Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: Type C

Abstract

Following the approach of Ding and I. Frenkel (1993) for type A, we showed in our previous work that the Gauss decomposition of the generator matrix in the R-matrix presentation of the quantum affine algebra yields the Drinfeld generators in all classical types. Complete details for type C were given therein, while the present paper deals with types B and D. The arguments for all classical types are quite similar so we mostly concentrate on necessary additional details specific to the underlying orthogonal Lie algebras.

Isomorphism between the R-Matrix and Drinfeld Presentations of Quantum Affine Algebra: Types B and D

“…We will embed the algebra U q ( o N ) into an extended quantum affine algebra which we denote by U ext q ( o N ); cf. [8,11] and [20]. Recalling the scalar function f (u) defined by (1.3) and (1.4) set…”

“…An isomorphism between the Drinfeld and R-matrix presentations of the algebras U q ( g) in type A was constructed by Ding and Frenkel [8]. In our previous work [20] we were able to extend this construction to the remaining classical types and gave detailed arguments in type C. The present article is concerned with types B and D, where we use the same approach as in [20] and mostly concentrate on necessary changes specific to the orthogonal Lie algebras o N and their root systems. In particular, this applies to low rank relations with the underlying Lie algebras o 3 and o 4 , and to formulas for the universal R-matrices.…”

“…To work with the quantum affine algebras in types B and D, we apply the Gauss decomposition of the generator matrices in the R-matrix presentation in the same way as in types A and C; see [8] and [20]. We show that the generators arising from the Gauss decomposition satisfy the required relations of the Drinfeld presentation.…”

Gauss Coordinates vs Currents for the Yangian Doubles of the Classical Types

“…For the type A algebras an explicit relations between generating series in the RT T -formulation and the currents in the "new" realization were found in [1]. Recent results published in the papers [7,8,9,10] describes similar equivalences for the other type algebras.…”

“…The main ingredients of this construction were the Gauss coordinates of T -operators. Recently, it was discovered in the papers [7,8,9,10] that the same mechanism allows to establish the corresponding isomorphisms between RT T and "current" realizations of the Yangian doubles and quantum affine algebras for B, C and D series. In this section we describe these isomorphisms for each of the algebras DY (gl N ), DY (o 2n+1 ), DY (sp 2n ) and DY (o 2n ) separately.…”