The Experts below are selected from a list of 278910 Experts worldwide ranked by ideXlab platform
Antun Milas - One of the best experts on this subject based on the ideXlab platform.
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Higher rank partial and false theta functions and Representation Theory
2017Co-Authors: Thomas Creutzig, Antun MilasAbstract:Abstract We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's partial theta functions coming from ADE root lattices, which are then linked to Representation Theory of W-algebras. We derive modular transformation properties of regularized higher rank partial and false theta functions as well as Kostant's version of these. Modulo conjectures in Representation Theory, as an application, we compute regularized quantum dimensions of atypical and typical modules of “narrow” logarithmic W -algebras associated to rescaled root lattices. This paper substantially generalize our previous work [19] pertaining to ( 1 , p ) -singlet W -algebras (the sl 2 case). Results in this paper are very general and are applicable in a variety of situations.
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Higher rank partial and false theta functions and Representation Theory.
2016Co-Authors: Thomas Creutzig, Antun MilasAbstract:We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's partial theta functions coming from ADE root lattices, which are then linked to Representation Theory of W-algebras. We derive modular transformation properties of regularized Kostant's partial and certain higher rank false theta functions. Modulo conjectures in Representation Theory, as an application, we compute regularized quantum dimensions of atypical and typical modules of "narrow" logarithmic W-algebras associated to rescaled root lattices. Results in this paper substantially generalize our previous work [19] pertaining to (1,p)-singlet W-algebras (the sl_2 case).
Thomas Creutzig - One of the best experts on this subject based on the ideXlab platform.
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Higher rank partial and false theta functions and Representation Theory
2017Co-Authors: Thomas Creutzig, Antun MilasAbstract:Abstract We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's partial theta functions coming from ADE root lattices, which are then linked to Representation Theory of W-algebras. We derive modular transformation properties of regularized higher rank partial and false theta functions as well as Kostant's version of these. Modulo conjectures in Representation Theory, as an application, we compute regularized quantum dimensions of atypical and typical modules of “narrow” logarithmic W -algebras associated to rescaled root lattices. This paper substantially generalize our previous work [19] pertaining to ( 1 , p ) -singlet W -algebras (the sl 2 case). Results in this paper are very general and are applicable in a variety of situations.
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Higher rank partial and false theta functions and Representation Theory.
2016Co-Authors: Thomas Creutzig, Antun MilasAbstract:We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's partial theta functions coming from ADE root lattices, which are then linked to Representation Theory of W-algebras. We derive modular transformation properties of regularized Kostant's partial and certain higher rank false theta functions. Modulo conjectures in Representation Theory, as an application, we compute regularized quantum dimensions of atypical and typical modules of "narrow" logarithmic W-algebras associated to rescaled root lattices. Results in this paper substantially generalize our previous work [19] pertaining to (1,p)-singlet W-algebras (the sl_2 case).
Corey Jones - One of the best experts on this subject based on the ideXlab platform.
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annular Representation Theory for rigid c tensor categories
2016Co-Authors: Shamindra Kumar Ghosh, Corey JonesAbstract:Abstract We define annular algebras for rigid C ⁎ -tensor categories, providing a unified framework for both Ocneanu's tube algebra and Jones' affine annular category of a planar algebra. We study the Representation Theory of annular algebras, and show that all sufficiently large (full) annular algebras for a category are isomorphic after tensoring with the algebra of matrix units with countable index set, hence have equivalent Representation theories. Annular algebras admit a universal C ⁎ -algebra closure analogous to the universal C ⁎ -algebra for groups. These algebras have interesting corner algebras indexed by some set of isomorphism classes of objects, which we call centralizer algebras. The centralizer algebra corresponding to the identity object is canonically isomorphic to the fusion algebra of the category, and we show that the admissible Representations of the fusion algebra of Popa and Vaes are precisely the restrictions of arbitrary (non-degenerate) ⁎ -Representations of full annular algebras. This allows approximation and rigidity properties defined for categories by Popa and Vaes to be interpreted in the context of annular Representation Theory. This perspective also allows us to define “higher weight” approximation properties based on other centralizer algebras of an annular algebra. Using the analysis of annular Representations due to Jones and Reznikoff, we identify all centralizer algebras for the TLJ ( δ ) categories for δ ≥ 2 .
