Homotopy Batalin-Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defind by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincare-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian-Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

André spectral sequences for Baues-Wirsching cohomology of categories

We construct spectral sequences in the framework of Baues-Wirsching cohomology and homology for functors between small categories and analyze particular cases including Grothendieck fibrations. We also give applications to more classical cohomology and homology theories including Hochschild-Mitchell cohomology and those studied before by Watts, Roos, Quillen and others

Homological and Hodge-theoretic aspects of matroids and polymatroids

In the first part of this thesis, we study the action of the automorphism group of a matroid on the homology space of the co-independent complex. This representation turns out to be isomorphic, up to tensoring with the sign representation, with that on the homology space associated with the lattice of flats. In the case of the cographic matroid of the complete graph, this result has application in algebraic geometry: indeed De Cataldo, Heinloth and Migliorini use this outcome to study the Hitchin fibration. In the second part, on the other hand, we use ideas from algebraic geometry to prove a purely combinatorial result. We construct a Leray model for a discrete polymatroid with arbitrary building set and we prove a generalized Goresky-MacPherson formula. The first row of the model is the Chow ring of the polymatroid; we prove Poincaré duality, Hard-Lefschetz theorem and Hodge-Riemann relations for the Chow ring.

On the Derived Categories and Quasitilted Algebras

In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X*, we consider the set J(X*) = {i is an element of Z vertical bar H(i)(X*) not equal 0} and we define the application l(X*) = maxJ(X*) - minJ(X*) + 1. We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras.

Una primera lección de Geometría algebraica

En este artículo se explica cómo aparece la Geometría Algebraica, partiendo del estudio de los conjuntos de soluciones de sistemas algebraicos

A Cartan-Eilenberg approach to Homotopical Algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.

Deterministic Genericity and the Computation of homological Invariants

The main goal of this thesis is to discuss the determination of homological invariants of polynomial ideals. Thereby we consider different coordinate systems and analyze their meaning for the computation of certain invariants. In particular, we provide an algorithm that transforms any ideal into strongly stable position if char k = 0. With a slight modification, this algorithm can also be used to achieve a stable or quasi-stable position. If our field has positive characteristic, the Borel-fixed position is the maximum we can obtain with our method. Further, we present some applications of Pommaret bases, where we focus on how to directly read off invariants from this basis. In the second half of this dissertation we take a closer look at another homological invariant, namely the (absolute) reduction number. It is a known fact that one immediately receives the reduction number from the basis of the generic initial ideal. However, we show that it is not possible to formulate an algorithm – based on analyzing only the leading ideal – that transforms an ideal into a position, which allows us to directly receive this invariant from the leading ideal. So in general we can not read off the reduction number of a Pommaret basis. This result motivates a deeper investigation of which properties a coordinate system must possess so that we can determine the reduction number easily, i.e. by analyzing the leading ideal. This approach leads to the introduction of some generalized versions of the mentioned stable positions, such as the weakly D-stable or weakly D-minimal stable position. The latter represents a coordinate system that allows to determine the reduction number without any further computations. Finally, we introduce the notion of β-maximal position, which provides lots of interesting algebraic properties. In particular, this position is in combination with weakly D-stable sufficient for the weakly D-minimal stable position and so possesses a connection to the reduction number.

Pseudodifferential operators with C*-algebra-valued symbols: Abstract characterizations

Given a separable unital C*-algebra C with norm parallel to center dot parallel to, let E-n denote the Banach-space completion of the C-valued Schwartz space on R-n with norm parallel to f parallel to(2)=parallel to < f, f >parallel to(1/2), < f, g >=integral f(x)* g(x)dx. The assignment of the pseudodifferential operator A=a(x,D) with C-valued symbol a(x,xi) to each smooth function with bounded derivatives a is an element of B-C(R-2n) defines an injective mapping O, from B-C(R-2n) to the set H of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E-n. In this paper, we construct a left-inverse S for O and prove that S is injective if C is commutative. This generalizes Cordes' description of H in the scalar case. Combined with previous results of the second author, our main theorem implies that, given a skew-symmetric n x n matrix J and if C is commutative, then any A is an element of H which commutes with every pseudodifferential operator with symbol F(x+J xi), F is an element of B-C(R-n), is a pseudodifferential operator with symbol G(x - J xi), for some G is an element of B-C(R-n). That was conjectured by Rieffel.

On super-RS algebra and Drinfeld realization of quantum affine superalgebras

We describe the realization of the super-Reshetikhin-Semenov-Tian-Shansky (RS) algebra in quantum affine superalgebras, thus generalizing the approach of Frenkel and Reshetikhin to the supersymmetric (and twisted) case. The algebraic homomorphism between the super-RS algebra and the Drinfeld current realization of quantum affine superalgebras is established by using the Gauss decomposition technique of Ding and Frenkel. As an application, we obtain Drinfeld realization of quantum affine superalgebra U-q [osp(1/2)((1))] and its degeneration - central extended super-Yangian double DY(h over bar) [osp(1/2)((1))].

Anisotropic correlated electron model associated with the Temperley-Lieb algebra

We present an anisotropic correlated electron model on a periodic lattice, constructed from an R-matrix associated with the Temperley-Lieb algebra. By modification of the coupling of the first and last sites we obtain a model with quantum algebra invariance.

Comments on the Drinfeld realization of the quantum affine superalgebra U-q[gl(m backslash n)(1)] and its Hopf algebra structure

By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to supersymmetric cases, we obtain the Drinfeld current realization for the quantum affine superalgebra U-q[gl(m\n)((1))]. We find a simple coproduct for the quantum current generators and establish the Hopf algebra structure of this super current algebra.

Decay chain differential equations: Solution through matrix algebra

A matricial method to solve the decay chain differential equations system is presented. The quantity of each nuclide in the chain at a time t may be evaluated by analytical expressions obtained in a simple way using recurrence relations. This method may be applied to problems of radioactive buildup and decay and can be easily implemented computationally. (C) 2009 Elsevier B.V. All rights reserved.