Steinberg-Leibniz algebras and superalgebras
The Steinberg Lie algebra \({\mathfrak s}{\mathfrak t}(n,A)\), \(n\geq 3\), over a unital associative algebra \(A\) is the universal central extension of the matrix Lie algebra \({\mathfrak s}{\mathfrak l}(n,A)\), and the Leibniz algebra \({\mathfrak s}{\mathfrak t}{\mathfrak l}(n,A)\) has a similar property in the category of Leibniz algebras. These facts motivate the author to study the Leibniz algebra \({\mathfrak s}{\mathfrak l}(n,D)\) over a unital associative dialgebra \(D\). He constructs the Steinberg Leibniz algebra \({\mathfrak s}{\mathfrak t}{\mathfrak l}(n,D)\) and establishes its basic properties. He proves that it is the universal central extension of \({\mathfrak s}{\mathfrak l}(n,D)\) and gives a description of the kernel which is a quotient of the first Hochschild homology group of the dialgebra \(D\). The results are connected with Leibniz algebras graded by finite root systems and with Leibniz \(K\)-theory. Finally, the author extends his results to the Steinberg Leibniz superalgebras \({\mathfrak s}{\mathfrak t}{\mathfrak l}(m,n,D)\).
scientific article; zbMATH DE number 2153360
scientific article