Steinberg–Leibniz algebras and superalgebras

As a universal central extension of the special linear Lie algebra sl(n,A) over a unital associative algebra A, the Steinberg algebras st(n,A) and stl(n,A) were studied in several papers. In this paper, we mainly study the Steinberg–Leibniz algebra stl(n,D) defined over a dialgebra D. We prove that it is the universal central extension of the special linear Leibniz algebra sl(n,D) with kernel HHS1(D), the quotient of the first Hochschild homology group HH1(D) of the dialgebra D by the ideal generated by a (b c)−a (b c) for all a,b,c D. We also obtain a similar theorem for the Steinberg–Leibniz superalgebra stl(m,n,D). This research plays a key role in studying the Leibniz algebras (superalgebras) graded by finite root systems and is also connected with ‘Leibniz K-theory.’