A new invariant for finite dimensional Leibniz/Lie algebras

[摘要] For an n-dimensional Leibniz/Lie algebra h over a field k we introduce a new invariant A(h), called the universal algebra of h, as a quotient of the polynomial algebra k[X-ij vertical bar i, j = 1, ..., n] through an ideal generated by n(3) polynomials. We prove that A(h) admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h. The new object A (h) is the key tool in answering two open problems in Lie algebra theory. First, we prove that the automorphism group Aut(Lbz), (h) of h is isomorphic to the group U(G(A(h)degrees)) of all invertible group-like elements of the finite dual A(h)degrees. Secondly, for an abelian group G, we show that there exists a bijection between the set of all G-gradings on h and the set of all bialgebra homomorphisms A(h) -> k[G]. Based on this, all G-gradings on h are explicitly classified and parameterized. A(h) is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra h. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-15

[关键词] Lie algebras;Universal constructions;Automorphism group;Gradings