On the Mahler measure of the Coxeter polynomial of tensor products of algebras

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• 作者：José A. de la Pe?a
• 关键词：Tensor product of algebras ; Tensor product of polynomials ; Coxeter polynomial ; Cyclotomic polynomial ; Primary 16G20 ; 16G50 ; 11R06 ; Secondary 14G60 ; 18E30
• 刊名：Bolet篓陋n de la Sociedad Matem篓垄tica Mexicana
• 出版年：2014
• 出版时间：October 2014
• 年：2014
• 卷：20
• 期：2
• 页码：257-275
• 全文大小：336 KB
• 参考文献：1.A’Campo, N.: Sur les valeurs propres de la transformation de Coxeter. Ivent. Math. 33(1), 61-7 (1976)
  2.Glasby, S.P.: On the tensor product of polynomials over a ring. Aust. Math. Soc. 71, 307-24 (2001)CrossRef MATH MathSciNet
  3.Happel, D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. Cambridge University Press, Cambridge (1988)CrossRef MATH
  4.Happel, D.: The trace of the Coxeter matrix and Hochschild cohomology. Linear Alg. Appl. 258, 169-77 (1997)CrossRef MATH MathSciNet
  5.Happel, D.: A characterization of hereditary categories with tilting object. Invent. Math. 144, 381-98 (2001)CrossRef MATH MathSciNet
  6.Happel, D.: Hochschild Cohomology of Finite Dimensional Algebras. Lect. Notes Math., vol. 1404, pp. 108-26. Springer, Berlin (1989)
  7.Kronecker, L.: Zwei S?tze über Gleichungen mit ganzzahligen Coefficienten. J. Crelle, Ouvres I, p. 105-08 (1857)
  8.Ladkani, S.: On the periodicity of Coxeter transformations and the non-negativity of their Euler forms. Linear Algebr. Appl. 428(4), 742-53 (2008)
  9.Lenzing, H., de la Pe?a, J.A.: A Chebysheff recursion formula for Coxeter polynomials. Linear Algebr. Appl. 430, 947-56 (2009)CrossRef MATH
  10.Lenzing, H., de la Pe?a, J.A.: Spectral analysis of finite dimensional algebras and singularities. In: Trends in Representation Theory of Algebras and Related Topics, pp. 541-88. EMS Publishing House, Zürich (2008)
  11.Lenzing, H., de la Pe?a, J.A.: Accessible algebras. Preprint. Paderborn, Germany and México (2010)
  12.Mahler, K.: Lectures on Trascendental Numbers. Lecture Notes in Mathematics, vol. 546. Springer, Berlin (1976)
  13.de la Pe?a, J.A.: Algebras whose Coxeter polynomials are product of cyclotomic polynomials. Algebr. Represent. Theory 17, 905-30 (2014).
  14.de la Pe?a, J.A.: On the Mahler measure of Coxeter polynomials of finite dimensional algebras. Adv. Math. (to appear)
  15.Prasolov, V.: Polynomials. Algorithms and Computation in Mathematics, vol. 11. Springer, Berlin (2001)
  16.Xi, Ch.: On wild hereditary algebras with small growth numbers. Commun. Algebr. 18(10), 3413-420 (1990)CrossRef MATH
  
• 作者单位：José A. de la Pe?a (1) (2)
  
  1. Centro de Investigación en Matemáticas, A.C., 36240, Guanajuato, Mexico
  2. Instituto de Matemáticas, UNAM, Cd. Universitaria, 04510, Mexico, D.F., Mexico
  
• 刊物类别：Mathematics, general;
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：2296-4495

文摘

Let \(A\) be a finite-dimensional algebra over an algebraically closed field \(k\). Assume \(A\) is basic and connected with \(n\) pairwise non-isomorphic simple modules. We consider the Coxeter transformation \(\phi _A\) as the automorphism of the Grothendieck group \(K_0(A)\) induced by the Auslander–Reiten translation \(\tau \) in the derived category \(\mathrm{Der}(\mathrm{mod}_A)\) of the module category \(\mathrm{mod}_A\) of finite-dimensional left \(A\)-modules. We say that \(A\) is an algebra of cyclotomic type if the characteristic polynomial \(\chi _A\) of \(\phi _A\) is a product of cyclotomic polynomials. In this paper we consider algebras of the form \(A\otimes B\) and the Mahler measure of their Coxeter polynomials. We show $$\begin{aligned} M(\chi _A)\,M(\chi _B) \le M(\chi _{A\otimes B}) \le M(\chi _A)^{r(B)+1} \, M(\chi _B)^{r(A)+1}, \end{aligned}$$