Representations of Quantum Affine Algebras in their R -Matrix Realization
aa r X i v : . [ m a t h . R T ] A ug Representations of quantum affine algebrasin their R -matrix realization Naihuan Jing, Ming Liu and Alexander Molev
Abstract
We use the isomorphisms between the R -matrix and Drinfeld presentations of the quan-tum affine algebras in types B, C and D produced in our previous work to describe finite-dimensional irreducible representations in the R -matrix realization. We also review the iso-morphisms for the Yangians of these types and use Gauss decomposition to establish anequivalence of the descriptions of the representations in the R -matrix and Drinfeld presen-tations of the Yangians. The Yangians and quantum affine algebras associated with simple Lie algebras comprise tworemarkable families of infinite-dimensional quantum groups, as introduced by V. Drinfeld [9]and M. Jimbo [17]. Both families have since found numerous connections with many areas inmathematics and physics, they possess rich and versatile representation theory.Finite-dimensional irreducible representations of the Yangians were classified by Drinfeld inhis paper [11] by relying on the pioneering work by V. Tarasov [27], [28]. The classification in[11] uses a new presentation of the Yangians and quantum affine algebras which is now oftenreferred to as the
Drinfeld presentation and which was used by V. Chari and A. Pressley toclassify finite-dimensional irreducible representations of quantum affine algebras [6, Ch. 12].A different kind of presentations of these algebras known as R -matrix presentations goesback to the work of the Leningrad school of L. Faddeev; see e.g. [12], [23], [25] and [26]. Inaccordance to Drinfeld [10], such presentations can be produced from the universal R -matrixassociated with a quantum group and they can be associated with arbitrary finite-dimensionalrepresentations. This approach was developed further in a more recent work [29].Explicit isomorphisms between the R -matrix and Drinfeld presentations of the Yangians intype A were given in the original work [11], while detailed proofs were produced by J. Brundanand A. Kleshchev [5]. An analogous isomorphism for the type A quantum affine algebras is dueto J. Ding and I. Frenkel [8].Despite of their importance in representation theory and applications, such isomorphisms hadremained unknown beyond type A until a recent work [15], [19], [20] and [21], where they wereproduced for the remaining classical types B , C and D .1ur goal in this paper is to give a brief review of Yangians and quantum affine algebrasin types B , C and D and apply the isomorphisms to describe finite-dimensional irreduciblerepresentations of these algebras in their R -matrix realization.In the case of Yangians, such a description was already given in [1], so that the isomorphismsconnect two sides of the representation theory and explain how additional symmetries of the rep-resentation parameters arise from the Gauss decomposition of the generator matrices. However,for the R -matrix realization of the quantum affine algebras the isomorphisms are essential toget a parametrization of their finite-dimensional irreducible representations. A key step in ourarguments relies on a consistency property for the triangular decomposition of the algebra andthe Gauss decomposition of the generator matrices, which we establish in Sec. 3.2.4 by derivingit from the defining relations. An alternative way can rely on explicit formulas for the universal R -matrices which, however, leads to more involved calculations. This property has been knownfor the quantum affine algebras of type A [22], so that the isomorphism of [8] connects thedescriptions of the representations given in [6, Sec. 12.2] and [14].In the Appendix we give a modified version of the Yangian isomorphisms produced in [19],which is based on the opposite Gauss decomposition of the generator matrix. This version canbe applied to make an alternative connection between the parameters of representations in thetwo realizations of the Yangians.We acknowledge the