On Higher-Order Sugawara Operators

Abstract

The higher Sugawara operators acting on the Verma modules over the affine Kac-Moody algebra at the critical level are related to the higher Hamiltonians of the Gaudin model due to work of Feigin, Frenkel and Reshetikhin. An explicit construction of the higher Hamiltonians in the case of gl n was given recently by Talalaev. We propose a new approach to these results from the viewpoint of the vertex algebra theory by proving directly the formulas for the higher order Sugawara operators. The eigenvalues of the operators in the Wakimoto modules of critical level are also calculated.

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.…”

“…We will twist this homomorphism by the involutive anti-automorphism 4) to get the anti-homomorphism For any λ ∈ h * , the Verma module M λ is defined as the quotient of U(g) by the left ideal generated by n + and the elements h i − λ(h i ) with i = 1, 2. We denote the image of 1 in M λ by 1 λ .…”

Higher-order Hamiltonians for the trigonometric Gaudin model

“…Explicit higher Hamiltonians for the rational Gaudin model associated with gl N were produced by Talalaev [19] by making use of the Bethe subalgebra of the Yangian for gl N and taking a classical limit; see also [13]. Some related families of higher Hamiltonians and their analogues for the orthogonal and symplectic Lie algebras were produced by using the center at the critical level following the general approach of Feigin, Frenkel and Reshetikhin [7]; see [3], [5] and also [14] for more details and references. In particular, such a family arises from the coefficients of the differential operators tr ∂ u + E(u) k with k = 1, 2, .…”

Quasi Modules for the Quantum Affine Vertex Algebra in Type A

“…We use the quasi module map to obtain further information on the algebra A c (gl N ). In our previous paper [9], coauthored with N. Jing, Molev and F. Yang, the center z(V c (gl N )) of the quantum VOA V c (gl N ) was described by providing explicit formulae for its algebraically independent topological generators, thus establishing the quantum analogue of the Feigin-Frenkel theorem in type A; see [2,3,5]. By considering the image of the center z(V −N (gl N )), with respect to the quasi module map Y W −N/2 (gl N ) , we find explicit formulae for families of central elements in the completed algebra A −N/2 (gl N ), which are, due to the fusion procedure originated in the work of A. Jucys [10], parametrized by arbitrary partitions with at most N parts.…”