Geodesic
Symmetric Hochschild extension algebras
Colloquium Mathematicum, Tome 80 (1999) no. 2, pp. 155-174
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By an extension algebra of a finite-dimensional K-algebra A we mean a Hochschild extension algebra of A by the dual A-bimodule $Hom_K(A,K)$. We study the problem of when extension algebras of a K-algebra A are symmetric. (1) For an algebra A= KQ/I with an arbitrary finite quiver Q, we show a sufficient condition in terms of a 2-cocycle for an extension algebra to be symmetric. (2) Let L be a finite extension field of K. By using a given 2-cocycle of the K-algebra L, we construct a 2-cocycle of the K-algebra LQ for an arbitrary finite quiver Q without oriented cycles. Then we show a criterion on L for all those K-algebras LQ to have symmetric non-splittable extension algebras defined by the 2-cocycles. 
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     title = {Symmetric {Hochschild} extension algebras},
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Yosuke Ohnuki; Kaoru Takeda; Kunio Yamagata. Symmetric Hochschild extension algebras. Colloquium Mathematicum, Tome 80 (1999) no. 2, pp. 155-174. doi: 10.4064/cm-80-2-155-174

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