Research Focus: Professor Losev studies Symmetry in Algebra

One manifestation of symmetry in algebra concerns algebraic varieties. High school students learn that the "roots"  of a polynomial p(x) are solutions of the equation p(x)=0. If we generalize this to a system of polynomial equations in several variables x1, x2,..., xn, then the solution set is called an "algebraic variety." These objects can be complicated with various kinds of singularities; even the simple equation xy=0 has the union of two lines as its solution set. However, Professor Losev uses symmetries in the algebraic varieties to study and classify them.

The symmetries of a variety are revealed through its transformations. Taking the composition of two symmetries, or inverting  a single symmetry, one again gets a symmetry, so the symmetries form a group. Of special interest is the case when the group of symmetries is algebraic, i.e, is itself an algebraic variety:  an example is the group of all matrices with nonzero determinant. Of course, one expects varieties with many symmetries to be interesting, and a "spherical variety" is one that is equipped with a particularly large algebraic group action. Examples of spherical varieties include toric varieties, that are very popular in algebraic geometry, as well Grassmann or more general flag varieties, and symmetric spaces. Spherical varieties appear in some enumerative problems of algebraic geometry, in representation theory, and in the theory of Hamiltonian systems with symmetries.

A very nice feature of spherical varieties is that, although their (algebraic) geometry is rich and complicated, they can be both classified and studied using some combinatorial-geometric invariants, like lattices or cones. Professor Losev proved a number of results that use combinatorial invariants to uniquely characterize a spherical variety in a given class. For example, he proved a conjecture of Knop on smooth affine spherical varieties; this conjecture is relevant for the relationship between spherical varieties and Hamiltonian systems.

Another manifestation of symmetry in algebra occurs in representation theory; here symmetries can take the form of various algebraic structures (such as groups, associative algebras, or Lie algebras) that are being represented as linear transformations of a vector space.  The goal is to classify representations of a given group or algebra and/or to compute important invariants of a representation such as the  dimension of the underlying vector space or a more subtle invariant called "character". In the case of a group representation, the character is the function that maps a group element to the trace of the corresponding transformation.

A very nice and  classical story from the beginning of the 20th century is the finite dimensional representation theory of complex semisimple Lie algebras (the Lie algebra of skew-symmetric matrices, for instance). Here the representations are fully classified and combinatorial formulas for their characters are known. By contrast, infinite dimensional representation theory is much more complicated; it has been studied   extensively since the 1960's, revealing connections to algebraic topology as well   as algebraic and symplectic geometry.

A representation of a Lie algebra is the same as a representation of a certain associative algebra, called the "universal enveloping algebra" that is  important in the finite dimensional theory and is really essential in the infinite dimensional one. It has a much more complicated generalization called a "W-algebra" that appeared in the work of Kostant in the late 1970's in a special case, with the general definition given by Premet in 2002. W-algebras are directly connected to the   representation theory of semisimple Lie algebras in the following way: one hopes to reduce "nice" infinite dimensional representations of the Lie algebra to a finite dimensional representation theory of a W-algebra. Professor Losev has made this connection precise in several cases that yield some down-to-earth results both for W-algebras (classification of finite dimensional irreducible modules and character formulas) and for universal enveloping algebras (computation of Goldie ranks for primitive ideals).

So far, Professor Losev has found symmetries in several different areas of algebra, and has made good use of them. He plans to keep looking.