Eivind Eriksen

Journal of Generalized Lie Theory and Applications 5 Doi: https://doi.org/10.4303/jglta/G110109

Let A be an associative algebra over a field k, and let be a finite family of right A-modules. A study of the noncommutative deformation functor Def of the family leads to the construction of the algebra A() of observables and the generalized Burnside theorem, due to Laudal (2002). In this paper, we give an overview of aspects of noncommutative deformations closely connected to the generalized Burnside theorem.

Canadian Journal of Mathematics (CJM) 62(3) p. 520-542 Doi: https://doi.org/10.4153/CJM-2010-015-6

We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases. In the last part of the paper, we describe a noncommutative deformation functor for quasi-coherent sheaves of modules on a ringed space (X,A) . We show that for any good A -affine open cover U of X , the forgetful functor QCohA→PreSh(U,A) induces an isomorphism of noncommutative deformation functors. Applications. We consider noncommutative deformations of quasi-coherent A -modules on X when (X,A)=(X,O X ) is a scheme or (X,A)=(X,D) is a D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of X closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left D X -module D X when X is an elliptic curve as an example.

Proceedings of the Estonian Academy of Sciences : Physics, Mathematics 59(4) p. 294-300 Doi: https://doi.org/10.3176/proc.2010.4.07

In this paper we study Lie-Rinehart cohomology for quotients of singularities by finite groups, and interpret these cohomology groups in terms of integrable connection on modules.

Journal of Algebra 322(12) p. 4283-4294

Let k be an algebraically closed field of characteristic 0, let R be a commutative k-algebra, and let M be a torsion free R-module of rank one with a connection . We consider the Lie-Rinehart cohomology with values in EndR(M) with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on M. When R is an isolated singularity of dimension d2, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when R is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology.

Journal of Algebra 322(12) p. 4283-4294 Doi: https://doi.org/10.1016/j.jalgebra.2009.09.015

Journal of Pure and Applied Algebra 212(7) p. 1561-1574 Doi: https://doi.org/10.1016/j.jpaa.2007.10.008

Generalized Lie Theory in Mathematics, Physics and Beyond p. 99-108

Journal of Generalized Lie Theory and Applications 2(3) p. 52-56

Communications in Algebra 36(8) p. 3032-3041

Springer

Generalized Lie Theory in Mathematics, Physics and Beyond p. 109-117

Journal of symbolic computation 42 Doi: https://doi.org/10.1016/j.jsc.2006.10.001

Journal of symbolic computation 42 p. 313-323

Studies in Health Technology and Informatics 111 p. 502-507

MODULI OPERADS DYNAMICS

Conference AGMP-8 Brno'12

Cornell University

Algebra, Geometry, and Mathematical Physics, 5th Baltic-Nordic AGMP Workshop

Arxiv http://arxiv.org

Algebra, Geometry, and Mathematical Physics. 4th Baltic-Nordic Workshop: Tartu, 09-11 October, 2008

Arxiv http://arxiv.org

Algebra, Geometry, and Mathematical Physics, Baltic-Nordic Workshop

Principles of Computer Security

Algebra-seminaret ved HiBu

ArXiv

Algebra-seminaret

Algebra, Geometry and Mathematical Physics Baltic-Nordic Network Workshop

ArXiv

ArXiv

ArXiv

Algebra, Geometry and Mathematical Physics Baltic-Nordic Network Workshop

Eivind Eriksen

Kompendiet er skrevet for og brukt i undervisningen ved kurset FO-008 Lineær Algebra (HiO/IU V2006).

Noncommutative Algebra and Geometry p. 90-126

In this paper, we give an introduction to the theory of noncommutative deformations of modules. This theory was introduced by O.A. Laudal.

utenTitteltekst

Let k be an algebraically closed field, X a topological space, A a sheaf of associative k-algebras on X, and F_1, ... , F_p a finite family of sheaves of left A-modules on X. In this general situation, we define a noncommutative deformation functor Def_F: a_p -> Sets, generalizing the noncommutative deformations of modules (Laudal) and deformations of a sheaf of modules on a scheme (Siqveland). Moreover, we show the following result: If X has a good A-affine open cover U, and F_i is a quasi-coherent left A-module for all i. Then Def_F has a pro-representing hull. If the global Hochschild cohomology groups H^n(U,F_j,F_i) are finite dimensional vector spaces over k for n=1,2 and for all i,j, then this hull is determined by an obstruction morphism. In particular, if X is a scheme over k, and F_i is coherent for all i, then H^n(U,F_j,F_i) is isomorphic to Ext^n_A(F_j,F_i) and we obtain a generalization of the usual deformation theory of coherent modules on a scheme.

utenTitteltekst

We introduced the notions of A_{\infty} algebras and A_{\infty} modules, and discussed some relations to Massey products and deformation theory.

utenTitteltekst

Let k be an algebraically closed field, X a topological space, A a sheaf of associative k-algebras on X, and F_1, ... , F a finite family of sheaves of left A-modules on X. In this general situation, we define a noncommutative deformation functor Def: a -> Sets, generalizing the noncommutative deformations of modules (Laudal) and deformations of a sheaf of modules on a scheme (Siqveland). Moreover, we show the following result: If X has a good A-affine open cover U, and F is a quasi-coherent left A-module for all i. Then Def has a pro-representing hull. If the global Hochschild cohomology groups H^n(U,F_j,F) are finite dimensional vector spaces over k for n=1,2 and for all i,j, then this hull is determined by an obstruction morphism. In particular, if X is a scheme over k, and F is coherent for all i, then H^n(U,F_j,F) is isomorphic to Ext^n(F_j,F) and we obtain a generalization of the usual deformation theory of coherent modules on a scheme.

Institut Mittag-Leffler

We introduce noncommutative deformation theory for finite families of quasi-coherent sheaves of modules on certain noncommutative ringed spaces (X,A), and describe the corresponding obstruction theory.

SMIT Conference

Cornell University

Let k be an algebraically closed field, let R be an associative k-algebra, and let F = {M_a: a in I} be a family of orthogonal points in R-Mod such that End_R(M_a) = k for all a in I. Then Mod(F), the minimal full sub-category of R-Mod which contains Fand is closed under extensions, is a full exact Abelian subcategory of R-Mod and a length category in the sense of Gabriel. In this paper, we use iterated extensions to relate the length category Mod(F) to noncommutative deformations of modules, and use some new methods to study Mod(F) via iterated extensions. In particular, we give a new proof of the characterization of uniserial length categories, which is constructive. As an application, we give an explicit description of some categories of holonomic and regular holonomic D-modules on curves which are uniserial length categories.