Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

Abstract

Abstract. 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 φ A as the automorphism of the Grothendieck group K 0 (A) induced by the Auslander-Reiten translation τ 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 χ A of φ 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).

On the Mahler measure of the Coxeter polynomial of an algebra

“…In [6], we showed that several of the common properties of classes of algebras presented in the above sections are consequence of the cyclotomic condition. In particular, properties of the structure of the Auslander-Reiten quiver of the derived category Der(A).…”

“…In a recent paper [6] we show that always e(φ A ) ≥ n and equality is reached when A is of cyclotomic type, that is, χ A (T ) factorizes as product of cyclotomic polynomials (equivalently ρ A = 1 or M (φ A ) = 1). Many finite dimensional algebras are known to be of cyclotomic type: hereditary algebras of finite or tame representation type, canonical algebras, some extended canonical algebras and many others, see [17,18,23,6] and Section 2 for more examples. For algebras A not of cyclotomic type, important results concerning the growth of the iterated Auslander-Reiten translation τ s X, for X an A-module, s ∈ N, have been shown, see [8,7,21].…”

“…In fact, R 2n = A n ⊗A 2 , where A s is the hereditary algebra associated to the linear quiver 1 → 2 → · · · → s, see [6].…”

“…Irreducible factorization of χ A ρ(ϕ A ) (2, 3, 11) 1 + T − T 3 − T 4 + T 6 + T 7 + T 8 + T 9 + T 10 − T 12 − T 13 + T 15 + T 16 1.1064 (2,4,9) Φ 2 (T )Φ 5 (T )(T 10 + T 9 − T 5 + T + 1) 1.1329 (2,5,8) Φ 2 (T )(1 + T 4 + T 6 + T 7 + T 8 + T 10 + T 14 ) 1.1574 (2, 6, 7) Φ 2 (T )(1 + T 4 + T 5 + T 6 + T 8 + T 9 + T 10 + T 14 ) 1.1669 (3,3,8) Φ 3 (T )(1 + T 5 + T 6 + T 7 + T 12 ) 1.1498 (3,4,7) 1 +T +T 2 +T 3 +T 4 +2T 5 +3T 6 +3T 7 +3T 8 +2T 9 +T 10 +T 11 +T 12 +T 13 +T 14 1.1847 (3,5,6) Φ 3 (T )(T 12 + T 9 + T 8 + T 7 + T 6 + T 5 + T 4 + T 3 + 1) 1.1966 (4,4,6) Φ 2 (T )Φ 4 (T )(T 10 − T 9 + T 8 + T 6 + T 4 + T 2 − T + 1) 1.2175 (4,5,5) Φ 5 (T )(T 10 + T 7 + T 6 + T 5 + T 4 + T 3 + 1) 1.2277 Table 2 Weights p with M (φ A ) = 1.…”

“…Three important measures are the following: the spectral radius of A is defined as ρ A = max{|λ| : λ ∈ Spec(φ A )}, the Mahler measure of χ A is M (χ A ) = n i=1 max{1, |λ i (A)|} and the energy function of φ A is e(φ A ) = n i=1 |λ i (A)|. In a recent paper [6] we show that always e(φ A ) ≥ n and equality is reached when A is of cyclotomic type, that is, χ A (T ) factorizes as product of cyclotomic polynomials (equivalently ρ A = 1 or M (φ A ) = 1). Many finite dimensional algebras are known to be of cyclotomic type: hereditary algebras of finite or tame representation type, canonical algebras, some extended canonical algebras and many others, see [17,18,23,6] and Section 2 for more examples.…”