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Subalgebras and discriminants of anticommutative algebras. (English. Russian original) Zbl 0948.14002

Izv. Math. 63, No. 3, 583-598 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 3, 169-184 (1999).
In the paper under review the author continues the study of anticommutative algebras generic in the sense of algebraic geometry which was started by himself [E. A. Tevelev, Transform. Groups 1, No. 1-2, 127-151 (1996; see the preceding review Zbl 0948.14001)]. He fixes integers \(k,n\) such that \(1<k<n-1\) and identifies the \(k\)-ary anticommutative \(n\)-dimensional algebras with the points of the vector space \({\mathcal A}_{n,k}=\bigwedge^kV^{\ast}\otimes V\) of \(k\)-linear anticommutative maps from \(V={\mathbb C}^n\) to \(V\). The main result of the paper shows that for a generic algebra \(A\in {\mathcal A}_{n,k}\) there are no \(m\)-dimensional subalgebras for \(k+1<m<n\), the number of \((k+1)\)-dimensional subalgebras is finite (and explicitly given) and the set of \(k\)-dimensional subalgebras is a smooth irreducible \((k-1)(n-1)\)-dimensional subvariety of the Grassmannian \(\text{ Gr}(k,A)\). Since the general approach of the paper does not allow to study the structure of subalgebras of concrete algebras, the author considers two classes of algebras called \(D\)-regular, when the variety of \(k\)-dimensional subalgebras is smooth and irreducible of dimension \((k-1)(n-k)\), and \(E\)-regular, for \(k=n-2\), again with a finite number of \(n-1\)-dimensional subalgebras. Finally, the author applies his results and studies in detail regular 4-dimensional anticommutative algebras.

MSC:

14A22 Noncommutative algebraic geometry
16S38 Rings arising from noncommutative algebraic geometry
15A75 Exterior algebra, Grassmann algebras
14M15 Grassmannians, Schubert varieties, flag manifolds

Citations:

Zbl 0948.14001
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