support of the Australian Research Council, grant DP180101825. B , C and D We will denote by g one of the simple Lie algebras of type B n , C n or D n . That is, g = o N iseither the orthogonal Lie algebra with N = 2 n + 1 and N = 2 n , or g = sp N is the symplecticLie algebra with N = 2 n . Choose simple roots in the form α i = ǫ i − ǫ i +1 for i = 1 , . . . , n − , (2.1)and α n = ǫ n if g = o n +1 ǫ n if g = sp n ǫ n − + ǫ n if g = o n , where ǫ , . . . , ǫ n is an orthonormal basis of a Euclidian space with the inner product ( · , · ) . TheCartan matrix [ A ij ] for g is given by A ij = 2( α i , α j )( α i , α i ) . (2.2)We will use the notation r i = 12 ( α i , α i ) , i = 1 , . . . , n. (2.3)2s defined in [11], the Drinfeld Yangian Y D ( g ) is generated by elements κ ir , ξ + ir and ξ − ir with i = 1 , . . . , n and r = 0 , , . . . subject to the defining relations [ κ ir , κ j s ] = 0 , [ ξ + ir , ξ − j s ] = δ ij κ i r + s , [ κ i , ξ ± j s ] = ± ( α i , α j ) ξ ± j s , [ κ i r +1 , ξ ± j s ] − [ κ ir , ξ ± j s +1 ] = ± ( α i , α j )2 (cid:16) κ ir ξ ± j s + ξ ± j s κ ir (cid:17) , [ ξ ± i r +1 , ξ ± j s ] − [ ξ ± ir , ξ ± j s +1 ] = ± ( α i , α j )2 (cid:16) ξ ± ir ξ ± j s + ξ ± j s ξ ± ir (cid:17) , X p ∈ S m [ ξ ± ir p (1) , [ ξ ± ir p (2) , . . . [ ξ ± ir p ( m ) , ξ ± j s ] . . . ]] = 0 , where the last relation holds for all i = j , and we denoted m = 1 − a ij .By the results of [11] (they apply to any simple Lie algebra g ), every finite-dimensionalirreducible representation of the algebra Y D ( g ) is generated by a nonzero vector ζ (called the highest vector ) which is annihilated by all ξ + ir and is a simultaneous eigenvector for all κ ir sothat κ ir ζ = d ir ζ , d ir ∈ C . Furthermore, there exist unique monic polynomials P ( u ) , . . . , P n ( u ) in u such that ∞ X r =0 d ir u − r − = P i ( u + r i ) P i ( u ) , i = 1 , . . . , n. Equivalently, every finite-dimensional irreducible representation of the algebra Y D ( g ) is gener-ated by a nonzero vector ζ ′ (the highest vector with respect to the opposite triangular decompo-sition) which is annihilated by all ξ − ir and is a simultaneous eigenvector for all κ ir so that κ ir ζ ′ = d ′ ir ζ ′ , d ′ ir ∈ C . Furthermore, there exist unique monic polynomials Q ( u ) , . . . , Q n ( u ) in u such that ∞ X r =0 d ′ ir u − r − = Q i ( u ) Q i ( u + r i ) , i = 1 , . . . , n. (2.4)All possible n -tuples of monic polynomials ( P ( u ) , . . . , P n ( u )) and ( Q ( u ) , . . . , Q n ( u )) arise inthis way. The equivalence of the two parametrizations is seen by using the automorphism of thealgebra Y D ( g ) defined by ξ + ir ( − r +1 ξ − ir , ξ − ir ( − r +1 ξ + ir , κ ir ( − r +1 κ ir . .2 Gaussian generators and isomorphisms Introduce the following elements of the endomorphism algebra
End C N ⊗ End C N : P = N X i,j =1 e ij ⊗ e ji and Q = N X i,j =1 ε i ε j e ij ⊗ e i ′ j ′ , where e ij ∈ End C N are the matrix units, and we use the notation i ′ = N + 1 − i and set ε i ≡ in the orthogonal case, and ε i = for i = 1 , . . . , n, − for i = n + 1 , . . . , n, in the symplectic case. Set κ = N/ − in the orthogonal case, N/ in the symplectic case . Following [30], consider the R - matrix R ( u ) R ( u ) = 1 − Pu + Qu − κ . (2.5)The extended Yangian X( g ) is defined as a unital associative algebra with generators t ( r ) ij ,where i, j N and r = 1 , , . . . , satisfying certain quadratic relations. Introduce theformal series t ij ( u ) = δ ij + ∞ X r =1 t ( r ) ij u − r ∈ X( g )[[ u − ]] (2.6)and set T ( u ) = N X i,j =1 e ij ⊗ t ij ( u ) ∈ End C N ⊗ X( g )[[ u − ]] . Consider the tensor product algebra
