Gelfand-Zeitlin theory from the perspective of classical mechanics. I

Abstract

Let M (n) be the algebra (both Lie and associative) of n × n matrices over C. Then M (n) inherits a Poisson structure from its dual using the bilinear form (x, y) = −tr xy. The Gl(n) adjoint orbits are the symplectic leaves and the algebra, P (n), of polynomial functions on M (n) is a Poisson algebra. In particular if f ∈ P (n) then there is a corresponding vector field ξ f on M (n). If m ≤ n then M (m) embeds as a Lie subalgebra of M (n) (upper left hand block) and P (m) embeds as a Poisson subalgebra of P (n). Then, as an analogue of the Gelfand-Zeitlin algebra in the enveloping algebra of M (n), let J(n) be the subalgebra of P (n) generated by P (m) Gl(m) for m = 1, . . . , n. One observes that J(n) ∼ = P (1) Gl(1) ⊗ · · · ⊗ P (n) Gl(n)

Fibers of characters in Gelfand-Tsetlin categories

“…Gelfand-Tsetlin subalgebras were considered in [FM] in connection with the solutions of the Euler equation, in [Vi] in connection with subalgebras of maximal Gelfand-Kirillov dimension in the universal enveloping algebra of a simple Lie algebra, in [KW1], [KW2] in connection with classical mechanics, and also in [Gr] in connection with general hypergeometric functions on the Lie group GL(n, C). A similar approach was used by Okunkov and Vershik in their study of the representations of the symmetric group S n [OV], with U being the group algebra of S n and Γ being the maximal commutative subalgebra generated by the Jucys-Murphy elements (1i) + .…”

Completeness of Determinantal Hamiltonian Flows on the Matrix Affine Poisson Space

“…Kostant and Wallach [10] have recently shown that all flows of the complexified Gelfand-Zeitlin integrable systems are complete everywhere on gl * n (C). (This is not true for the flows of the classical system on u n .)…”

Gelfand–Tsetlin modules of quantum gln defined by admissible sets of relations

“…By Lemma 2.4 we conclude that W is a Gelfand-Tsetlin module. Therefore, V C (T (v)) is a Gelfand-Tsetlin module with action of Γ given by (22) and (23).…”