Colloquium Mathematics - Dr. Arthemy Kiselev, University of Groningen

Title: The Wronskians over multidimension and homotopy Lie algebras.

Abstract:

The Wronakian determinant of N functions in one variable x helps us verify that they are linearly independent on an interval (a,b) in R. Can we have an equally convenient procedure over multidimensional spaces R^d with Cartesian coordinates x,y,z,...? Yes we can; let us study the definition of Wronskians for functions in many variables, and let us explore which differential-algebraic identities these structures satisfy.

To see why the Wronskian determinants actually do satisfy a set of quadratic, Jacobi-type identities, we observe first that the Wronskian of size 2x2 results from commutation of vector fields on the real line R. From differential geometry we know that vector fields are differential operators of strict order p=1. By taking the alternated composition of N=2p differential operators of strict order p>0 on the affine line, we obtain the operator of same order p with the Wronskian determinant for coefficient. As we had the Jacobi identity for the Lie algebra of vector fields, so we establish the (table of) quadratic, Jacobi-type identities for higher-order Wronskians of N>1 arguments. (In string theory from Theoretical Physics, these identities govern homotopy deformations of Lie algebras, here of vector fields.) We prove that the new Wronskians over multidimensional base with d coordinates x,y,z,... do satisfy the (table of) identities for strongly homotopy Lie algebras.

The problem is to understand how fast the dimension grows under iterated N-ary brackets. We spot a countable chain of finite-dimensional homotopy Lie algebras that generalize the vector field realization of sl(2) on R; we explicitly calculate all the structure constants. Yet the four-dimensional analogue of sl(2) over the plane R^2 with Cartesian coordinates (x,y), now with ternary bracket from the Wronskian, is so far the only known finite-dimensional homotopy Lie algebra of this type over base dimension > 1. The hunt is on; we conclude that Lie algebra sl(2) is the prototype not only for semisimple complex Lie algebras encoded by root systems but also for countably many N-ary homotopy Lie algebras.