Cyclotomic Temperley–Lieb Algebra of Type D and Its Representation Theory

Abstract

We define a new class of algebras, cyclotomic Temperley–Lieb algebras of type D, in a diagrammatic way, which is a generalization of Temperley–Lieb algebras of type D. We prove that the cyclotomic Temperley–Lieb algebras of type D are cellular. In fact, an explicit cellular basis is given by means of combinatorial methods. After determining all the irreducible representations of these algebras, we give a necessary and sufficient condition for a cyclotomic Temperley–Lieb algebra of type D to be quasi-hereditary.

Key Words:

• Cellular algebras
• Cyclotomic Temperley–Lieb algebras of type D
• Irreducible representations
• Quasi-hereditary algebras

2000 Mathematics Subject Classification:

• Primary 16G30
• Secondary 20G05, 57M25

ACKNOWLEDGMENT

The author would like to thank the referee for his/her useful comments. The author is also grateful to Professor Changchang Xi for his encouragement and advice.

Notes

Present address: Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1 N GN5. [email protected]

Communicated by D. Happel.