28,867 research outputs found
Modular Representation Theory of Symmetric Groups
We review some recent advances in modular representation theory of symmetric
groups and related Hecke algebras. We discuss connections with
Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group
algebras , which these connections reveal; graded categorification
and connections with quantum groups and crystal bases; modular branching rules
and the Mullineaux map; graded cellular structure and graded Specht modules;
cuspidal systems for affine KLR algebras and imaginary Schur-Weyl duality,
which connects representation theory of these algebras to the usual Schur
algebras of smaller rank.Comment: This is an expository paper for ICM proceeding
Global algebras of nonlinear generalized functions with applications in general relativity
We give an overview of the development of algebras of generalized functions
in the sense of Colombeau and recent advances concerning diffeomorphism
invariant global algebras of generalized functions and tensor fields. We
furthermore provide a survey on possible applications in general relativity in
light of the limitations of distribution theory
Andre-Quillen cohomology of algebras over an operad
We study the Andre-Quillen cohomology with coefficients of an algebra over an
operad. Using resolutions of algebras coming from Koszul duality theory, we
make this cohomology theory explicit and we give a Lie theoretic
interpretation. For which operads is the associated Andre-Quillen cohomology
equal to an Ext-functor ? We give several criterion, based on the cotangent
complex, to characterize this property. We apply it to homotopy algebras, which
gives a new homotopy stable property for algebras over cofibrant operads.Comment: Corrections in Sections 5 and 6, to appear in Advances in Mathematic
Advances in the theory of μŁΠ algebras
Recently an expansion of ŁΠ1/2 logic with fixed points has been considered [23]. In the present work we study the algebraic semantics of this logic, namely μŁΠ algebras, from algebraic, model theoretic and computational standpoints. We provide a characterisation of free μŁΠ algebras as a family of particular functions from [0,1]n to [0,1]. We show that the first-order theory of linearly ordered μŁΠ algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered ŁΠ1/2 algebras. Furthermore, we give a functional representation of any ŁΠ1/2 algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of μŁΠ algebras is in PSPACE. © The Author 2010. Published by Oxford University Press. All rights reserved.Marchioni acknowledges partial support of the Spanish projects MULOG2 (TIN2007-68005-C04), Agreement Technologies (CONSOLIDER CSD2007-0022, INGENIO 2010), the Generalitat de Catalunya grant 2009-SGR-1434, and Juan de la Cierva Program of the Spanish MICINN, as well as the ESF Eurocores-LogICCC/MICINN project (FFI2008-03126-E/FILO). Spada acknowledges partially supported of the FWF project P 19872-N18.Peer Reviewe
An overview of fine gradings on simple Lie algebras
This paper presents a survey of the results and ideas behind the
classification of the fine gradings, up to equivalence, on the simple finite
dimensional Lie algebras over an algebraically closed field of characteristic
zero. It provides an expanded version of the mini course delivered by the
second author at the Conference "Advances in Group Theory and Applications
AGTA-2015".Comment: 14 page
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