End C N ⊗ End C N ⊗ X( g ) and introduce the series withcoefficients in this algebra by T ( u ) = N X i,j =1 e ij ⊗ ⊗ t ij ( u ) and T ( u ) = N X i,j =1 ⊗ e ij ⊗ t ij ( u ) . (2.7)The defining relations for the algebra X( g ) are then written in the form R ( u − v ) T ( u ) T ( v ) = T ( v ) T ( u ) R ( u − v ) . (2.8)The Yangian Y( g ) is defined as the subalgebra of X( g ) which consists of the elements stableunder the automorphisms µ f : T ( u ) f ( u ) T ( u ) , (2.9)for all series f ( u ) = 1 + f u − + f u − + · · · with f i ∈ C . Equivalently, Y( g ) is isomorphic tothe quotient of the algebra X( g ) by the relation T t ( u + κ ) T ( u ) = 1 , (2.10)where t denotes the matrix transposition with e t ij = ε i ε j e j ′ ,i ′ .4 .2.1 Isomorphisms Apply the Gauss decomposition to the matrix T ( u ) , T ( u ) = F ( u ) H ( u ) E ( u ) , (2.11)where F ( u ) , H ( u ) and E ( u ) are uniquely determined matrices of the form F ( u ) = . . . f ( u ) 1 . . . ... ... . . . ... f N ( u ) f N ( u ) . . . , E ( u ) = e ( u ) . . . e N ( u )0 1 . . . e N ( u ) ... ... . . . ... . . . , and H ( u ) = diag h h ( u ) , . . . , h N ( u ) i . Define the series with coefficients in Y( g ) by κ i ( u ) = h i (cid:16) u − ( i − / (cid:17) − h i +1 (cid:16) u − ( i − / (cid:17) for i = 1 , . . . , n − , and κ n ( u ) = h n (cid:16) u − ( n − / (cid:17) − h n +1 (cid:16) u − ( n − / (cid:17) for o n +1 h n (cid:16) u − n/ (cid:17) − h n +1 (cid:16) u − n/ (cid:17) for sp n h n − (cid:16) u − ( n − / (cid:17) − h n +1 (cid:16) u − ( n − / (cid:17) for o n . Furthermore, set ξ + i ( u ) = f i +1 i (cid:16) u − ( i − / (cid:17) , ξ − i ( u ) = e i i +1 (cid:16) u − ( i − / (cid:17) for i = 1 , . . . , n − , ξ + n ( u ) = f n +1 n (cid:16) u − ( n − / (cid:17) for o n +1 f n +1 n (cid:16) u − n/ (cid:17) for sp n f n +1 n − (cid:16) u − ( n − / (cid:17) for o n and ξ − n ( u ) = e n n +1 (cid:16) u − ( n − / (cid:17) for o n +112 e n n +1 (cid:16) u − n/ (cid:17) for sp n e n − n +1 (cid:16) u − ( n − / (cid:17) for o n . Introduce elements of Y( g ) by the respective expansions into power series in u − , κ i ( u ) = 1 + ∞ X r =0 κ ir u − r − and ξ ± i ( u ) = ∞ X r =0 ξ ± ir u − r − (2.12)for i = 1 , . . . , n . According to [19, Main Theorem], the mapping which sends the generators κ ir and ξ ± ir of Y D ( g ) to the elements of Y( g ) with the same names defines an isomorphism Y D ( g ) ∼ = Y( g ) . 5 .2.2 Central elements of the extended Yangian A presentation of the extended Yangian X( g ) in terms of the Gaussian generators is given in [19,Thm. 5.14]. By [19, Thm. 5.8], all coefficients of the series z ( u ) = n Y i =1 h i ( u + κ − i ) − n Y i =1 h i ( u + κ − i + 1) · h n +1 ( u ) h n +1 ( u − / if N = 2 n + 1 n − Y i =1 h i ( u + κ − i ) − n Y i =1 h i ( u + κ − i + 1) · h n +1 ( u ) if N = 2 n, belong to the center C of the extended Yangian X( g ) . Moreover, these coefficients generate thecenter and we have the tensor product decomposition X( g ) ∼ = Y( g ) ⊗ C . The following identity holds in X( g ) : T t ( u + κ ) T ( u ) = z ( u ) , so that the Yangian Y( g ) is isomorphic to the quotient of X( g ) by the relation z ( u ) = 1 . We willrecord the relations which follow from the arguments in [19, Sec. 5.3]. Lemma 2.1.
In the algebra X( g ) we have h ( u + κ ) h ′ ( u ) = z ( u ) , (2.13) h i ( u + κ − i ) h i ′ ( u ) = h i +1 ( u + κ − i ) h ( i +1) ′ ( u ) , (2.14) where i = 1 , . . . , n for N = 2 n + 1 , and i = 1 , . . . , n − for N = 2 n . Definition 2.2.
A representation V of the algebra Y( g ) (or X( g ) ) is called a highest weightrepresentation if there exists a nonzero vector ζ ∈ V such that V is generated by ζ and thefollowing relations hold: t ij ( u ) ζ = 0 for i < j N, and (2.15) t ii ( u ) ζ = λ i ( u ) ζ for i = 1 , . . . , N, (2.16)for some formal series λ i ( u ) ∈ u − C [[ u − ]] . The vector ζ is called the highest vector of therepresentation V . Note that the formulas in [19, (5.4) and (5.47)] should be corrected by swapping the order of the factors on theirright hand sides. B , C and D was proved in [1] in terms of their R -matric presentation. Wewill use the isomorphisms of [19] which we recalled in Sec. 2.2.1, to make an explicit connectionbetween this theorem and the results of [11]. Note that such a connection was already establishedin [15], where isomorphisms between three presentations of the orthogonal and symplectic Yan-gians were constructed. However, those results did not use the Gaussian presentation which wewill rely on in our arguments. Theorem 2.3.
