On the R-Matrix Realization of Yangians and their Representations

Abstract

We study the Yangians Y(a) associated with the simple Lie algebras a of type B, C or D. The algebra Y(a) can be regarded as a quotient of the extended Yangian X(a) whose defining relations are written in an R-matrix form. In this paper we are concerned with the algebraic structure and representations of the algebra X(a). We prove an analog of the Poincaré-Birkhoff-Witt theorem for X(a) and show that the Yangian Y(a) can be realized as a subalgebra of X(a). Furthermore, we give an independent proof of the classification theorem for the finite-dimensional irreducible representations of X(a) which implies the corresponding theorem of Drinfeld for the Yangians Y(a). We also give explicit constructions for all fundamental representation of the Yangians.

Representations of twisted Yangians of types B, C, D: I

“…By [AMR,Thm. 3.1], the algebra X(g N ) is isomorphic to the tensor product of its subalgebras ZX(g N ) and Y (g N ).…”

“…The Yangian Y (g N ) is a Hopf subalgebra of X(g N ) with the coproduct, antipode and counit obtained by restricting those of X(g N ); see [AMR,Prop. 3.3].…”

“…, n − 1} otherwise. (To obtain the exact relation in [AMR,Prop. 5.14], replace i by n − j, u by u + κ − j and then take the reciprocal on both sides.)…”

“…5.14], replace i by n − j, u by u + κ − j and then take the reciprocal on both sides.) Moreover, if M (λ(u)) is non-trivial, then it has a unique irreducible (non-zero) quotient L(λ(u)), and any irreducible highest weight X(g N )-module with highest weight (λ(u)) is non-trivial) then, by [AMR,Thm. 5.16], it is finite-dimensional if and only if there exist monic polynomials…”

“…The classification is stated in terms of certain polynomials as in the case of Yangians of simple Lie algebras [Dr3], twisted Yangians for symmetric pairs of type A [Mo5] and twisted q-Yangians [GM]. To obtain our classification theorems, we follow a well established approach (see [Mo1,Mo2,Mo5,MR,AMR]). As a first step, we prove that any finite-dimensional irreducible module is of highest weight type, hence the quotient of a Verma module: see Theorem 4.5.…”