The Wedderburn decomposition in finite-dimensional alternative superalgebras
An alternative superalgebra \(A= A_0\oplus A_1\) over a field \(k\) is a \(\mathbb{Z}_2\)-graded algebra satisfying identities \[ (x,y, z)= (-1)^{ij+1} (y,x,z)= (-1)^{jk+1} (x,z,y), \] where \(x\in A_i\), \(y\in A_j\), \(z\in A_k\), \(i,j,k\in \{0,1\}\) and \((x,y,z)= (xy)z- x(yz)\). Every graded simple finite dimensional superalgebra \(A\) is an alternative algebra. If \(A_1\neq 0\) then \(A\) is associative [\textit{E. I. Zelmanov} and \textit{I. P. Shestakov}, Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 4, 676-693 (1990; Zbl 0724.17022)]. \textit{A. Kemer} [Algebra Logika 26, No. 5, 597-641 (1987; Zbl 0664.16017)]\ classified all simple alternative superalgebras \(A\) over an algebraically closed field \(k\) into 4 classes: (1) \(A= A_0= C\) -- a Cayley-Dickson \(k\)-algebra; (2) \(A= \text{Mat} (n,k) \oplus \text{Mat} (n,k)\); (3) \(A= \text{Mat}^{p,q} (n,k)\); (4) \(A= A_0= \text{Mat} (n,k)\). Suppose that \(\text{char } k\neq 2,3\) and let \(N\) be the solvable radical of \(A\) such that \(A/N\) is a separable algebra which is a direct sum of finitely many simple superalgebras. Assume that \(M(1,k) \oplus \text{Mat} (1,k)\) and \(\text{Mat}^{1,1} (2,k)\) do not occur as simple constituents of \(A/N\). Then \(A\) contains a subalgebra \(S\) such that \(S\simeq A/N\). There are given two examples of alternative superalgebras whose solvable radical does not split off. A similar result for alternative algebras was proved by \textit{R. D. Schafer} [Introduction to nonassociative algebras (London, Academic Press, 1966; Zbl 0145.25601)].
scientific article; zbMATH DE number 738956