1. Any finite-dimensional irreducible representation of the algebra Y( g ) is ahighest weight representation. Its parameters satisfy the relations λ i ( u ) λ i +1 ( u ) = P i ( u + 1) P i ( u ) , i = 1 , . . . , n − , (2.17) and λ n ( u ) λ n +1 ( u ) = P n ( u + 1 / P n ( u ) for type B n ,λ n ( u ) λ n +1 ( u ) = P n ( u + 2) P n ( u ) for type C n ,λ n − ( u ) λ n +1 ( u ) = P n ( u + 1) P n ( u ) for type D n , for some monic polynomials P i ( u ) in u .2. Every n -tuple ( P ( u ) , . . . , P n ( u )) of monic polynomials in u arises in this way.3. The series λ i ( u ) satisfy the relations λ i ( u + κ − i ) λ i ′ ( u ) = λ i +1 ( u + κ − i ) λ ( i +1) ′ ( u ) , (2.18) where i = 0 , , . . . , n for N = 2 n + 1 , and i = 0 , , . . . , n − for N = 2 n , and we set λ ( u ) = λ ′ ( u ) := 1 .Proof. Using the isomorphism Y D ( g ) ∼ = Y( g ) and the classification results recalled in Sec. 2.1,we find that any finite-dimensional irreducible representation V of the algebra Y( g ) in types B n and C n is generated by a vector ζ ′ such that e i,i +1 ( u ) ζ ′ = 0 for i = 1 , . . . , n, (2.19) h i ( u ) ζ ′ = λ i ( u ) ζ ′ for i = 1 , . . . , n + 1 , (2.20)for some formal series λ i ( u ) ∈ u − C [[ u − ]] . For type D n , the same conditions hold, exceptthat relation (2.19) with i = n should be replaced with e n − ,n +1 ( u ) ζ ′ = 0 . Indeed, for all types,relation (2.19) is clear from the definition of the highest vector ζ ′ , while (2.20) follows from the7ondition that ζ ′ is an eigenvector for all series κ i ( u ) with i = 1 , . . . , n and z ( u ) ζ ′ = ζ ′ . Now,Lemma 2.1 and [19, Lem. 5.15] imply that e ij ( u ) ζ ′ = 0 for i < jh i ( u ) ζ ′ = λ i ( u ) ζ ′ for i = 1 , . . . , N, for certain formal series λ i ( u ) ∈ u − C [[ u − ]] satisfying identities (2.18). Finally, note thatthe values of the parameters (2.3) are found by r i = 1 for i = 1 , . . . , n − , while r n = 1 / fortype B n , r n = 2 for type C n , and r n = 1 for type D n , so that conditions in Part 1 of the theoremfollow from (2.4). Corollary 2.4.
All statements of Theorem hold in the same form for the extended Yangian X( g ) , except that the value i = 0 is excluded for the conditions (2.18) .Proof. The proof of the theorem obviously extends to the algebra X( g ) . Relation (2.18) with i = 0 is now replaced by the property that the series z ( u ) acts in the highest weight representationas multiplication by λ ( u + κ ) λ ′ ( u ) . Remark . It is clear that the arguments used in the proof of the theorem can be reversed, sothat the classification theorem for the Yangian representations in types B , C and D proved in [1]implies the corresponding results of [11]. We will suppose that q is a transcendental complex number and set q i = q r i for i = 1 , . . . , n ,with r i defined in (2.3). The Cartan matrix of the simple Lie algebra g of type B n , C n or D n isgiven by (2.2). We will use the standard notation [ k ] q = q k − q − k q − q − (3.1)for a nonnegative integer k , together with [ k ] q ! = k Y s =1 [ s ] q and " kr q = [ k ] q ![ r ] q ! [ k − r ] q ! . The quantum affine algebra U q ( b g ) (with the trivial central charge) in its Drinfeld presentationis the associative algebra with generators x ± i,m , a i,l and k ± i for i = 1 , . . . , n and m, l ∈ Z with l = 0 , subject to the following defining relations: k i k j = k j k i , k i a j,l = a j,l k i , a i,m a j,l = a j,l a i,m ,k i x ± j,m k − i = q ± A ij i x ± j,m , [ a i,m , x ± j,l ] = ± [ mA ij ] q i m x ± j,m + l , ± i,m +1 x ± j,l − q ± A ij i x ± j,l x ± i,m +1 = q ± A ij