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Representations of Yang-Mills algebras. (English) Zbl 1271.17012

Let \(k\) be an algebraically closed field of characteristic zero. Given any positive integer \(n\geq 2\), let \(\mathcal{F}(n)\) be the free Lie algebra generated by \(x_{1},\cdots, x_{n}\). The Yang-Mills Lie algebra \(\mathcal{YM}(n)\) is defined by \( \mathcal{YM}(n)=\mathcal{F}(n)/\langle \{\sum_{i=1}^{n}[x_{i}, [x_{i}, x_{j}]]: j=1,\cdots, n\}\rangle.\) The Yang-Mills algebra \(\text{YM}(n)\) is the associative enveloping algebra \(\mathcal{U}(\mathcal{YM}(n))\) of the Lie algebra \(\mathcal{YM}(n)\). Note that \(\text{YM}(n)\) is a cubic Koszul algebra of global dimension \(3\), and it is noetherian if and only if \(n=2\). In this paper, the authors study representations of the Yang-Mills algebras \(\text{YM}(n)\). To this end, they establish a relationship between the Yang-Mills algebras \(\text{YM}(n)\) and the Weyl algebras \(A_{r}(k)\). In particular, they prove that the Weyl algebras \(A_{r}(k)\) are homomorphic images of the Yang-Mills algebras \(\text{YM}(n)\). As a result, they obtain many families of infinite dimensional representations of \(\text{YM}(n)\) via pulling back the representations of \(A_{r}(k)\). Besides, they also describe some nilpotent finite dimensional irreducible representations of \(\text{YM}(n)\).

MSC:

17B55 Homological methods in Lie (super)algebras
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)

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