Finite-Dimensional Irreducible Representations of the Very Non Standard Quantum so(2N-1)-Algebra

Abstract

Quantum groups arose in the mid 1980s in the study of certain integrable models in mathematical physics. Fundamental objects in the theory of quantum groups, as developed by V. Drinfeld and M. Jimbo, are certain deformations of universal enveloping algebras of semisimple Lie algebras. The resulting quantised enveloping algebras depend on a deformation parameter q which, for the purpose of this thesis, is assumed not to be a root of unity. Crucially, quantised enveloping algebras retain the structure of a Hopf algebra from universal enveloping algebras. Moreover, the finite-dimensional representations of quantised enveloping algebras are classified in terms of highest weights, similar to the classification of finite-dimensional representations of semisimple Lie algebras. There exist other deformations of universal enveloping algebras which are not Drinfeld-Jimbo quantum groups. One of the earliest classes of examples is that of the non-standard quantum sonalgebras introduced by A. Gavrilik and A. Klimyk. These algebras appear as special examples in the theory of quantum symmetric pairs developed by G. Letzter in the late 1990s. This theory provides quantum group analogues of Lie subalgebras fixed under an involutive automorphism. Quantum symmetric pairs are given in terms of a coideal subalgebra of a quantised enveloping algebra. The representation theory of these coideal subalgebras is not known in general, having only been determined for certain classes of examples. The present thesis is devoted to the representation theory of the coideal subalgebra corresponding to the symmetric pair of type DII in the Cartan classification. In this case, so2N−1 is realised as a Lie subalgebra of so2N fixed under an involutive automorphism. The resulting coideal subalgebras are not isomorphic to the Gavrilik-Klimyk algebras, and we hence call them very non-standard quantum so2N−1-algebras. In this thesis, we classify the finite-dimensional irreducible representations of the very nonstandard quantum so2N−1-algebras. Importantly, these algebras have a very simple analogue of a Cartan subalgebra, and every finite-dimensional module is a weight module. We show that (up to a choice of signs) the irreducible representations of very non-standard quantum so2N−1- algebras are uniquely determined by a highest weight. We construct root vectors and prove a Poincar´e-Birkhoff-Witt theorem which supports a triangular decomposition. The root vectors then allow us to introduce a notion of Verma modules, and we show that every simple module is obtained as a quotient of a Verma module. The arguments to this point mimic the known approach to the representation theory of quantised enveloping algebras. The weights also need to be dominant integral for the simple quotients of Verma modules to be finite-dimensional. However, in the coideal case, it is harder to show that dominant integral weights are actually sufficient to obtain finite-dimensional simple quotients. The reason for this is a missing sl2-triple which, when found, acts only on a subspace of the representation, and hence cannot be used to obtain Weyl-group invariance. Instead, we use a filtered-graded argument to show that a (possibly larger) quotient of the Verma module, which can be considered as a module for the ambient Hopf algebra, is finite-dimensional.