i x ± i,m x ± j,l +1 − x ± j,l +1 x ± i,m , [ x + i,m , x − j,l ] = δ ij ψ i,m + l − ϕ i,m + l q i − q − i , X π ∈ S r r X l =0 ( − l " rl q i x ± i,s π (1) . . . x ± i,s π ( l ) x ± j,m x ± i,s π ( l +1) . . . x ± i,s π ( r ) = 0 , i = j, where in the last relation we set r = 1 − A ij . The elements ψ i,m and ϕ i, − m with m ∈ Z + aredefined by ψ i ( u ) := ∞ X m =0 ψ i,m u − m = k i exp (cid:16) ( q i − q − i ) ∞ X s =1 a i,s u − s (cid:17) , (3.2) ϕ i ( u ) := ∞ X m =0 ϕ i, − m u m = k − i exp (cid:16) − ( q i − q − i ) ∞ X s =1 a i, − s u s (cid:17) , (3.3)whereas ψ i,m = ϕ i, − m = 0 for m < . Remark . For the Lie algebras g of types C n and D n , it will be convenient to work withan extended quantum algebra obtained by adjoining the square roots k ± / n and ( k n − k n ) ± / ,respectively. Accordingly, we need to add the defining relations k / n x ± j,m k − / n = q ± A nj x ± j,m for type C n , and ( k n − k n ) / x ± j,m ( k n − k n ) − / = q ± ( A n − ,j + A nj ) / x ± j,m for type D n , while the new elements commute with all the remaining generators.As explained in [6, Sec. 12.2], the algebra U q ( b g ) admits a family of sign automorphismssuch that the composition of any finite-dimensional irreducible representation with a suitableautomorphism of this kind is isomorphic to a type representation . Such a representation isgenerated by a vector ζ which is annihilated by all x + i,m and is a simultaneous eigenvector for all k i and a i,l . Furthermore, if the series Φ i ( u ) ∈ C [[ u ]] and Ψ i ( u ) ∈ C [[ u − ]] are defined by ϕ i ( u ) ζ = Φ i ( u ) ζ and ψ i ( u ) ζ = Ψ i ( u ) ζ , then there exist unique polynomials P ( u ) , . . . , P n ( u ) in u all with constant term such that Φ i ( u ) = q − deg P i i P i ( u q i ) P i ( u ) = Ψ i ( u ) , i = 1 , . . . , n, where the equalities are understood for the expansions of the rational functions in u as series in u and u − , respectively . Every n -tuple of polynomials ( P ( u ) , . . . , P n ( u )) in u , where each P i ( u ) has constant term , arises in this way. The roles of u and u − are swapped in our notation as compared to [6, Sec. 12.2]. U q ( b g ) is validfor any simple Lie algebra g . Note also that the corresponding results of [6, Sec. 12.2] applyto centrally extended algebras U q ( b g ) , which show that the central element acts trivially in anyfinite-dimensional representation. For this reason we only consider the quotients of the quantumaffine algebras by the relation specifying the value of the central element as equal to . To define R -matrix realizations of the quantum affine algebras, introduce elements of the endo-morphism algebra End C N ⊗ End C N by P = N X i,j =1 e ij ⊗ e ji and Q = N X i,j =1 q ¯ ı − ¯ ε i ε j e i ′ j ′ ⊗ e ij , where i ′ = N + 1 − i , as before, and ( 1 , , . . . , N ) = (cid:16) n − , . . . , , , , − , − , . . . , − n + (cid:17) for g = o n +1 ( n, . . . , , , − , − , . . . , − n ) for g = sp n ( n − , . . . , , , , − , . . . , − n + 1) for g = o n . Furthermore, introduce the R -matrix by R = q N X i =1 ,i = i ′ e ii ⊗ e ii + e n +1 ,n +1 ⊗ e n +1 ,n +1 + X i = j,j ′ e ii ⊗ e jj + q − X i = i ′ e ii ⊗ e i ′ i ′ + ( q − q − ) X i
