Denote by D and G two simply-laced Dynkin types, that is, disjoint unions of simply-laced Dynkin diagrams, and by g(D) and g(G) the semi-simple Lie algebras of that type and recall that they are graduated by the root spaces g(D)_a.

When does there exist a graduated inclusion j:g(D) \hookrightarrow g(G) (here graduated means: there exists a linear map f such that j( g(D)_a) Í g(G)_f(a))?

We translate this question into the language of unit forms, that is, integer quadratic forms q : Zⁿ ®Z satisfying g(c_i)=1 for each canonical base vector c_i. This enables us to give a complete answer to the previous question.

The talk will present results from a joint work with José Antonio de la Peña.

In the following we use the following notation. If B is a finite-dimensional algebra over a field F, we denote by B-mod the category of finitely generated left B-modules. By F(x) we denote the field of rational functions over x. We put F(x,y) = F(x)(y).

Let A be a finite-dimensional algebra over the algebraically closed field k. We put A^k(x) = A Ä_k k(x) and A^k(x,y) = AÄ_k k(x,y).

We prove the following result:

Theorem    The algebra A is of tame representation type if and only if for any indecomposable object M in A^k(x,y)-mod such that _A^k(y)M is an indecomposable A^k(y)-module, there is an indecomposable object N in A^k(x)-mod with _A N an indecomposable A-module such that

Joint work with Leonardo Salmeron.

In this talk we will discuss a method of constructing graded-simple modules from ungraded simple modules over graded algebras (associative or Lie). This method works both in finite-dimensional and infinite-dimensional quasifinite set-ups. The key ingredient in our construction is the action of a cyclotomic quantum torus on the module. We apply this method to get a description of irreducible representations for the twisted toroidal Lie algebras.

This is based on a joint work with Michael Lau.

Cluster algebras, introduced by Fomin and Zelevinsky a few years ago, have gained a lot of interest by now. Acylic cluster algebras have been shown to be related to cluster categories and tilted algebras. Cyclic cluster algebras, however, are less well understood.

We consider the first non-trivial case, cluster algebras of rank three (square, coefficient-free), and study which of them are cyclic. Rank three cluster algebras are given by triples of integers (x,y,z), and we provide an answer which involves the hyperplanes defined by

This is joint work with Ibrahim Assem, Martin Blais and Lutz Hille.

It is a celebrated result by Gerstenhaber from 1964 that (classical) Hochschild cohomology is graded commutative for any associative algebra. Suarez-Alvarez's elegant treatment of the categorical Eckmann-Hilton argument yields easily the same property for derived Hochschild cohomology as defined by Quillen. There is a canonical algebra homomorphism from classical to derived Hochschild cohomology that factors through the Yoneda algebra of the self-extensions of the given algebra as a bimodule over itself.

The question addressed here is whether the latter Yoneda algebra might also always be graded commutative. That is known under mild "Tor-transversality" conditions, when that algebra already coincides with the derived version of Hochschild cohomology. Here we give a simple example, the normalization of a plane cusp singularity, where the answer is negative. Indeed, the Yoneda Ext-algebra in that case is essentially an infinitely generated tensor algebra.

We present a new method of computing the Chow group of zero cycles on projective homogeneous varieties which is based on an idea of parametrization of splitting fields.

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A has finite global dimension. The Auslander-Reiten translation t_A defines an automorphism in the derived category of the module category mod_A. The linear transformation induced on the Grothendieck group is called the Coxeter transformation and the associated characteristic polynomial f_A is the Coxeter polynomial of A. The spectra of the Coxeter polynomial is related with important properties of the algebra: the structure of the Auslander-Reiten quiver of A, the growth of the iterated translations tⁿ [X] for indecomposable modules X and other facts. For hereditary algebras A = kQ with Q a quiver, f_A is known to be closely related to the characteristic polynomial of the adjacency matrix of the underlying graph of Q. We study new classes of algebras where the spectra of f_A can be described by means of characteristic polynomials of adjacency matrices of graphs.

