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Vesselin Stoyanov Drensky | Institute of Mathematics Bulgarian Academy of Sciences - bulgaria


The project is devoted to the study of algebras with polynomial identities. More precisely we shall study generic matrix algebras and the corresponding algebras of invariants under the action of classical groups. The algebra of generic matrices is one of the most importantobjects of study in PI theory. One of the most successful approaches to the polynomial identities satisfied by matrix algebras is viageneric matrices and invariants. Thus the general linear group acts by simultaneous conjugations on the d-tuples of matrices of order n; the corresponding algebra of invariants coincides with the algebra generated by the traces of d generic matrices of order n. If Gis a subgroup of the general linear group one may consider the restriction of the above action to G. Two very interesting cases arise when one fixes G to be the orthogonal or the symplecticgroup. The corresponding algebras of invariant are very closely related to the polynomial identities with involution of the matrixalgebra. In the case of a transpose involution one considers the orthogonal group while for a symplectic involution one has the symplectic group. The main goal of the project is to describe minimal systems of generators and defining relations for the algebras oforthogonal invariants for matrices of order 3 (any d), and 4 (possibly for small values of d). When n=4 we are planning to describe the symplectic invariants for small d's. The results will be then applied to the study of the cocharacters of algebras with involution, with a special attention to the matrix algebras and their two involutions. (AU)

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