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Universal associative envelopes of \((n+1)\)-dimensional \(n\)-Lie algebras. (English) Zbl 1260.17005

Summary: For \(n\) even, we prove Pozhidaev’s conjecture on the existence of associative enveloping algebras for simple \(n\)-Lie (Filippov) algebras. More generally, for \(n\) even and any \((n+1)\)-dimensional \(n\)-Lie algebra \(L\), we construct a universal associative enveloping algebra \(U(L)\) and show that the natural map \(L\to U(L)\) is injective. We use noncommutative Gröbner bases to present \(U(L)\) as a quotient of the free associative algebra on a basis of \(L\) and to obtain a monomial basis of \(U(L)\). In the last section, we provide computational evidence that the construction of \(U(L)\) is much more difficult for \(n\) odd.

MSC:

17A42 Other \(n\)-ary compositions \((n \ge 3)\)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
17B35 Universal enveloping (super)algebras
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