3 research outputs found
Feigin-Frenkel center in types B, C and D
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]).Comment: 29 pages, constructions of Pfaffian-type Sugawara operators and
commutative subalgebras in universal enveloping algebras are adde