End C N ⊗ End C N ⊗ U Rq ( b g ) and introduce the series withcoefficients in this algebra by L ± ( u ) = N X i,j =1 e ij ⊗ ⊗ l ± ij ( u ) and L ± ( u ) = N X i,j =1 ⊗ e ij ⊗ l ± ij ( u ) . The defining relations then take the form R ( u, v ) L ± ( u ) L ± ( v ) = L ± ( v ) L ± ( u ) R ( u, v ) , (3.9) R ( u, v ) L +1 ( u ) L − ( v ) = L − ( v ) L +1 ( u ) R ( u, v ) , (3.10)together with the relations L ± ( u ) DL ± ( u ξ ) t D − = 1 , (3.11)where D is the diagonal matrix D = diag h q , . . . , q N i . (3.12) The condition i < j was erroneously replaced by the opposite inequality in [20, 21]. .2.1 Isomorphisms By applying the Gauss decomposition to L + ( u ) and L − ( u ) introduce matrices F ± ( u ) = . . . f ± ( u ) 1 . . . ... ... . . . ... f ± N ( u ) f ± N ( u ) . . . , E ± ( u ) = e ± ( u ) . . . e ± N ( u )0 1 . . . e ± N ( u ) ... ... . . . ... . . . , and H ± ( u ) = diag h h ± ( u ) , . . . , h ± N ( u ) i , such that L ± ( u ) = F ± ( u ) H ± ( u ) E ± ( u ) . (3.13)Their entries are found by the quasideterminant formulas [13]: h ± i ( u ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ± ( u ) . . . l ± i − ( u ) l ± i ( u ) ... . . . ... ... l ± i − ( u ) . . . l ± i − i − ( u ) l ± i − i ( u ) l ± i ( u ) . . . l ± i i − ( u ) l ± ii ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , i = 1 , . . . , N, (3.14)whereas e ± ij ( u ) = h ± i ( u ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ± ( u ) . . . l ± i − ( u ) l ± j ( u ) ... . . . ... ... l ± i − ( u ) . . . l ± i − i − ( u ) l ± i − j ( u ) l ± i ( u ) . . . l ± i i − ( u ) l ± ij ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.15)and f ± ji ( u ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ± ( u ) . . . l ± i − ( u ) l ± i ( u ) ... . . . ... ... l ± i − ( u ) . . . l ± i − i − ( u ) l ± i − i ( u ) l ± j ( u ) . . . l ± j i − ( u ) l ± j i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ± i ( u ) − (3.16)for i < j N . Set X + i ( u ) = e + i,i +1 ( u ) − e − i,i +1 ( u ) , X − i ( u ) = f + i +1 ,i ( u ) − f − i +1 ,i ( u ) , for i = 1 , . . . , n − , and X + n ( u ) = e + n,n +1 ( u ) − e − n,n +1 ( u ) for types B n and C n e + n − ,n +1 ( u ) − e − n − ,n +1 ( u ) for type D n ,X − n ( u ) = f + n +1 ,n ( u ) − f − n +1 ,n ( u ) for types B n and C n f + n +1 ,n − ( u ) − f − n +1 ,n − ( u ) for type D n . x ± i,m of the algebra U q ( b g ) into the series x ± i ( u ) = X m ∈ Z x ± i,m u − m . (3.17)By the Main Theorems of [20] and [21], the maps x ± i ( u ) ( q i − q − i ) − X ± i ( uq i ) ,ψ i ( u ) h − i +1 ( uq i ) h − i ( uq i ) − ,ϕ i ( u ) h + i +1 ( uq i ) h + i ( uq i ) − , (3.18)for i = 1 , . . . , n − , and x ± n ( u ) ( q n − q − n ) − [2] − / q n X ± n ( uq n ) for type B n ( q n − q − n ) − X ± n ( uq n +1 ) for type C n ( q n − q − n ) − X ± n ( uq n − ) for type D n ,ψ n ( u ) h − n +1 ( uq n ) h − n ( uq n ) − for type B n h − n +1 ( uq n +1 ) h − n ( uq n +1 ) − for type C n h − n +1 ( uq n − ) h − n − ( uq n − ) − for type D n ,ϕ n ( u ) h + n +1 ( uq n ) h + n ( uq n ) − for type B n h + n +1 ( uq n +1 ) h + n ( uq n +1 ) − for type C n h + n +1 ( uq n − ) h + n − ( uq n − ) − for type D n , define an isomorphism U q ( b g ) → U Rq ( b g ) . The extended quantum affine algebra is defined by the same presentation as the algebra U Rq ( b g ) ,except that the relation (3.11) is omitted. It was shown in [20] and [21], that this algebra canbe explicitly described in terms of the Gaussian generators by producing complete sets of rela-tions. We will denote the extended algebra by U ext q ( b g ) and identify its R -matrix and Gaussianpresentations.The maps described above can be understood as an embedding ι : U q ( b g ) ֒ → U ext q ( b g ) so thatwe can regard U q ( b g ) as a subalgebra of U ext q ( b g ) . This subalgebra can also be described with theuse of the multiplication automorphisms µ f : U ext q ( b g ) → U ext q ( b g ) , L ± ( u ) f ± ( u ) L ± ( u ) , (3.19)where f ± ( u ) = ∞ X m =0 f ± [ ∓ m ] u ± m , f ± [ ∓ m ] ∈ C , f + [0] f − [0] = 1 . (3.20)13amely, the image of U q ( b g ) under the embedding ι consists of the elements in U ext q ( b g ) which arefixed by all automorphisms of the form (3.19).All coefficients of the series z ± ( u ) given by z ± ( u ) = n Y i =1 h ± i ( uξq i ) − n Y i =1 h ± i ( uξq i − ) · h ± n +1 ( u ) h ± n +1 ( uq ) for N = 2 n + 1 , n − Y i =1 h ± i ( uξq i ) − n Y i =1 h ± i ( uξq i − ) · h ± n +1 ( u ) for N = 2 n, belong to the center of the algebra U ext q ( b g ) . If N = 2 n then the constant terms of the series z ± ( u ) are the central elements z ± [0] = l ± nn [0] l ± n ′ n ′ [0] . In this case we will extend this algebra byadjoining the square roots z ± [0] / . Then in all three cases there exist power series ζ ± ( u ) withcoefficients in the center of U ext q ( b g ) such that ζ ± ( u ) ζ ± ( u ξ ) = z ± ( u ) . Under the automorphism(3.19) we have µ f : ζ ± ( u ) f ± ( u ) ζ ± ( u ) . This implies that the coefficients of the entries of the matrices ζ ± ( u ) − L ± ( u ) belong to thesubalgebra U q ( b g ) ⊂ U ext q ( b g ) . Therefore, if C denotes the subalgebra of U ext q ( b g ) generated by thecoefficients of the series ζ ± ( u ) , then we have the tensor product decomposition U ext q ( b g ) = U q ( b g ) ⊗ C , assuming that the algebra U q ( b g ) is extended by adjoining square roots in types C n and D n as inRemark 3.1. In the algebra U ext q ( b g ) we have L ± ( u ) DL ± ( u ξ ) t D − = z ± ( u ) , (3.21)so that the subalgebra U q ( b g ) can also be regarded as the quotient of U ext q ( b g ) by the relations z ± ( u ) = 1 . The following relations are implied by (3.21), and they were essentially derived inSec. 4.5 of [20] and [21]. Lemma 3.2.
In the algebra U ext q ( b g ) we have h ± ( uξ ) h ± ′ ( u ) = z ± ( u ) , (3.22) h ± i ( uξq i ) h ± i ′ ( u ) = h ± i +1 ( uξq i ) h ± ( i +1) ′ ( u ) , (3.23) where i = 1 , . . . , n for N = 2 n + 1 , and i = 1 , . . . , n − for N = 2 n .Remark . Note that for the parameters d ij ( u, v ) defined in (3.6) we have d j ′ i ′ ( u, v ) = d ij ( u, v ) .Therefore the R -matrix (3.5) possesses the symmetry property R T T ( u, v ) = R ( u, v ) , (3.24)where R ( u, v ) = P R ( u, v ) P , while T denotes the standard matrix transposition with e Tij = e ji and T a is the partial transposition applied to the a -th copy of the endomorphism algebra End C N .14e can use the R -matrix R ( u, v ) instead of R ( u, v ) to define the extended quantum affinealgebra e U ext q ( b g ) in a way similar to U ext q ( b g ) , by using the relations R ( u, v ) e L ± ( u ) e L ± ( v ) = e L ± ( v ) e L ± ( u ) R ( u, v ) , (3.25) R ( u, v ) e L +1 ( u ) e L − ( v ) = e L − ( v ) e L +1 ( u ) R ( u, v ) , (3.26)where we impose the opposite zero mode conditions e l + ij [0] = e l − ji [0] = 0 for i > j and e l + ii [0] e l − ii [0] = e l − ii [0] e l + ii [0] = 1 . (3.27)The symmetry property (3.24) implies that the mapping e L ± ( u ) L ± ( u ) T defines an anti-isomorphism e U ext q ( b g ) → U ext q ( b g ) . Using the definition of quasideterminants, weobtain the following formulas for the images of the respective Gaussian generators e h ± i ( u ) h ± i ( u ) , e e ± ij ( u ) f ± ji ( u ) and e f ± ij ( u ) e ± ji ( u ) . The quantum affine algebra U ext q ( b g ) possesses a Hopf algebra structure defined by the coproduct ∆ : l ± ij ( u ) N X k =1 l ± ik ( u ) ⊗ l ± kj ( u ) , (3.28)the antipode S : L ± ( u ) L ± ( u ) − (3.29)and the counit ε : L ± ( u ) . (3.30) Proposition 3.4.
