Summary
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
. 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
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.
-
(1)
Let x ∈ M(n). Then A · x is an orbit of maximal dimension (d(n − 1)) if and only if the differentials (dp i)x, i = 1, . . ., d(n), are linearly independent.
-
(2)
The orbits, O x, of the adjoint action of Gl(n) on M(n) are A-stable, and if O x is an orbit of maximal dimension (n(n − 1)), that is, if x is regular, then the A-orbits of dimension d(n − 1) in O x are the leaves of a polarization of a Zariski open dense subset of the symplectic manifold O x.
The results of the paper are related to the theory of orthogonal polynomials. Motivated by the interlacing property of the zeros of neighboring orthogonal polynomials on ℝ, we introduce a certain Zariski open subset M Ω(n) of M(n) and prove
-
(3)
MΩ(n) has the structure of (ℂ×)d(n−1) bundle over a (d(n)-dimensional) variety of Hessenberg matrices. Moreover, the fibers are maximal A-orbits. The variety of Hessenberg matrices plays a major role in this paper.
In Part II of this two-part paper, we deal with a commutative analogue of the Gelfand-Kirillov theorem. The fibration in (3) leads to the construction of n 2 + 1 functions (including a constant function) in an algebraic extension of the function field of M(n) which, under Poisson bracket, satisfies the commutation relations of the direct sum of a 2 d(n − 1) + 1 dimensional Heisenberg Lie algebra and an n-dimensional commutative Lie algebra.
With admiration, To a dear friend and brilliant colleague. B.K.
To Tony on his 60th birthday: We honor your past accomplishments and anticipate your future successes. N.W.
Research supported in part by NSF grant DMS-0209473 and in part by the KG & G Foundation.
Research supported in part by NSF grant DMS-0200305.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Borel, Linear Algebraic Groups, W. A. Benjamin, Inc., 1969.
C. Chevalley, Fondements de la Géométrie Algébrique, Faculté des Sciences de Paris, Mathématiques approfondies, 1957/1958.
N. Chris and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
V. Guillemin and S. Sternberg, On the collective integrability according to the method of Thimm, Ergod. Th. & Dynam. Sys., 3 (1983), 219–230.
R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. Vol. 52, Springer-Verlag, 1977.
D. Jackson, Fourier Series and Orthogonal Polynomials, The Carus Mathematical Monographs, no. 6, Math. Assoc. of America, 1948.
B. Kostant, Quantization and Unitary Representations, Lecture Notes in Math, Vol. 170, Springer-Verlag, 1970.
B. Kostant, Lie group representations on polynomial rings, AJM, 85 (1963), 327–404.
B. Kostant and N. Wallach, Gelfand-Zeitlin from the Perspective of Classical Mechanics. II, Prog. Math., Vol. 244 (2005), pp. 387–420.
F. Knop, Automorphisms, Root Systems and Compactifications of Homogeneous Varieties, JAMS, 9 (1996), no. 1, 153–174.
D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math., Vol. 1358, Springer, 1995.
T. Springer, Linear Algebraic Groups, 2nd edition, Prog. Math., Vol. 9, Birkhäuser Boston, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Kostant, B., Wallach, N. (2006). Gelfand-Zeitlin theory from the perspective of classical mechanics. I. In: Bernstein, J., Hinich, V., Melnikov, A. (eds) Studies in Lie Theory. Progress in Mathematics, vol 243. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4478-4_12
Download citation
DOI: https://doi.org/10.1007/0-8176-4478-4_12
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4342-3
Online ISBN: 978-0-8176-4478-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)