Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

作者：de la Pena Jose A^*

来源：Algebras and Representation Theory, 2014, 17(3): 905-930.

DOI：10.1007/s10468-013-9424-0

• Finite dimensional algebra
• Eigenvalues of the Coxeter transformation
• Coxeter polynomial
• Kronecker's theorem
• Mahler measure of a polynomial

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is basic connected with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation I center dot (A) as the automorphism of the Grothendieck group K (0)(A) induced by the Auslander-Reiten translation tau in the derived category Der(modA) of the module category modA of finite dimensional left A-modules. We say that A is an algebra of cyclotomic type if the characteristic polynomial chi (A) of I center dot (A) is a product of cyclotomic polynomials. There are many examples of algebras of cyclotomic type in the representaton theory literature: hereditary algebras of Dynkin and extended Dynkin types, canonical algebras, some supercanonical and extended canonical algebras. Among other results, we show that: (a) algebras satisfying the fractional Calabi-Yau property have periodic Coxeter transformation and are, therefore, of cyclotomic type, and (b) algebras whose homological form h (A) is non-negative are of cyclotomic type. For an algebra A of cyclotomic type we describe the shape of the Auslander-Reiten components of Der(modA).

• 出版日期2014-6