Naruhiko Aizawa
Prefecture University of Osaka, Japan
Recent developments of representation theory of nonrelativistic conformal algebras.
Abstract: Nonrelativistic conformal algebra is a particular class of non-semisimple Lie algebras. The member of the class is a finite or an infinite dimensional Lie algebra. The semisimple part of the finite dimensional algebras is the direct sum of sl(2) and so(d), while the Virasoro algebra is the semisimple part of the infinite dimensional algebras. This class of Lie algebra appears in various kind of problems in theoretical and mathematical physics. For instance, one can find them in connection with fluid dynamics, gravity theory, AdS/CFT correspondence and vertex operator algebras. This motivate us to study representations of the nonrelativistic conformal algebras. In the beginning of this talk, I introduce various members of the nonrelativistic conformal algebra. Then I pick up some members of physical interest and study them in some more detail. Our first problem is the central extensions of the algebras. The list of possible central extensions is given. Our second problem is the irreducible representations of highest (lowest) weight type. We start with the Verma module and study its irreducibility. This is done by calculating the Kac determinant and explicit construction of singular vectors. If the Verma module is reducible, then it will be shown that how to obtain the irreducible modules.