On moduli spaces for abelian categories

We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.

Simple components of the center of FG/J(FG)

Let G be a finite group, F a field, FG the group ring of G over F, and J(FG) the Jacobson radical of FG. Using a result of Berman and Witt, we give a method to determine the structure of the center of FG/J(FG), provided that F satisfies a field theoretical condition.

Nuclear elements of degree 6 in the free alternative algebra

We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known clegree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory and time. The calculations can be done in characteristic zero or any prime greater than 6, and similar results are expected. We generated the nuclear elements using prime 103. We check our answer using five other primes.

Torsion theories induced from commutative subalgebras

We begin a study of torsion theories for representations of finitely generated algebras U over a field containing a finitely generated commutative Harish-Chandra subalgebra Gamma. This is an important class of associative algebras, which includes all finite W-algebras of type A over an algebraically closed field of characteristic zero, in particular, the universal enveloping algebra of gl(n) (or sl(n)) for all n. We show that any Gamma-torsion theory defined by the coheight of the prime ideals of Gamma is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in Spec Gamma have the same coheight. Hence, the coheight of these associated prime ideals is an invariant of a given simple U-module. This implies the stratification of the category of U-modules controlled by the coheight of the associated prime ideals of Gamma. Our approach can be viewed as a generalization of the classical paper by Block (1981) [4]; it allows, in particular, to study representations of gl(n) beyond the classical category of weight or generalized weight modules. (C) 2011 Elsevier B.V. All rights reserved.

FROM TRISECTIONS IN MODULE CATEGORIES TO QUASI-DIRECTED COMPONENTS

In this paper, we define and study a special type of trisections in a module category, namely the compact trisections which characterize quasi-directed components. We apply this notion to the study of laura algebras and we use it to define a class of algebras with predictable Auslander-Reiten components.

ASSOCIATIVE IDENTITIES OF OCTONIONS

We work to find a basis of identities for an octonion algebra modulo an associator ideal of a free alternative algebra, or, in other words, a basis for an associative replica of an ideal of identities of an octonion algebra.

Star-group identities and groups of units

Analogous to *-identities in rings with involution we define *-identities in groups. Suppose that G is a torsion group with involution * and that F is an infinite field with char F not equal 2. Extend * linearly to FG. We prove that the unit group U of FG satisfies a *-identity if and only if the symmetric elements U(+) satisfy a group identity.

Galois orders in skew monoid rings

We introduce a new class of noncommutative rings - Galois orders, realized as certain subrings of invariants in skew semigroup rings, and develop their structure theory. The class of Calms orders generalizes classical orders in noncommutative rings and contains many important examples, such as the Generalized Weyl algebras, the universal enveloping algebra of the general linear Lie algebra, associated Yangians and finite W-algebras (C) 2010 Elsevier Inc All rights reserved

Structurable superalgebras of Cartan type

We describe the simple Lie superalgebras arising from the unital structurable superalgebras of characteristic 0 and construct four series of the unital simple structurable superalgebras of Cartan type. We give a classification of simple structurable superalgebras of Cartan type over an algebraically closed field F of characteristic 0. Together with the Faulkner theorem on the classification of classical such superalgebras, it gives a classification of the simple structurable superalgebras over F. Crown Copyright (C) 2010 Published by Elsevier Inc. All rights reserved.

Partial projective representations and partial actions

The concept of a partial projective representation of a group is introduced and studied. The interaction with partial actions is explored. It is shown that the factor sets of partial projective representations over a field K are exactly the K-valued twistings of crossed products by partial actions. (C) 2009 Elsevier B.V. All rights reserved.

Noncommutative Jordan superalgebras of degree n > 2

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a ""nodal"" case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.

Boutet de Monvel`s calculus and groupoids I

Can Boutet de Monvel`s algebra on a compact manifold with boundary be obtained as the algebra Psi(0)(G) of pseudodifferential operators on some Lie groupoid G? If it could, the kernel G of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C*(G). While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C*(G) possesses an ideal I isomorphic to G. In fact, we prove first that G similar or equal to Psi circle times K with the C*-algebra Psi generated by the zero order pseudodifferential operators on the boundary and the algebra K of compact operators. As both Psi circle times K and I are extensions of C(S*Y) circle times K by K (S*Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

On the Relative Socle for Stratifying Systems

Let A be a finite dimensional k-algebra, (, ) be a stratifying system in mod(A) and F() be the class of -filtered A-modules. In this article, we give the definition and also study some of the properties of the relative socle in F(). We approach the relative socle in three ways. Namely, we view it as (1) a -semisimple subobject of M having the largest -length, (2) a maximal -semisimple subobject of M, and (3) a minimal -essential subobject of M.

Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data

In [H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure Appl. (9) (1983) 73-97.] Brezis and Friedman prove that certain nonlinear parabolic equations, with the delta-measure as initial data, have no solution. However in [J.F. Colombeau, M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl (1990) 186-196.] Colombeau and Langlais prove that these equations have a unique solution even if the delta-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais` result proving that we may take any generalized function as the initial data. Our approach relies on recent algebraic and topological developments of the theory of Colombeau generalized functions and results from [J. Aragona, Colombeau generalized functions on quasi-regular sets, Publ. Math. Debrecen (2006) 371-399.]. (C) 2009 Elsevier Ltd. All rights reserved.

Group identities on symmetric units

Let F be an infinite field of characteristic different from 2, G a group and * an involution of G extended by linearity to an involution of the group algebra FG. Here we completely characterize the torsion groups G for which the *-symmetric units of FG satisfy a group identity. When * is the classical involution induced from g -> g(-1), g is an element of G, this result was obtained in [ A. Giambruno, S. K. Sehgal, A. Valenti, Symmetric units and group identities, Manuscripta Math. 96 (1998) 443-461]. (C) 2009 Elsevier Inc. All rights reserved.