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.
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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
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
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