Algebras and orthogonal groups. I
The authors investigate nonassociative algebras \(A\) over a field \(K\) with large automorphism groups \(G=\Aut A\); more precisely, they call \(A\) malleable if the stabilizer \(G_{x,y}\) of \(x,y\in A\) fixes precisely the elements in \(Kx+Ky\) (or in \(Kx+Ky+Ke\), if \(A\) has a unit element \(e\)). \(A\) is called linear if its multiplication is given by \(xy= f(y)x+g(x)y\) with linear functionals \(f,g\). The authors prove the following: Theorem 1.2: Up to one exception with \(|A|=4\), a nonassociative algebra \(A\) is linear if and only if every subspace of \(A\) is a subalgebra of \(A\) (these algebras are called \(S\)-algebras in \textit{I. Kaplansky} [Algebras with many derivations, in: Aspects Math. Appl., 431-438 (1986; Zbl 0592.17005)]). Furthermore, the authors describe the isomorphism type of these algebras in terms of orbits of linear groups. By Theorem 3.3, a non-zero algebra \(A\) is malleable if and only if \(A\) is linear, with only few explicitly described exceptions (the exceptions \(A\) have dimension 2, or \(|A|=8\)). An algebra \(A\) with unit element \(e\) is called bilinear if its multiplication is given by \(xy= \beta(e,y)x+ \beta(x,e)y- \beta(x,y)e\) for some bilinear form \(\beta\). The authors characterize bilinear algebras by the requirement that each 3-dimensional subspace containing \(e\) is a subalgebra (Theorem 4.3). Then they describe the isomorphism classes of bilinear algebras of commutative type and of Lie type in terms of orbits of some linear groups, and they determine which algebras of these types are quasi-alternative [in the sense of \textit{E. Becker}, Abh. Math. Semin. Hamb. 38, 88-105 (1972; Zbl 0233.17002)]. Furthermore, they determine the ideals, the units, and the idempotent and nilpotent elements of bilinear algebras. Part II of this paper appears in vol. 67 (1997) of the same journal.
scientific article; zbMATH DE number 977855
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