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Register a new userOn the classification of graded twisted planes
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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.
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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 loops in extension categories, and the algebraic definition involves homotopy liftings as introduced by the first author. As a consequence of our description, we prove that the topological definition indeed yields a Gerstenhaber algebra structure in the monoidal category setting, answering a question of Hermann. For use in proofs, we generalize $A_{infty}$-coderivation and homotopy lifting techniques from bimodule categories to some exact monoidal categories.
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