A.17)We also need to the asymptotic behavior of the function G b along the imaginary axis, G b ( x ) ∼ ζ b , Im x → + ∞ ,G b ( x ) ∼ ζ − e iπx ( x − Q ) , Im x → −∞ . (A.18)– 28 –he Fadeev’s quantum dilogarithm function in addition to the relation with the G b functionhas the following integral representation g b (cid:18) π b log x (cid:19) = exp (cid:20)(cid:90) R + i d w e − ixw w b) sinh( w/ b) (cid:21) , (A.19)and g b ( e π b r ) = (cid:90) d t e πitr e − iπt G b ( Q + it ) ,g − ( e π b r ) = (cid:90) d t e πitr e − πtQ G b ( Q + it ) , The shift and reflection relations that it satisfies are as follows g b ( e − iπ b x ) = (1 + x ) g b ( e + iπ b x ) ,g b ( e πi b x ) g b ( e − πi b x ) = e iπQ ζ e iπx . Also, for non-commutative variables
U, V such that
U V = q V U where q = e iπ b it satisfies thepentagon relation g b ( U ) g b ( V ) = g b ( V ) g b ( q − U V ) g b ( U ) . (A.20)The pentagon equation can be equivalently expressed as the Ramanujan summation formula[19, 35, 36] (cid:90) i ∞− i ∞ d τi e πiτβ G b ( τ + α ) G b ( τ + Q ) = G b ( α ) G b ( β ) G b ( α + β ) . (A.21)Moreover, for the function e b we have the following shift and reflection relations e b (cid:18) x − i b ± (cid:19) = (1 + e π b ± x ) e b (cid:18) x + i b ± (cid:19) , (A.22) e b ( x ) e b ( − x ) = e − iπ (1 − Q / / e iπx . (A.23)The asymptotic behaviour of the function e b along the real axis e b ( x ) = (cid:40) , x → −∞ e − iπ (1 − Q / / e iπx , x → + ∞ (A.24)Also, we know that for self-adjoint operators p , x such that [ p , x ] = πi we have the followingvariant of the pentagon relation e b ( p ) e b ( x ) = e b ( x ) e b ( x + p ) e b ( p ) . (A.25) A.2 Supersymmetric non-compact quantum dilogarithm
Now, we will consider special functions related to the supersymmetric analogue of the Faddeev’squantum dilogarithm. We can define the supersymmetric analogues of double gamma functionsΓ NS ( x ) = Γ b (cid:16) x (cid:17) Γ b (cid:18) x + Q (cid:19) , Γ R ( x ) = Γ b (cid:18) x + b2 (cid:19) Γ b (cid:18) x + b − (cid:19) . – 29 –urthermore, let us define S NS ( x ) = Γ NS ( x )Γ NS ( Q − x ) , G NS ( x ) = ζ e − iπ x ( Q − x ) S NS ( x ) ,S R ( x ) = Γ R ( x )Γ R ( Q − x ) , G R ( x ) = e − iπ ζ e − iπ x ( Q − x ) S R ( x ) , (A.26)where ζ = exp( − iπQ / S b , the functions S R ( x ) and S NS ( x ) are meromorphic withpoles and zeros in S R ( x ) = 0 ⇔ x = Q + n b + m b − , n, m ∈ Z ≥ , m + n ∈ Z + 1 ,S NS ( x ) = 0 ⇔ x = Q + n b + m b − , n, m ∈ Z ≥ , m + n ∈ Z ,S R ( x ) − = 0 ⇔ x = − n b − m b − , n, m ∈ Z ≥ , m + n ∈ Z + 1 ,S NS ( x ) − = 0 ⇔ x = − n b − m b − , n, m ∈ Z ≥ , m + n ∈ Z . We state the shift and reflection properties of the functions G NS and G R G R ( x + b ± ) = (1 − e πi b ± x ) G NS ( x ) , (A.27) G NS ( x + b ± ) = (1 + e πi b ± x ) G R ( x ) , (A.28) G R ( x ) G R ( Q − x ) = e − iπ ζ e πi x ( x − Q ) , (A.29) G NS ( x ) G NS ( Q − x ) = ζ e πi x ( x − Q ) . (A.30)We define the supersymmetric analogues of Faddeev’s quantum dilogarithm function as e R ( x ) = e b (cid:18) x i − b − ) (cid:19) e b (cid:18) x − i − b − ) (cid:19) ,e NS ( x ) = e b (cid:18) x i − ) (cid:19) e b (cid:18) x − i − ) (cid:19) . (A.31)and relate them to the double sine function in a way as follows e R ( x ) = ζ G R ( − ix + Q ) ,e NS ( x ) = ζ G NS ( − ix + Q ) . (A.32)In addition, the functions e R and e NS have an integral representation e − ( r ) + e − ( r ) = ζ (cid:90) d te πitr e − πtQ G NS ( Q + it ) ,e − ( r ) − e − ( r ) = ζ (cid:90) d te πitr e − πtQ G R ( Q + it ) . (A.33)– 30 –he shift and reflection relations that they satisfy are as follows e R (cid:18) x − i b ± (cid:19) = (1 + ie π b ± x ) e NS (cid:18) x + i b ± (cid:19) ,e NS (cid:18) x − i b ± (cid:19) = (1 − ie π b ± x ) e R (cid:18) x + i b ± (cid:19) ,e NS ( x ) e NS ( − x ) = e − iπQ / e − iπ (1 − Q / / e iπx / ,e R ( x ) e R ( − x ) = e iπ/ e − iπQ / e − iπ (1 − Q / / e iπx / . Asymptotically, the functions e NS and e R behave as e NS ( x ) = (cid:40) , x → −∞ e − iπQ / e − iπ (1 − Q / / e iπx / , x → + ∞ (A.34) e R ( x ) = (cid:40) , x → −∞ e iπ/ e − iπQ / e − iπ (1 − Q / / e iπx / , x → + ∞ (A.35)Also, we know that for self-adjoint operators p , x such that [ p , x ] = πi they satisfy four pentagonrelations f + ( p ) f + ( x ) = f + ( x ) f + ( x + p ) f + ( p ) − if − ( x ) f − ( x + p ) f − ( p ) , (A.36a) f + ( p ) f − ( x ) = − if + ( x ) f − ( x + p ) f − ( p ) + f − ( x ) f + ( x + p ) f + ( p ) , (A.36b) f − ( p ) f + ( x ) = f + ( x ) f + ( x + p ) f − ( p ) − if − ( x ) f − ( x + p ) f + ( p ) , (A.36c) f − ( p ) f − ( x ) = if + ( x ) f − ( x + p ) f + ( p ) − f − ( x ) f + ( x + p ) f − ( p ) , (A.36d)where f ± ( x ) = e R ( x ) ± e NS ( x ). The equations (A.36) are equivalent to the supersymmetricanalogues of the Ramanujan integral identities [37] (cid:90) i ∞− i ∞ d τi e πiτβ (cid:20) G R ( τ + α ) G NS ( τ + Q ) + G NS ( τ + α ) G R ( τ + Q ) (cid:21) = 2 ζ − G R ( α ) G NS ( β ) G R ( α + β ) , (cid:90) i ∞− i ∞ d τi e πiτβ (cid:20) G NS ( τ + α ) G NS ( τ + Q ) + G R ( τ + α ) G R ( τ + Q ) (cid:21) = 2 ζ − G NS ( α ) G NS ( β ) G NS ( α + β ) , (cid:90) i ∞− i ∞ d τi e πiτβ (cid:20) G R ( τ + α ) G NS ( τ + Q ) − G NS ( τ + α ) G R ( τ + Q ) (cid:21) = 2 ζ − G R ( α ) G R ( β ) G NS ( α + β ) , (cid:90) i ∞− i ∞ d τi e πiτβ (cid:20) G NS ( τ + α ) G NS ( τ + Q ) − G R ( τ + α ) G R ( τ + Q ) (cid:21) = 2 ζ − G NS ( α ) G R ( β ) G NS ( α + β ) . (A.37)Finally, we define g R ( x ) = e R ( e πbx ) , g NS ( x ) = e NS ( e πbx ) . (A.38)– 31 – .3 Binomial and q-binomial identites The ordinary binomial and q-binomial formule have a continuous analogues. The continuousbinomial formula for x, y ∈ R ≥ is given by( x + y ) is = (cid:90) d t π (cid:18) − is − it (cid:19) Γ y it x i ( s − t ) , ( x − y ) is = (cid:90) d t π (cid:18) − is − it (cid:19) Γ e ∓ πt y it x i ( s − t ) , ( − x − y ) is = e ∓ πs (cid:90) d t π (cid:18) − is − it (cid:19) Γ y it x i ( s − t ) , (A.39)where s ∈ R , the sign depends on the choice of the branch of the logarithm and the continuousbinomial coefficient is given explicitly in terms of gamma functions (cid:18) st (cid:19) Γ = Γ( t )Γ( s − t )Γ( s ) . (A.40)Moreover, for the non-commutative elements u, v , which satisfy uv = q vu for q = e iπ b , one hasthe countinuous q-binomial formula( u + v ) it = b (cid:90) d τ (cid:18) tτ (cid:19) b u i ( t − τ ) v iτ , (A.41)where t ∈ R and a continuous version of the q-binomial coefficient has the form (cid:18) tτ (cid:19) b = G b ( − τ ) G b ( − t + τ ) G b ( − t ) . (A.42) B Toy model: continuous monomial algebra
As an instructive case on the road to the analysis of the continuous version of U q ( sl (2)) and U q ( osp (1 | { y n } n ∈ N with the multiplication and co-multiplication relations as follows y n y m = y n + m , ∆( y m ) = ( y + y ) n . Where it comes to the continuous case, let us start by considering a Hopf algebra A composed ofthe basis elements { e ( s, (cid:15) ) } s ∈ R ,(cid:15) = ± given as follows e ( s, +) = 12 π Γ( − is ) e πs ( y ) ispv , (B.1) e ( s, − ) = 12 π Γ( − is ) e πs ( − y ) ispv , (B.2)– 32 –here we use the following definitions of the principal value prescription for yy ispv = | y | is Θ( y ) + e − πs | y | is Θ( − y ) , ( − y ) ispv = | y | is Θ( − y ) + e − πs | y | is Θ(+ y ) , and where Θ is the step function. This definition is dictated by the fact that we do assume that y is not a positive operator, and therefore its complex power needs to be made well-defined.For the continuous version of the algebra of monomials A the multiplication is as follows e ( s, +) e ( s (cid:48) , +) = 12 π (cid:18) − i ( s + s (cid:48) ) − is (cid:19) Γ e ( s + s (cid:48) , +) ,e ( s, − ) e ( s (cid:48) , − ) = 12 π (cid:18) − i ( s + s (cid:48) ) − is (cid:19) Γ e ( s + s (cid:48) , − ) ,e ( s, +) e ( s (cid:48) , − ) = 12 π (cid:18) − i ( s + s (cid:48) ) − is (cid:19) Γ (cid:34) e − πs (cid:48) (1 − e − πs )1 − e − π ( s + s (cid:48) ) e ( s + s (cid:48) , +) + e − πs (1 − e − πs (cid:48) )1 − e − π ( s + s (cid:48) ) e ( s + s (cid:48) , − ) (cid:35) , = e ( s (cid:48) , − ) e ( s, +) , and the co-multiplication ∆( e ( s, +)) = (cid:90) d τ e ( τ, +) ⊗ e ( s − τ, +) , ∆( e ( s, − )) = (cid:90) d τ e ( τ, − ) ⊗ e ( s − τ, − ) . In addition to algebra A we consider also a dual Hopf algebra A ∗ composed of the basiselements { ˆ e ( s, (cid:15) ) } s ∈ R ,(cid:15) = ± defined by the equationsˆ e ( s, +) = | ˆ y | is Θ(ˆ y ) , (B.3)ˆ e ( s, − ) = | ˆ y | is Θ( − ˆ y ) . (B.4)The multiplication of these elements areˆ e ( s, +)ˆ e ( s (cid:48) , +) = ˆ e ( s + s (cid:48) , +) , ˆ e ( s, − )ˆ e ( s (cid:48) , − ) = ˆ e ( s + s (cid:48) , − ) , ˆ e ( s, − )ˆ e ( s (cid:48) , +) = 0 , ˆ e ( s, +)ˆ e ( s (cid:48) , − ) = 0 , and co-multiplicationˆ∆(ˆ e ( s, +)) = (cid:90) d τ π (cid:18) − is − it (cid:19) Γ (cid:34) ˆ e ( τ, +) ⊗ ˆ e ( s − τ, +) + ˆ e ( τ, +) ⊗ ˆ e ( s − τ, − ) e − π ( s − τ ) (1 − e − πτ )1 − e − πs ++ˆ e ( τ, − ) ⊗ ˆ e ( s − τ, +) e − πτ (1 − e − π ( s − τ ) )1 − e − πs (cid:35) , ˆ∆(ˆ e ( s, − )) = (cid:90) d τ π (cid:18) − is − it (cid:19) Γ (cid:34) ˆ e ( τ, − ) ⊗ ˆ e ( s − τ, − ) + ˆ e ( τ, +) ⊗ ˆ e ( s − τ, − ) e − πτ (1 − e − π ( s − τ ) )1 − e − πs ++ˆ e ( τ, − ) ⊗ ˆ e ( s − τ, +) e − π ( s − τ ) (1 − e − πτ )1 − e − πs (cid:35) . – 33 –y observation one can clearly see that the coefficients of multiplication and co-multiplication forthe algebras A and A ∗ defined by the following equations e ( s, (cid:15) ) e ( s (cid:48) , (cid:15) (cid:48) ) = (cid:88) (cid:15) (cid:48)(cid:48) (cid:90) R d σ m σ,(cid:15) (cid:48)(cid:48) s,(cid:15) ; s (cid:48) ,(cid:15) (cid:48) e ( σ, (cid:15) (cid:48)(cid:48) ) , ∆( e ( s, (cid:15) )) = (cid:88) (cid:15) (cid:48) ,(cid:15) (cid:48)(cid:48) (cid:90) R d σ d σ (cid:48) µ σ,(cid:15) (cid:48) ; σ (cid:48) ,(cid:15) (cid:48)(cid:48) s,(cid:15) e ( σ, (cid:15) (cid:48) ) ⊗ e ( σ (cid:48) , (cid:15) (cid:48)(cid:48) ) , ˆ e ( s, (cid:15) )ˆ e ( s (cid:48) , (cid:15) (cid:48) ) = (cid:88) (cid:15) (cid:48)(cid:48) (cid:90) R d σ ˆ m s,(cid:15) ; s (cid:48) ,(cid:15) (cid:48) σ,(cid:15) (cid:48)(cid:48) ˆ e ( σ, (cid:15) (cid:48)(cid:48) ) , ˆ∆(ˆ e ( s, (cid:15) )) = (cid:88) (cid:15) (cid:48) ,(cid:15) (cid:48)(cid:48) (cid:90) R d σ d σ (cid:48) ˆ µ s,(cid:15)σ,(cid:15) (cid:48) ; σ (cid:48) ,(cid:15) (cid:48)(cid:48) ˆ e ( σ, (cid:15) (cid:48) ) ⊗ ˆ e ( σ (cid:48) , (cid:15) (cid:48)(cid:48) ) , satisfy the relations m σ,(cid:15) s,(cid:15) ; s (cid:48) ,(cid:15) = ˆ µ σ,(cid:15) s,(cid:15) ; s (cid:48) ,(cid:15) , ˆ m s,(cid:15) ; s (cid:48) ,(cid:15) σ,(cid:15) = µ s,(cid:15) ; s (cid:48) ,(cid:15) σ,(cid:15) , i.e. we see that those two algebras are indeed dual to each other. This allows us to define themultiplication relations between the elements of the Heisenberg algebra H ( A )(1 ⊗ e ( s, ± ))(ˆ e ( s (cid:48) , ± ) ⊗
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