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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.

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