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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.

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.

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\textit{C. Schneider} and \textit{H. Usefi}, J. Algebra 337, No. 1, 126--140 (2011; Zbl 1263.17013)

### References:

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