José Antonio de la Peña

Instituto de Matemáticas, UNAM and

Centro de Investigación en Matemáticas,

Guanajuato.

Mérida. December 2014.

On the occasion of the 60. Anniversary of José Seade

o Representations of algebras: general concepts.

o The Coxeter polynomial of an algebra.

o Mahler measure of polynomials and Lehmer’s conjecture

o Algebras associated to singularities.

o Algebras of cyclotomic type.

o Examples.

Lenzing-de la Peña: Extended canonical algebras and

Fuchsian singularities. Math Z. (2010).

de la Peña: Algebras whose Coxeter polynomial are

products of cyclotomic polynomials. Algebras and

Representation Th. (2014)

de la Peña: On the Mahler measure of the Coxeter

polynomial of a finite dimensional algebra. Adv. Math.

(2014)

de la Peña: Cyclotomicity of the Coxeter matrix and the

representation type of algebras. In preparation.

Mroz-de la Peña: Tubes in derived categories. J. Algebra.

>

>

>

José Antonio de la Peña

José Antonio de la Peña

A finite dimensional algebra kQ accepts only finitely many

indecomposable representations (up to isomorphisms) if and

only if Q has an underlying Dynkin diagram:

After work of Tits and Gelfand et al. there is a bijection:

X dim X

between indecomposable modules and roots of the (Tits) form

qA(u) =u,uA

José Antonio de la Peña

Auslander-Reiten theory.

Consider a finite dimensional algebra A and X an

indecomposable non-projective module. There exists an exact

sequence

0 X E X 0

Such that

• X is an indecomposable non-injective module;

• the sequence almost split, that is, for an indecomposable

module Y non-isomorphic to X, the following is exact:

0 Hom (Y,X) Hom (Y, E) Hom (Y, X) 0

In particular, there is a natural isomorphism:

Ext1(Z,X) D Hom (X, Z),

where D=Homk( - , k) is the natural duality.

José Antonio de la Peña

Spectral theory of Coxeter. transformations

Let A be a finite dimensional algebra of finite global dimension

and C the associated Cartan matrix. The Coxeter matrix is

= - C C-T

In case A= kQ

o for X an indecomposable non-projective module holds

[X] = [X], for the classes in the Grothendieck group.

o let be the spectral radius of then:

• if Q is of Dynkin type =1 but 1 is not eigenvalue;

• If Q is extended Dynkin =1 is an eigenvalue;

• If Q is wild, then > 1 and defines the growth rate of

(dim n X )n

for non-preprojective indecomposable modules

Singularities of Dynkin type

Derived categories and invariant transformations.

Notation and definitions.

Let A be a triangular finite dimensional algebra.

𝜑𝐴 denotes the Coxeter transformation of 𝐾0 𝐴 = ℤ𝑛

𝜒𝐴(T) denotes the Coxeter polynomial (=characteristic polynomial of 𝜑𝐴)

Write:

𝜒𝐴(T) =𝑎0 + 𝑎1 𝑇 + 𝑎2𝑇2 +⋯+ 𝑎𝑛𝑇

𝑛 an integral polynomial

𝜒𝐴 T = (𝑇 − 𝜆𝑖)𝑛𝑖=1 for the eigenvalues 𝜆𝑖

Then

• 𝑎𝑖 = 𝑎𝑛−𝑖 we say that 𝜒𝐴(T) is self-reciprocal;

• 𝑎0 = 𝑎𝑛 = (−1)𝑛det𝜑𝐴 = 1

• 𝑎1 = − 𝜆𝑖 = −𝑡𝑟𝑛𝑖=1 𝜑𝐴

We say that A is of cyclotomic type if all eigenvalues satisfy || 𝜆𝑖 ||=1

Derived categories and invariant transformations.

Serre duality: Ext1 𝑋, 𝑌 ≅ 𝐷Hom(𝑌, 𝜏𝑋)

Euler bilinear form:

< [𝑋], [𝑌] >= (−1)𝑗dimExt𝑗(𝑋, 𝑌)∞𝑗=0

Therefore

< [𝑋], [𝑌] >= −< 𝑌 , 𝜏𝑋 >

1 0 0 0 0 0 0 -1

C= 1 1 0 0 = 0 0 1 1 (t)= t4+t3+t+1= (t+1)2(t2-t+1)

1 0 1 0 0 1 0 1

1 1 1 1 -1 -1 -1 -1

José Antonio de la Peña

Hereditary algebras

Algebras A=kQ are:

representation finite if Q is Dynkin;

Tame if Q is extended Dynkin and there are one

parametric families of modules of dimension u if qA(u) = 0

(Dlab-Ringel);

• Wild, else.

V. Kac: There are infinitely many indecomposable modules

X with dim X=u if u is a connected vector with qA(u) ≤ 0.

