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The isomorphism problem for universal enveloping algebras of nilpotent Lie algebras. (English) Zbl 1263.17013

The isomorphism problem for universal enveloping algebras \(U(L)\) of Lie algebras \(L\) is well known. This problem has a negative answer when formulated in its most general form (see, for instance, an example of a Lie algebra over a field of odd characteristic constructed by A. A. Mikhalev and A. Zolotykh [Combinatorial aspects of Lie superalgebras. Boca Raton, FL: CRC Press (1995; Zbl 0871.17002)]). Nevertheless, the isomorphism problem has had a positive solution when considered on specific classes of Lie algebras. As well several invariants of \(L\) are known to be determined by \(U(L)\). Inspired by the positive group-ring-theoretic results, D. Riley and H. Usefi proved that the nilpotence of \(L\) is determined by \(U(L)\) [Algebr. Represent. Theory 10, No. 6, 517–532 (2007; Zbl 1192.17005)].
The paper under review shows that the isomorphism type of a nilpotent Lie algebra of dimension at most 6 over a field of characteristic different from 2 and 3 is determined by the isomorphism type of its universal enveloping algebra. The proof uses the recent classification due to W. A. de Graaf of 6-dimensional nilpotent Lie algebras over underlying fields of characteristic not 2 [J. Algebra 309, No. 2, 640–653 (2007; Zbl 1137.17012)]. Examples show that the restriction on the characteristic of the field is necessary.

MSC:

17B35 Universal enveloping (super)algebras
17B30 Solvable, nilpotent (super)algebras

Software:

GAP; ModIsom; Sophus
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Full Text: DOI arXiv

References:

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