Generic algebras (Q5917845)

The author considers either commutative or anti-commutative finite-dimensional algebras \(A\) over \(\mathbb{C}\) which are generic (in the sense of algebraic geometry). He proves many interesting results on the structure of sub-algebras of a given dimension \(k\). For example, if \(\dim(A) >k>3\) then there are no \(k\)-dimensional sub-algebras, the number of 3-dimensional sub-algebras is finite and equal to the Chern number of some explicit vector bundle on the Grassmannian \(\text{Gr}(3,n)\). Next, it can be attached a projective cubic surface \(K\) to any 4-dimensional \(A\) (such that the operators of the left multiplication have zero trace). It is shown for generic such \(A\) that there is a natural isomorphism between the variety of non-commutative 2-dimensional sub-algebras in \(A\) and the surface \(K\) minus two projective lines.