Iwahori - Hecke type algebras associated with the Lie superalgebras A ( m , n ) , B ( m , n ) , C ( n ) and D ( m , n ) Hiroyuki Yamane

In this paper we give Iwahori-Hecke type algebras Hq(g) associated with the Lie superalgebras g = A(m,n), B(m,n), C(n) and D(m,n). We classify the irreducible representations of Hq(g) for generic q. Introduction Recently, motivated by a question posed by V. Serganova [S] and study of the Weyl groupoids [H1][H2] associated with Nichols algebras [AS1][AS2] including generalizations of quantum groups, I. Heckenberger and the author [HY] introduced a notion of ‘Coxeter groupoids’ (in fact they can be defined as semigroups), and showed that a Matsumoto-type theorem holds for the groupoids, so they have the solvable word problem. We mention that the Coxeter groupoid associated with the affine Lie superalgebra D(2, 1; x) was used in the study [HSTY], where Drinfeld second realizations of Uq(D (2, 1; x)) was analized by physical motivation in recent study of AdS/CFT correspondence. It would be able to be said that one of the main purposes at present of the representation theory is to study the Kazhdan-Lusztig polynomials (cf. [Hu, 7.9]) and their versions. The polynomials are defined by using the standard and canonical bases of the Iwahori-Hecke algebras. The existence of those bases is closely related to the Matsumoto theorem of the Coxeter groups. So it would be natural to ask what to be the Iwahori-Hecke algebras of the Coxeter groupoids. In this paper, we give a tentative answer to this question for the Coxeter groupoids W associated with the Lie superalgebras g = A(m,n), B(m,n), C(n) and D(m,n). We introduce the Iwahori-Hecke type algebra Hq(g) (in the text, it is also denoted by Hq(W )) as q-analogue of the semigroup algebra CW/C0, where 0 is the zero element of W . We also show that if q is nonzero and not any root of unity, Hq(g) is semisimple and there exists a natural one-to-one correspondence between the equivalence classes of the irreducible representations of Hq(g) and