On Lie Algebras Generated by Two Differential Operators

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


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.


Associative Algebra Metaplectic Group Simple Jordan Algebra Heisenberg Commutation Relation Nagoya Mathematical Journal 
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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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