Manifolds and Lie Groups pp 187-195 | Cite as

# On Lie Algebras Generated by Two Differential Operators

Chapter

## Abstract

We shall denote by K a field of characteristic 0, by K[x] the ring of polynomials in r variables x with a \({a_{{i_l} \cdot \cdot \cdot {i_r}{j_l} \cdot \cdot \cdot {j_r}}}\)
in K; it is a linear differential operator with polynomial coefficients.

_{1},...,x_{r}with coefficients in K, and by D. the K-derivation in K[x] defined by D_{i}x_{j}. = δ_{ij}.. for 1 ≦ i, j ≦ r; then the multiplications by x_{1},...,x_{r}in K[x] and D_{1}.,...,D_{r}generate a subalgebra A of the associative K-algebra of all K-linear transformations in K[x]. An element X of A can be written uniquely in the form$$x = \sum {{a_{{i_l} \cdot \cdot \cdot {i_r}{j_l} \cdot \cdot \cdot {j_r}}}x_l^{{i_l}} \cdot \cdot \cdot x_r^{{i_r}}} D_l^{{j_l}} \cdot \cdot \cdot D_r^{{j_r}}$$

## Keywords

Associative Algebra Metaplectic Group Simple Jordan Algebra Heisenberg Commutation Relation Nagoya Mathematical Journal
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1981