# Semisimple Lie Algebras of Differential Operators

Author: Richter, D.A.

Source: Acta Applicandae Mathematicae, Volume 66, Number 1, March 2001 , pp. 41-65(25)

Publisher: Springer

OR

Price: $47.00 plus tax (Refund Policy) Abstract: Starting from the commutation relations in a complex semisimple Lie algebra$\mathfrak{g}$, one may obtain a space$\hat{\mathfrak{g}}$of vector fields on Euclidean space such that$\mathfrak{g}$and$\hat{\mathfrak{g}}$are isomorphic when$\hat{\mathfrak{g}}$is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of$\hat{\mathfrak{g}}$is a Borel subalgebra$\mathfrak{b}$. Furthermore, one may adjoin to the vector fields in$\hat{\mathfrak{g}}$multiplication operators to obtain an$\mathfrak{h}^{*}$-parameter family of distinct presentations of$\mathfrak{g}$as spaces of differential operators, where$\mathfrak{h}^{*}$is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of$\mathfrak{g}$can be presented in this way. All of this is carried out explicitly for arbitrary$\mathfrak{g}\$. In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.

Document Type: Regular Paper

Affiliations: Mathematics Department, Southeast Missouri State University, Cage Girardeau, MO 63701, U.S.A. e-mail: drichter@semovm.semo.edu

Publication date: 2001-03-01

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