prof. Vladimir D. Lyakhovsky, Sankt-Petersburg State University

Algebraic symmetry problems in quantum theory. New methods to deal with representations

Quantum theory has a long and fruitful coexistence with Lie groups and Lie algebras representation theory. Both partners had certain benefits in this union. Numerous tasks born in quantum field theory and especially in model building provided an intensive development of representation theory during last decades. More and more complicated algebraic constructions are involved. In this report I want to demonstrate that old chapters of this theory have not yet obtained their final form and there are problems that are to be reconsidered and new methods that are to be applied. Common problems of applied symmetry will be reconsidered, such as 1. how to find eigenfunctions of mutually commuting operators and their multiplicities, 2. how to decompose operators representing symmetry transformations and find irreducible subspaces, 3. how tensors (and especially tensor powers) could be simplified, 4. what new symmetric polynomials are to be introduced. To solve these problems new tools are used. They are mostly based on geometric approach. It will be demonstrated how new presentation reduces complexity of problems makes some solutions obvious.