Symbolic Computation Group

The factorization of differential operators is an old problem. In the case of ordinary differential operators (1D case), a factorization into first- order factors exists for every operator, and the structure is known owing to a theorem of Frobenius. Already in the case of two variables (2D case), the situation is very different and a factorization may not exist at all or there may be two different factorizations into different numbers of irreducible factors for the same operator.

There is a lot of purely algebraic work on the subject. In this talk we would like to take more care about the geometric properties of the operators.

Specifically, we shall talk about two possible directions: the case of operators acting on the superline (one even and one odd variable); and the case of operators acting on the algebra of densities (introduced in 2004 by H. Khudaverdian and Th. Voronov in connection with Batalin-Vilkovisky geometry) on the line. Both are in a sense cases of “1 and 1/2” dimensions. In the case of the superline, a result analogous to the Frobenius result can be proved. In the case of the algebra of densities on the line, operators behave differently from what we know for the 1D case. We discovered that one can explicitly describe the obstruction to factorization of the generalized Sturm-Liouville operator in terms of a solution of the corresponding classical Sturm-Liouville equation.

The talk is based on the joint work with Th. Voronov and my student Simon Li.

The related papers are:
Li, Shemyakova, Voronov, Lett. Math. Phys.,Volume 107, Issue 9, 2017;
Shemyakova, Voronov, Lett. Math. Phys., 2018.