Let G be a graph on n vertices. Let l₁,l₂,...,l_n be its eigenvalues (i.e., the eigenvalues of the adjacency matrix of G). The energy of G is defined as [1]

The quantity

References

[1]

I. Gutman, The energy of a graph. Ber. Math.-Statist. Sekt. Forsch. Graz 103(1978), 1-22.

[2]

The energy of a graph: Old and new results. Algebraic Combinatorics and Applications, Springer, Berlin, 2001, 196-211; Spectra and energies of iterated line graphs of regular graphs. Appl. Math. Lett. 18(2005), 679-682; Laplacian energy of a graph. Lin. Algebra Appl. 414(2006), 29-37; Note on the Coulson integral formula. J. Math. Chem. 39(2006), 259-266.

[3]

E. Estrada and J. A. Rodríguez-Velázquez, Subgraph centrality in complex networks. Phys. Rev. E71(2005), 056103.

A classical result of Hochschild, Kostant, and Rosenberg characterizes smoothness of commutative algebras essentially of finite type over a field in terms of its Hochschild cohomology. I will discuss a similar characterization of the Gorenstein property.

This is joint work with L. L. Avramov.

In this joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh, we show that the hypersurface ring of the generic determinant admits a non-commutative crepant resolution by a "quiverized Clifford algebra".

Let A be a finite dimensional elementary algebra over a field with rad(A)⁰ = 0. The objective is to study D^b(A), the derived category of bounded complexes in the category of finite dimensional left A-modules. Our technique is to find a proper covering of the ordinary quiver of A so that the complexes of projective A-modules are determined by the representations of the covering. In this way, we are able to give a complete description of the indecomposables, the almost split triangles, the shapes of the components of the Auslander-Reiten quiver of D^b(A) as well as the derived type of A.

This is a joint work with Raymundo Bautista.

A major result in Algebraic Geometry is the theorem of Bernstein-Gelfand-Gelfand that states the existence of an equivalence of triangulated categories: gr_L @ D^b (CohPⁿ), where gr_L denotes the stable category of finitely generated graded modules over the n+1 exterior algebra and D^b (CohPⁿ) is the derived category of bounded complexes of coherent sheaves on projective space Pⁿ.

Generalizations of this result were obtained in a paper by Martínez-Villa and Saorín and from a different point of view, the theorem has been extended by Yanagawa to Zⁿ-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra.

The aim of the talk is to show how to extend the results to group graded algebras in order to obtain a generalization of Yangawa's results having in mind the application to other settings.

Given a commutative square of finite free O-modules, we construct a double complex of O-modules, that we have called the Gobelin. (A Gobelin is a richly embroidered French wall tapestry.) The Gobelin is weaved with vertical and horizontal strands of the Buchsbaum-Eisenbud type, constructed each from a Koszul complex of half of the commutative square. We apply the Gobelin to compute the homological index of a germ of a holomorphic vector field on a complete intersection variety, having both an isolated singularity. The first spectral sequence of the Gobelin provides free resolutions of the modules of Kähler differential forms on the complete intersection, and for small degree the homology of the Gobelin coincides with the homology of the complex obtained by contracting differential forms on the complete intersection with the vector field. The second spectral sequence of the Gobelin provides formulas to compute the homology groups of the Gobelin with local linear algebra.

Almost commuting subgroups of Lie groups appear naturally in many areas of Mathematics and Physics (e.g. flat connections on tori). The main purpose of this talk is to explain how these subgroups also arise in the Galois cohomology attached to toroidal Lie Algebras.

The adjacency matrix A = (a_ij) of a graph G with set of vertices {v₁,...,v_n} and set of edges E_G is defined as

We construct a canonical family of irreducible representations of a Sylow p-subgroup of the symplectic group Sp_2n(q), where q is a power of an odd prime p. Some of these representations appear in the Weil and Steinberg modules of Sp_2n(q), and a connection between the 2-modular reduction of these will be discussed.