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

Abstract

Abstract. We initiate a theory of highest weight representations for twisted Yangians of types B, C, D and we classify the finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types CI, DIII and BCD0.

Nested algebraic Bethe ansatz for orthogonal and symplectic open spin chains

“…The symmetric pairs that we are interested in are of the form (g 2n , g ρ 2n ), where ρ is an involution of g 2n and g ρ 2n denotes the ρ-fixed subalgebra of g 2n . The involution ρ is given by ρ(F ) = GF G −1 for a particular matrix G ∈ End(C 2n ); we will use the matrices G in agreement with those in Section 2.2 of [GRW1]. This allows us to view g ρ 2n as the subalgebra of g 2n generated by the elements F ρ ij = F ij + (GF G −1 ) ij .…”

“…• CD0 : (g 2n , g ρ 2n ) = (g 2n , g 2n ) and G = I. Note that we have excluded the DI case, when both p and q are odd (called DI(b) in [GRW1]). In this case the matrix G can not be chosen to be diagonal.…”

“…Let us now focus on all the remaining symmetric pairs. By Lemma 2.2 in [GRW1] the matrix G(u) itself satisfies the symmetry relation (3.18), namely…”

“…The twisted Yangian X ρ (g 2n , g θ 2n ) tw . We now focus on the Extended twisted Yangian X ρ (g 2n , g θ 2n ) tw and its representation theory adhering closely to [GR,GRW1,GRW2]. As before, we drop the "Extended" part of the name to simplify the notation.…”

“…Next, we focus on the lowest weight representations. We will rephrase some of the statements given in Section 4 of [GRW1], where the highest weight representation theory of X ρ (g 2n , g θ 2n ) tw was introduced.…”