Pseudo-composition superalgebras
Let \(G=G_0\oplus G_1\) be the Grassmann superalgebra with the countable set of standard generators \(e_i\), \(i\geq 1,\) such that \(e_i^2=e_ie_j+e_je_i=0\) for all \(i,j\geq 1\). Let \(A=A_0\oplus A_1\) be a supercommutative superalgebra over a field of characteristic different from 2 and 3 with a supersymmetric bilinear form \(\langle\;|\;\rangle\) which is symmetric on \(A_0\) and skew-symmetric on \( A_1\). It is supposed also that \(\langle A_0 |A_1 \rangle=0\). The superalgebra \(A\) is a pseudo-composition superalgebra if the bilinear form \(\langle\;|\;\rangle\) being extended to the Grassmann envelope \(G(A)=(G_0\otimes A_0) \oplus (G_1\otimes A_1)\) becomes a supersymmetric bilinear form over \(G_0\) on \(G(A)\) and (i) \(\langle G_0\otimes A_0 |G_1\otimes A_1\rangle=0\); (ii) \(\langle\;|\;\rangle\) is symmetric on \(G_0\otimes A_0\) and skew-symmetric on \(G_1\otimes A_1\); (iii) \(x^3=\langle x|x\rangle x\) for all \(x\in G(A)\). The main result of the paper presents a classification of finite dimensional superalgebras \(A\) of characteristic different from 2 and 3 such that the form \(\langle\;|\;\rangle\) is non-degenerate on \(A\). This result generalizes a corresponding of pseudo-composition algebras obtained by \textit{K. Meyberg} and \textit{J. M. Osborn} [Math. Z. 214, 67-77 (1993; Zbl 0788.17003)].
scientific article; zbMATH DE number 1448635
scientific article