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

trace(𝜑𝑘)

N.B. given two cyclotomic polynomials Φ𝑛 and Φ𝑚

o Φ𝑛⨂Φ𝑚 is product of cyclotomic polynomials;

o Φ𝑛⨂Φ𝑚 = Φ𝑘𝑒𝑘

𝑘 its cyclotomic decomposition: open problem

o Mahler measure is multiplicative, ie 𝑀 𝑓𝑔 = 𝑀 𝑓 𝑀 𝑔 .

L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, Crelle, Oeuvres I (1857), 105-108

𝐴𝑚+1 = 𝐴𝑚[𝑀𝑛] one-point extension s.t. 𝐸𝑥𝑡𝑘 𝑀𝑛, 𝑀𝑛 = 0, for 𝑘 ≥ 1

where 𝐵 𝑀 =𝐵 𝑀0 𝑘

with the usual matrix operations.

0 10 20 30 40 50 60 70 80 90 10010

0

101

102

103

10,22,30,42,50,62,70,82,90,102 are not cyclotomic

José-Antonio de la Peña

José-Antonio de la Peña

José-Antonio de la Peña

Finite dimensional algebras and singularities José Antonio de la Peña

José Antonio de la Peña

m=2

Spectral radius = 1.106471… < 1.1762… Mahler measure= 1.224278…

joint work with Helmut Lenzing

• Dolgachev: Math. Annalen 265 (1983)

• Wagreich: Proc. Symp. Pure Math. AMS 40 (1983),

José Antonio de la Peña

Exhaustive Search: It is possible to determine all polynomials of a given degree D

having bounded Mahler measure. Searches performed using measure bound 1.3:

1980 D ≤ 16 (Boyd) 1989 D = 18 and 20 (Boyd)

Jan 1996 D = 22 May 1996 D = 24 2003 D ≤ 40 (Flammang, Rhin, Sac-Epee, Wu)

2008 D ≤ 44 (Rhin, Mossinghoff, Wu)

2008 D ≤ 54 (Rhin, Mossinghoff, Wu) (From Mossinghoff’s Web page).

Lehmer added the following remark in his 1933 paper (using Ω to denote

the measure):

“We have not made an examination of all 10th degree symmetric

polynomials, but a rather intensive search has failed to reveal a better

polynomial than x10 + x9 − x7 − x6 − x5 − x4 − x3 + x + 1, Ω = 1.176280818.

“All efforts to find a better equation of degree 12 and 14 have been

unsuccessful.” Despite extensive searches, Lehmer's polynomial remains

the world champion

8 9 10

A the corresponding hereditary algebra has 𝑀 𝜒𝐴 = 1.1762… minimal

known.

=

Happel (1997):

Clearly, if 𝐴 is a cyclotomic algebra with 𝑛 vertices, then 𝑡𝑟 𝜑𝑘 ≤ 𝑛, for any 𝑘 ≥ 0

Thank you!

José Antonio de la Peña

jap@matem.unam.mx

jap@cimat.mx

Let B be an accessible critical of non cyclotomic type, then either

𝜒𝐵 has a unique root outside the unit circle (Pissot polynomial), M(𝜒𝐵 )=𝜌𝐵

or

𝜒𝐵 has two roots outside the unit circle, M(𝜒𝐵 )=𝜌𝐵2

Theorem 1: A an accessible algebra, then

either A is of cyclotomic type

or

there is a convex subcategory B of A such that Mahler measure

M(𝜒𝐵) ≥ 𝜇0 (Lehmer’s number).

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80 90 10010

0

101

102

103

10,22,30,42,50,62,70,82,90,102 are not cyclotomic

m the m-th cyclotomic polynomial has as roots the primitive roots of unity, hence it has degree (m).

Finite dimensional algebras and singularities José Antonio de la Peña

There is a functor modZ k[x0, x1] coh P1,

that takes each graded k[x0, x1]-module M to the triple ((Mx0)0, (Mx1)0, σM),

where y acts on the degree zero part of Mx0 via the identification y = x1/x0,

the variable y−1 acts on the degree zero part of Mx1 via the identification y−1

=x0/x1, and the isomorphism σM equals the obvious identification

[(Mx0)0]x1/x0 =[(Mx1)0]x0/x1 .

Proposition (Serre). The above functor induces an equivalence

modZ k[x0, x1]/mod0Z k[x0, x1] coh P1

The category coh P1 is a k-linear hereditary category satisfying Serre

duality. More precisely, there is a functorial k-linear isomorphism

DExt1(F, G ) ≈ Hom(G,F(−2)) for all F, G coh P1.

Finite dimensional algebras and singularities José Antonio de la Peña

Theorem: (1) [Geigle-Lenzing]: coh(X) is a hereditary category with Serre

duality. (2) [Happel]: if H is a hereditary category then the bounded derived

category Der(H) is triangulated equivalent to Der(mod H) for some

hereditary algebra H or Der(coh(X)) for some weighted projective line X.

R is the translation algebra

Finite dimensional algebras and singularities José Antonio de la Peña