Gelfand-Zeitlin Theory from the Perspective of Classical Mechanics. II

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 $$ J\left( n \right) \cong P\left( 1 \right)^{Gl\left( 1 \right)} \otimes \cdots \otimes P\left( n \right)^{Gl\left( n \right)} $$ . We prove that J(n) is a maximal Poisson commutative subalgebra of P(n) and that for any p ∈ J(n) the holomorphic vector field ξ p is integrable and generates a global one-parameter group σ p(z) of holomorphic transformations of M(n). If d(n) = n(n + 1)/2, then J(n) is a polynomial ring ℂ[p 1, . . ., p d(n)] and the vector fields \( \xi _{p_i } \) , i = 1, . . ., d(n − 1), span a commutative Lie algebra of dimension d(n − 1). Let A be a corresponding simply-connected Lie group so that A ≅ ℂd(n−1). Then A operates on M(n) by an action σ so that if a ∈ A, then $$ \sigma \left( a \right) = \sigma _{p_1 } \left( {z_1 } \right) \cdots \sigma _{pd\left( {n - 1} \right)} \left( {z_{d\left( {n - 1} \right)} } \right) $$ where a is the product of exp z i \( \xi _{p_i } \) for i = 1, . . ., d(n − 1). We prove that the orbits of A are independent of the choice of the generators p i. Furthermore, for any matrix the orbit A · x may be explicitly given in terms of the adjoint action of a n − 1 abelian groups determined by x. In addition we prove the following results about this rather remarkable group action.