Contractions and Polynomial Lie Algebras

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 C ₀ 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.