On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras. (English) Zbl 1213.17011
Let \(\mathfrak{g}\) be a finite dimensional simple complex Lie algebra and \(R\) be the exterior algebra of \(\mathfrak{g}\oplus\mathfrak{g}\). There are three standard copies of the adjoint representation of \(\mathfrak{g}\) in \(R^2\). Consider the algebra \(A=R/J\), where \(J\) is the ideal generated by these three copies. In this paper the author shows that the algebra \(A^{\mathfrak{g}}\) of \(\mathfrak{g}\)-invariants in \(A\) is generated by a certain element \(S\). The author also proposes a conjecture, the validity of which would imply that the \(S^h=0\), where \(h\) is the Coxeter number of \(\mathfrak{g}\). The proof is based on various results in Lie algebra cohomology.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 17B56 | Cohomology of Lie (super)algebras |
| 22E67 | Loop groups and related constructions, group-theoretic treatment |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
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