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.

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.

Reviewer: Pham Hũ’u Tiêp (Essen)

##### 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 |