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We use a representation of a graded twisted tensor product of $K[x]$ with $K[y]$ in $L(K^{Bbb{N}_0})$ in order to obtain a nearly complete classification of these graded twisted tensor products via infinite matrices. There is one particular example and three main cases: quadratic algebras classified by Conner and Goetz, a family called $A(n,d,a)$ with the $n+1$-extension property for $nge 2$, and a third case, not fully classified, which contains a family $B(a,L)$ parameterized by quasi-balanced sequences.
For some exact monoidal categories, we describe explicitly a connection between topological and algebraic definitions of the Lie bracket on the extension algebra of the unit object. The topological definition, due to Schwede and Hermann, involves loo
2-local derivation is a generalized derivation for a Lie algebra, which plays an important role to the study of local properties of the structure of the Lie algebra. In this paper, we prove that every 2-local derivation on the twisted Heisenberg-Virasoro algebra is a derivation.
We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.
In this paper we explore the possibility of endowing simple infinite-dimensional ${mathfrak{sl}_2(mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied in [arXiv:1308.6089] and [arXiv:1601.03008].
In this work the description up to isomorphism of complex naturally graded quasi-filiform Zinbiel algebras is obtained.