Coideal Subalgebras and Quantum Symmetric Pairs

By Gail Letzter. 2001

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Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half discusses various well known quantum coideal subalgebras and the implications of the abstract theory on these examples. The focus is on the locally finite part of the quantized enveloping algebra, analogs of enveloping algebras of nilpotent Lie subalgebras, and coideals used to form quantum symmetric pairs. The last family of examples is explored in detail. Connections are made to the construction of quantum symmetric spaces. The introduction of quantum groups in the early 1980’s has had a tremendous influence on the theory of Hopf algebras. Indeed, quantum groups provide a source of new and interesting examples. We shall discuss the reverse impact: the theory of quantum groups uses the Hopf structure extensively. This special structure is often hidden in the classical setting, while it is prominent and fundamental for quantum analogs. Let g be a semisimple Lie algebra and write G for the corresponding connected, simply connected algebraic group. There are two standard types of quantum groups associated to g and G. The first is the quantized enveloping algebra which is a quantum analog of the enveloping algebra of g. The second is the quantized function algebra which is a quantum analog of the algebra of regular functions on G. We will be focusing on a particular aspect of the Hopf theory of both types of quantum groups: the study of (one-sided) coideal subalgebras. One of the reasons coideal subalgebras are so important in the study of quantum groups is that quantum groups do not have “enough” Hopf subalgebras. This shortage of Hopf subalgebras is especially notable for quantized enveloping algebras. Consider a Lie subalgebra t of the Lie algebra g. The enveloping algebra U(t) of t is a Hopf subalgebra of U(g). However, upon passage to the Supported by NSA grant MDA 904-99-1-0033. AMS subject classification: 17B37.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

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