On The Finite Basis Problem For Semigroups And Representations Of Quantum Affine Algebras

The finite basis problem for some semigroups and some problems in the represen-tation theory of quantum affine algebras and its applications are studied in this disser-tation. We study the finite basis problem for ε4,(?)(?)n(n=2,3),(?)2(F)(|F|=2), the extended T-system of type G2, the correspondence between XXZ type Bethe ansatz equations and some quasi-polynomial spaces. We explain them more precisely as fol-lows.1. We show that the monoid of extensive transformations of a chain of order four is finitely based and a finite basis for this monoid is given. This completes the descrip-tion of the equational property of the monoids of all full extensive transformations, partial extensive transformations, and partial order-preserving extensive transforma-tions over any finite chain.2. Let (?)(?)n denote the monoid of all upper triangular boolean n x n matrices. It is shown by Volkov and Goldberg that (?)(?)n is nonfinitely based if n>3, but the cases when n=2,3remain open. We show that the monoid (?)(?)2is finitely based and give a finite identity basis for the monoid (?)(?)2.We also show that (?)(?) is inherently nonfinitely based. Hence (?)(?)n is finitely based if and only if n<2.3. Let (?)n(F) denote the submonoid of all upper triangular n×n matrices over a finite field F. It is shown by Volkov and Goldberg that (?)n(F) is nonfinitely based if|F|>2and n≥4, but the cases when|F|>2and n=2,3or when|F|=2remain open. We show that the monoid (?)2(F) is finitely based when|F|=2, and a finite identity basis for (?)2(F) is given. Moreover, all maximal subvarieties of the variety generated by (?)2(F) with|F|=2are determined.4We define the modules Bk,l(s),Ck,l(s),Dk,l(s),εk,l(s),Fk,l(s)Bk,l(s),Ck,l(s),Dk,l(s)εk,l(s),Fk,l(s) of the quantum affine algebra Uq(g) of type G2. We show that these modules satisfy a set of3-term recurrence relations, called extended T-system. This extended T-system contains the celebrated T-system of type G2.We show that the modules in the ex-tended T-system of type G2are special or anti-special. Therefore the FM algorithm applies to these modules. We define left modules, right modules, top modules, bot-tom modules, and sources. We show that the tensor product of a top module and the corresponding bottom module is irreducible and the sources are irreducible. Us-ing the extended T-system, we compute the dimension formulas for the modules in the extended T-system. In particular, we obtain the dimension formulas for the G2minimal affinizations Bkl(s), Bkl(s).5. We study XXZ type Bethe ansatz equations. On the one hand, starting from some quasi-polynomial spaces, we construct the solutions of Bethe ansatz equations. On the other hand, starting from the solutions of Bethe ansatz equations, we construct some quasi-polynomial spaces. In this way, we show that the orbits of solutions to XXZ type Bethe ansatz equations correspond to some spaces of quasi-polynomials.