IRREDUCIBLE REPRESENTATIONS OF A CLASS OF FINITE-DIMENSIONAL ALGEBRAS

Abstract

ABSTRACT In the previous papers1-2 we constructed an -dimensional Hopf algebra which is isomorphic to Drinfeld quantum double of Taft's Hopf algebra if and is a primitive nth root of unity, and studied the irreducible representations of . In this paper, we continue our study and examine the irreducible representations of when q is any nth root of 1. We give the structures of all simple -modules, and then classify them. We show that there are only distinct simple -modules up to isomorphism. Though is not a bialgebra, we show that the tensor product of two simple -modules possesses an -module structure in away similar to the case that q is a primitive nth root of 1. Then we give a sufficient and necessary condition for the tensor product of two simple -modules to be semisimple. More generally, we describe the structure of socal of the tensor product. Quantum groups arose from the study of quantum inverse scattering method, especially the Yang-Baxter equation. Quantum groups added new aspects to representation theory. Let be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. 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 (see[3], [5]. The module and comodule structures satisfy a natural compatibility condition. M together with this structure is called a quantum Yang-Baxter H-module, which is also called a Yetter-Drinfeld module over H (see6-7. A quantum Yang-Baxter H-module is a (left) crossed H-bimodule.[5] Let H be a finite-dimensional Hopf algebra over afield k, and let be the Drinfeld's quantum double derived from H.[8] It is well known9-10 that a vector space M possesses a crossed H-bimodule structure if and only if M possesses a left -module structure. Hence the Yetter-Drinfeld module category is the same as the left -module category Suppose that and that is a primitive nth root of unity. Then is also a primitive nth root of unity. Taft constructed an -dimensional Hopf algebra in[11]. The 's form an interesting class of pointed Hopf algebras from a combinatorial point of view. When n is odd, provides an invariant of three-manifolds.[12] Generally, the double of is of interest in connection with knot theory. Kauffman and Radford showed[13] that is a ribbon Hopf algebra if and only if n is odd. In the paper,[1] the author constructed an infinite-dimensional noncommutative and noncocommutative Hopf algebra for any with . When q is a root of the nth cyclotomic polynomial over Z, has an -dimensional quotient Hopf algebra . If q is a primitive nth root of unity, then is isomorphic to as a Hopf algebra for any . If q is just a nth root of 1, then is merely an algebra.In[2], the author examined the irreducible representations of , equivalently, of , where and q is a primitive nth root of 1. In this paper, we will consider the more general case that q is any nth root of 1, and examine the irreducible representations of the algebra . In this case, let m be the order of q, then , and is an algebra which is not necessarily a bialgebra (Hopf algebra). Let . In Sec. 2, we give the classification of the simple -modules. We show that for any and there exists a unique l-dimensional simple -module , and that for any and , there exists a unique m-dimensional simple -module . We also show that any simple -module is either isomorphic to some or isomorphic to some . Upto isomorphisms, there are only distinct simple -modules. Though is not a bialgebra, we will show that the tensor product of two simple -modules possesses an -module structures which is similar to the case[2] that is a Hopf algebra. In Sec. 3, we consider the tensor product of two simple -modules U and V, and give a sufficient and necessary condition for the tensor product to be semisimple. We first show that if U and V are simple -module with dim or dim then is also a simple -module. Then we prove that is a semisimple -module if and only if , that with is semisimple if and only if , and that is semisimple if and only if or (mod m). Generally, we describe the structure of socal of the tensor product. In particular, if we take , then we can get all the results described in[2].