Yangian Characters and Classical $$\mathcal{W}$$ -Algebras

Abstract

The Yangian characters (or q-characters) are known to be closely related to the classical W-algebras and to the centers of the affine vertex algebras at the critical level. We make this relationship more explicit by producing families of generators of the W-algebras from the characters of the Kirillov-Reshetikhin modules associated with multiples of the first fundamental weight in types B and D and of the fundamental modules in type C. We also give an independent derivation of the character formulas for these representations in the context of the RT T presentation of the Yangians. In all cases the generators of the W-algebras correspond to the recently constructed elements of the Feigin-Frenkel centers via an affine version of the Harish-Chandra isomorphism.

Eigenvalues of Bethe vectors in the Gaudin model

“…The W-algebra W( L g) can be defined as a subalgebra of U( h − ) which consists of the elements annihilated by the screening operators; see [10, Sec. 8.1] and also [16] for explicit formulas in the classical types. Given any element χ ∈ g * and a nonzero z ∈ C, the mapping…”

“…Simple explicit formulas for generators of this algebra were found in [5] for type A and in [15] for types B, C and D; see also [4] and [17] for simpler arguments in type A and extensions to Lie superalgebras. The calculation of the Harish-Chandra images of the generators in type A is straightforward, whereas types B, C and D require a rather involved application of the q-characters; see [16]. Our goal in this paper is to apply these results to get the action of the higher Gaudin Hamiltonians on tensor products of representations of g in an explicit form and calculate the corresponding eigenvalues of the Bethe vectors.…”

“…A connection between the Yangian characters (or q-characters) and the Segal-Sugawara operators played an essential role in the calculation of the Harish-Chandra images in [16]. We will explore this connection further by constructing a map gr taking a character to an element of the associated classical W-algebra.…”

Segal–Sugawara vectors for the Lie algebra of type G2

“…Their explicit form in the classical types goes back to [1] and [15] in type A and to [7] in types B, C and D. For type G 2 such generators were calculated in [21]; see also [3], where they appear in a different context. The images of explicit generators of z( g) under the isomorphism (4.2) are found in [4] and [5] in type A and in [18] for types B, C and D; see also [24] for a direct calculation for the Pfaffian-type vector. In the next section we calculate the images of the Segal-Sugawara vectors in type G 2 constructed in Sec.…”

The R-Matrix Presentation for the Yangian of a Simple Lie Algebra

“…In this setting, the evaluation morphism Y (sl N ) ։ U (sl N ), which only exists for g = sl N , is particularly simple to describe and this phenomena gives rise to many interesting results. The R-matrix presentation of Y (g) has also been studied for orthogonal and symplectic Lie algebras [AAC + , AMR], and this has led to a more explicit description of the relationship between Yangian characters, classical W -algebras, and the centers of vertex algebras at the critical level [MM1]. It has also served as the catalyst for the study of twisted Yangians of type B-C-D and their representations [GR,GRW1,GRW2,GRW3,IMO].…”