Steinberg–Leibniz algebras and superalgebras

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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 HHS 1 ( D ) , the quotient of the first Hochschild homology group HH 1 ( 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.’