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Invariant algebras. (English) Zbl 0805.17006
Lie groups, their discrete subgroups, and invariant theory, Adv. Sov. Math. 8, 57-64 (1992).
[For the entire collection see Zbl 0742.00088.]
This paper is devoted to questions of construction and classification of finite-dimensional algebras over algebraically closed fields of characteristic zero, whose automorphism group contains a Levi subgroup of nonzero rank.
Concerning classification, the following result, which is proved by M. Brion and F. Knop, is established: If $$G$$ is a connected, simply connected, semisimple Lie group with Lie algebra $$L$$, then the set $$\Gamma(G)$$ of all highest weights $$\Lambda$$ such that the simple $$G$$- module with the highest weight $$\Lambda$$ has the structure of a nontrivial $$G$$-invariant algebra is a finitely generated monoid with respect to addition of weights. Explicit generators for $$\Gamma(G)$$ are given for $$G+A_ 4$$, $$B_ n$$, $$E_ 7$$ and $$E_ 8$$. Furthermore, a class of invariant simple algebra structures on direct sums $$\sum_ i A_ i$$ of certain simple nonisomorphic modules $$A_ i$$ for a reductive Lie algebra $$L$$ is constructed. Sufficient conditions for the existence of suitable components $$A_ i$$ for this construction are given in the case where $$L$$ is simple.

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.) 22E60 Lie algebras of Lie groups