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On Lie Algebras Generated by Two Differential Operators

  • Jun-ichi Igusa
Part of the Progress in Mathematics book series (PM, volume 14)

Abstract

We shall denote by K a field of characteristic 0, by K[x] the ring of polynomials in r variables x1,...,xr with coefficients in K, and by D. the K-derivation in K[x] defined by Dixj. = δij.. for 1 ≦ i, j ≦ r; then the multiplications by x1,...,xr in K[x] and D1.,...,Dr 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
$$x = \sum {{a_{{i_l} \cdot \cdot \cdot {i_r}{j_l} \cdot \cdot \cdot {j_r}}}x_l^{{i_l}} \cdot \cdot \cdot x_r^{{i_r}}} D_l^{{j_l}} \cdot \cdot \cdot D_r^{{j_r}}$$
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.

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

Authors and Affiliations

  • Jun-ichi Igusa
    • 1
  1. 1.The Johns Hopkins UniversityBaltimoreUSA

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