Second Homology Groups and Universal Coverings of Steinberg Leibniz Algebras of Small Characteristic

Abstract

It is known that the second Leibniz homology group HL 2 (stl n (R)) of the Steinberg Leibniz algebra stl n (R) is trivial for n ≥ 5. In this paper, we determine HL 2 (stl n (R)) explicitly (which are shown to be not necessarily trivial) for n = 3, 4 without any assumption on the base ring.

Universal Central Extensions of Leibniz Superalgebras Over Superdialgebras

“…For the special linear superalgebra sl(m, n, A) it was worked out for A an associative algebra in [26] and [29]; and for A an associative superalgebra in [4] and [10]. In the category of Leibniz algebras, the universal central extension of sl(n, A) (seen as a Leibniz algebra), where A is an associative algebra, was found in [24] when n ≥ 5 and in [13] when n ≥ 3. For the Leibniz superalgebra sl(m, n, A), when m + n ≥ 5 and A is an associative algebra it was calculated in [21].…”