On graded Frobenius algebras (Q1408749)

Let \(A\) be a finite dimensional algebra over a field \(k\). The author introduces an admissible system for a graded Frobenius algebra as a triple \(\Phi=(\sigma,\gamma,\theta)\) where \(\sigma\) is a \(k\)-algebra automorphism, \(\gamma\colon{_AA_A}\to{_\sigma X_\sigma}\) is an isomorphism and \(\theta\colon X^{\otimes d}\to k\) is a linear map satisfying \(\theta(x_1\otimes\cdots\otimes x_d)=\theta(x_2\otimes\cdots\otimes x_d\gamma(x_1))\). It is shown that any basic graded Frobenius algebra \(\Gamma=\bigoplus_{i=0}^d\Gamma_i\) with the properties (i) \(\Gamma_i\Gamma_j=\Gamma_{i+j}\) for any \(i,j\) and (ii) \(\text{soc}(\Gamma)\subseteq\Gamma_d\) can be obtained from some admissible system and from any admissible system a graded Frobenius algebra, \(\Gamma(\Phi)\), satisfying conditions (i) and (ii) is constructed. The rest of the paper is concentrated on the case \(A=k\). For two nondegenetated admissible systems \(\Phi=(1_k,\gamma,\theta)\) and \(\Phi'=(1_k,\gamma',\theta')\), the associated \(k\)-algebras are isomorphic if and only if \(\theta'=\theta\circ s^{\otimes d}\) for some \(s\in\text{GL}(X)\). In the final section of the paper, some concrete admissible systems are considered.