Algebraic Aspects of the Quantum Yang–Baxter Equation

In this paper we examine a variety of algebraic contexts in which the quantum Yang–Baxter equation arises, and derive methods for generating new solutions from given ones. The solutions we describe are encoded in objects which have a module and a comodule structure over a bialgebra. Our work here is based in part on the ideas of [DR1, DR2]. Suppose that R : M ⊗M −→ M ⊗M is a solution to the quantum Yang–Baxter equation, where M is a finite-dimensional vector space over a field k. In [DR2] it was shown that R can be derived from a left H-module structure and a right H-comodule structure on M for some bialgebra H over k (also see [Yet], [Maj1], [Maj2], [FRT2]). The module and comodule structure satisfy a natural compatibility condition. M together with these structures is called a left quantum Yang– Baxter H-module. Every left quantum Yang–Baxter module gives rise to a solution to the quantum Yang–Baxter equation in a natural way. We study the category HYB of left quantum Yang–Baxter H-modules, and show that many of the familiar constructions in the module category HM can be made in HYB. Through these constructions solutions to the quantum Yang–Baxter equation can be combined in various ways to generate new ones. We place particular emphasis on the tensor product of objects of HYB, and ways in which the tensor algebra T (M) of an object M of HYB can be viewed as an object of HYB. Using the tensor algebra we are able to extend certain solutions when dim M = 2 to the quantum plane. We turn the tensor algebra T (M) of an object M of HYB into an object of HYB by giving it a module algebra and a comodule algebra structure. Module algebra and comodule algebra structures play a very important role in this paper. In Section 9 we develop a calculus for Uq(sl2) starting with a right k[SLq(2)]-comodule algebra structure on the quantum plane k[x, y]q, where k[SLq(2)] is the coordinate ring of quantum SLq(2) over k. This calculus has been described in other ways [M-S,Sud]. The right comodule algebra structure on k[x, y]q accounts for the left Uq(sl2)-module algebra structure on k[x, y]q described in [M-S]. We show how the basic irreducible representations of [Tak] follow by the calculus. A module algebra action, or comodule algebra action, is a natural vehicle for extending structure on a submodule, or subcomodule, respectively to the subalgebra it generates. The calculus, and the solutions to the quantum Yang–Baxter equation we describe in this paper, are determined in this manner on small structures, two-dimensional in this case. We believe that many more results will follow from this circle of ideas. In Section 2 we discuss basic aspects of coalgebra and bialgebras used in the sequel. A good general reference for this is Sweedler’s book [SB] on Hopf algebras. In Section 3 we review the quantum Yang–Baxter equation in connection with quantum Yang– Baxter modules. The A(R) construction of Faddeev, Reshetikhin and Takhtajan [FRT] is considered in this light, and examined in detail in when R arises from two commuting operators. Even though this particular R is rather simple, the algebra structure of A(R) is surprisingly complex. Using the Scratchpad computer algebra system, we determine the algebra structure of A(R) in an interesting specific case, showing that our particular example has a finite (non-commutative) Groeb-