Hochschild Cohomology of Quiver Algebras Defined by Two Cycles and a Quantum-Like Relation

Abstract

We consider quiver algebras A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k, and describe the minimal projective bimodule resolution of A _q . In particular, in the case q = 1, we determine the Hochschild cohomology ring of A ₁ and show that it is a finitely generated k-algebra. Moreover the Hochschild cohomology ring of A ₁ modulo nilpotence is isomorphic to the polynomial ring of two variables.

Key Words:

• Finiteness conditions
• Hochschild cohomology
• Projective bimodule resolution
• Quantum-like relation

2000 Mathematics Subject Classification:

• 16E40
• 16G20
• 18G05

ACKNOWLEDGMENTS

The author is grateful to Professor K. Sanada and Dr. T. Furuya for many valuable suggestions and encouragement. Furthermore, the author would like to thank Professor N. Snashall for giving him some piece of advice during her visit to the Tokyo University of Science.

Notes

Communicated by D. Zacharia.