Feigin-Frenkel center in types B, C and D

Abstract

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).

29 pages, constructions of Pfaffian-type Sugawara operators and commutative subalgebras in universal enveloping algebras are added

References in corpus (3)

• Bethe eigenvectors of higher transfer matrices
• On higher order Sugawara operators
• A new fusion procedure for the Brauer algebra and evaluation homomorphisms

Cited by in corpus (19)

• Gaudin Models and Multipoint Conformal Blocks: General Theory
• From Gaudin Integrable Models to $d$-dimensional Multipoint Conformal Blocks
• Equivalences between three presentations of orthogonal and symplectic Yangians
• Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras
• Center of the quantum affine vertex algebra in type A
• The $R$-matrix presentation for the Yangian of a simple Lie algebra
• Higher Sugawara operators for the quantum affine algebras of type A
• Cyclotomic Gaudin models: construction and Bethe ansatz
• Quantization of the shift of argument subalgebras in type A
• Classical W-algebras in types A, B, C, D and G
• $h$-adic quantum vertex algebras associated with rational $R$-matrix in types $B$, $C$ and $D$
• Invariants of the vacuum module associated with the Lie superalgebra gl(1|1)
• Bethe subalgebras in Yangians and the wonderful compactification
• Segal-Sugawara vectors for the Lie algebra of type $G_2$
• Casimir elements from the Brauer-Schur-Weyl duality
• On Segal--Sugawara vectors and Casimir elements for classical Lie algebras
• Center of the Yangian double in type A
• Center of affine ${\mathfrak{sl}}_{2|1}$ at the critical level
• Segal-Sugawara vectors for orthosymplectic Lie superalgebras