Contractions and Polynomial Lie Algebras

Part of the Progress in Mathematics book series (PM, volume 295)

Abstract

Let \(\mathfrak{g}\) denote a Lie algebra over \(\Bbbk\), and let B denote a commutative unital \(\Bbbk\)-algebra. The tensor product \(\mathfrak{g}\otimes_{\Bbbk}B\) carries the structure of a Lie algebra over \(\Bbbk\) with Lie bracket
$$[x \otimes a, y\otimes b] = [x,y]\otimes ab, \quad x,y \in\mathfrak {g}, \ a,b \in B.$$
If C0 denotes the quotient of the polynomial algebra \(\Bbbk[t]\) by the ideal generated by some power of t, then \(\mathfrak{g} \otimes C_{0}\) is called a polynomial Lie algebra.

In this contribution, \(\mathfrak{g} \otimes C_{0}\) is shown to be a contraction of \(\mathfrak{g} \otimes C\), where C is a semisimple commutative unital algebra. The contraction is exploited to derive a reducibility criterion for the universal highest-weight modules of \(\mathfrak{g} \otimes C_{0}\), via contraction of the Shapovalov form. This yields an alternative derivation of the reducibility criterion, obtained by the author in previous work.

Keywords

Highest-weight theory Contraction Deformation Polynomial Lie algebra 

Mathematics Subject Classification (2010)

17B10 17B99 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.BerlinGermany

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