Feigin–Frenkel center in types B, C and D

Abstract

For each simple Lie algebra $\mathfrak{g}$ consider the corresponding affine vertex algebra $V_{\mathrm{crit}}(\mathfrak{g})$ 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 $V_{\mathrm {crit}}(\mathfrak{g})$ associated with the simple Lie algebras $\mathfrak{g}$ 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 $\mathfrak{sl}_{2}$ in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras $\mathrm{U}(\mathfrak{g}[t])$ and $\mathrm{U}(\mathfrak{g})$, and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra $\mathfrak{g}$. We also introduce analogues of the Bethe subalgebras of the Yangians $\mathrm{Y}(\mathfrak{g})$ and show that their graded images coincide with the respective commutative subalgebras of $\mathrm{U}(\mathfrak{g}[t])$.