In the Hopf algebra U ext q ( b g ) we have ∆ : z ± ( u ) z ± ( u ) ⊗ z ± ( u ) and S : z ± ( u ) z ± ( u ) − . (3.31) In particular, U q ( b g ) is a Hopf subalgebra of U ext q ( b g ) . Moreover, S : L ± ( u ) z ± ( u ) z ± ( u ξ ) L ± ( u ξ ) . (3.32)15 roof. The formulas for the images of the series z ± ( u ) under the maps ∆ and S follow easilyfrom the definition of z ± ( u ) and the Hopf algebra axioms. For the proof of (3.32) use the relation L ± ( u ) − = z ± ( u ) − DL ± ( u ξ ) t D − and apply (3.31). Furthermore, for the power series ζ ± ( u ) we have ∆ : ζ ± ( u ) ζ ± ( u ) ⊗ ζ ± ( u ) , and so ∆ (cid:16) U q ( b g ) (cid:17) ⊂ U q ( b g ) ⊗ U q ( b g ) , S (cid:16) U q ( b g ) (cid:17) ⊂ U q ( b g ) , thus proving that U q ( b g ) is a Hopf subalgebra of U ext q ( b g ) . Denote by U q ( b g ) + (respectively, U q ( b g ) − ) the subalgebra of U q ( b g ) generated by the elements x + i,m (respectively, x − i,m ), and denote by U q ( b g ) the subalgebra generated by k ± i and a i,l together withthe additional elements in types C n and D n , introduced in Remark 3.1. The multiplication mapprovides the triangular decomposition isomorphism U q ( b g ) − ⊗ U q ( b g ) ⊗ U q ( b g ) + ∼ = U q ( b g ) , as proved in [4]; see also [16] for a generalization to quantum affinizations of symmetrizableKac–Moody algebras.Here we aim to establish a key property of the Gauss decomposition by showing that it isconsistent with the triangular decomposition (see Proposition 3.8 below). We will rely on a fewrelations for the Gaussian generators described in the following lemmas.We will use a standard notation [ x, y ] q = xy − q y x for q -commutators and begin by provingsome q -commutator formulas; cf. [22, Lem. 5.6]. Lemma 3.5.
For any k < i < j < k ′ such that i = j ′ we have h e ± k i ( u ) , e − ij [0] i q = (1 − q ) e ± k j ( u ) . (3.33) Proof.
Due to the consistency property of Gauss decompositions for subalgebras as stated inProp. 4.2 in [20] and [21], we may assume without loss of generality, that k = 1 . Write thedefining relations (3.9) and (3.10) in terms of the series l ± ij ( u ) to get ( u q δ ij − v q − δ ij ) l ± ia ( u ) l ± jb ( v ) + ( q − q − ) ( u δ i
The defining relations give ( u − v ) l ± ( u ) l − ′ ′ ( v ) + ( q − q − ) u l ± ′ ( u ) l − ′ ( v )= ( u − v ) l − ′ ′ ( v ) l ± ( u ) + ( q − q − ) v l − ′ ( v ) l ± ′ ( u ) − u − vu − v ξ N X c =1 d c ′ ( u, v ) l − ′ c ( v ) l ± c ′ ( u ) . By comparing the coefficients of v on both sides we come to the relation l − ′ ′ [0] l ± ( u ) = q − l ± ( u ) l − ′ ′ [0] − ( q − q − ) l − ′ ′ [0] l ± ( u ) . (3.41)On the other hand, taking i = 2 ′ and j = 1 ′ in (3.34) we get l ± ′ ( u ) l − ′ ′ [0] = l − ′ ′ [0] l ± ′ ( u ) − ( q − q − ) l − ′ ′ [0] l ± ′ ( u ) . Write l ± i ( u ) = l ± ( u ) e ± i ( u ) for i = 1 ′ , ′ and use (3.41) together with (3.35) to bring this to theform e ± ′ ( u ) l − ′ ′ [0] = q − l − ′ ′ [0] e ± ′ ( u ) − ( q − q − ) l − ′ ′ [0] (cid:16) e ± ′ ( u ) + e ± ( u ) e ± ′ ( u ) (cid:17) . Finally, write l − ′ ′ [0] = l − ′ ′ [0] e − ′ ′ [0] and apply (3.37) with i = 2 ′ . Lemma 3.7.
For any i < j < i ′ we have the relations e ± ij ( uξq i ) = q ¯ − ¯ ı +1 j − i − X s =0 ( − s +1 X j ′ = a