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annular Representation Theory for rigid c tensor categories
2015Co-Authors: Shamindra Kumar Ghosh, Corey JonesAbstract:We define annular algebras for rigid $C^{*}$-tensor categories, providing a unified framework for both Ocneanu's tube algebra and Jones' affine annular category of a planar algebra. We study the Representation Theory of annular algebras, and show that all sufficiently large (full) annular algebras for a category are isomorphic after tensoring with the algebra of matrix units with countable index set, hence have equivalent Representation theories. Annular algebras admit a universal $C^{*}$-algebra closure analogous to the universal $C^{*}$-algebra for groups. These algebras have interesting corner algebras indexed by some set of isomorphism classes of objects, which we call centralizer algebras. The centralizer algebra corresponding to the identity object is canonically isomorphic to the fusion algebra of the category, and we show that the admissible Representations of the fusion algebra of Popa and Vaes are precisely the restrictions of arbitrary (non-degenerate) $*$-Representations of full annular algebras. This allows approximation and rigidity properties defined for categories by Popa and Vaes to be interpreted in the context of annular Representation Theory. This perspective also allows us to define "higher weight" approximation properties based on other centralizer algebras of an annular algebra. Using the analysis of annular Representations due to Jones and Reznikoff, we identify all centralizer algebras for the $TLJ(\delta)$ categories for $\delta\ge 2$.
Thomas W Kephart - One of the best experts on this subject based on the ideXlab platform.
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lieart a mathematica application for lie algebras and Representation Theory
2015Co-Authors: Robert Feger, Thomas W KephartAbstract:Abstract We present LieART 2.0 which contains substantial extensions to the Mathematica application LieART ( Lie A lgebras and R epresentation T heory) for computations frequently encountered in Lie algebras and Representation Theory, such as tensor product decomposition and subalgebra branching of irreducible Representations. The basic procedure is unchanged—it computes root systems of Lie algebras, weight systems and several other properties of irreducible Representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be user friendly. New extended tables of properties, tensor products and branching rules of irreducible Representations are included in the supplementary material for use without Mathematica software. LieART 2.0 now includes the branching rules to special subalgebras for all classical and exceptional Lie algebras up to and including rank 15. Program summary Program Title: LieART 2.0 CPC Library link to program files: http://dx.doi.org/10.17632/8vm7j67bwt.1 Licensing provisions: GNU Lesser General Public License Programming language: Mathematica External routines/libraries: Wolfram Mathematica 8–12 Nature of problem: The use of Lie algebras and their Representations is widespread in physics, especially in particle physics. The description of nature in terms of gauge theories requires the assignment of fields to Representations of compact Lie groups and their Lie algebras. Mass and interaction terms in the Lagrangian give rise to the need for computing tensor products of Representations of Lie algebras. The mechanism of spontaneous symmetry breaking leads to the application of subalgebra decomposition. This computer code was designed for the purpose of Grand Unified Theory (GUT) model building, (where compact Lie groups beyond the U ( 1 ) , SU ( 2 ) and SU ( 3 ) of the Standard Model of particle physics are needed), but it has found use in a variety of other applications. Tensor product decomposition and subalgebra decomposition have been implemented for all classical Lie groups SU ( N ) , SO ( N ) and Sp ( 2 N ) and all the exceptional groups E 6 , E 7 , E 8 , F 4 and G 2 . This includes both regular and irregular (special) subgroup decomposition of all Lie groups up through rank 15, and many more. Solution method: LieART generates the weight system of an irreducible Representation (irrep) of a Lie algebra by exploiting the Weyl reflection groups, which is inherent in all simple Lie algebras. Tensor products are computed by the application of Klimyk’s formula, except for SU ( N ) ’s, where the Young-tableaux algorithm is used. Subalgebra decomposition of SU ( N ) ’s are performed by projection matrices, which are generated from an algorithm to determine maximal subalgebras as originally developed by Dynkin [1] , [2] . We generate projection matrices by the Dynkin procedure, i.e., removing dots from the Dynkin or extended Dynkin diagram, for regular subalgebras, and we implement explicit projection matrices for special subalgebras. Restrictions: Internally irreps are represented by their unique Dynkin label. LieART’s default behavior in TraditionalForm is to print the dimensional name, which is the labeling preferred by physicist. Most Lie algebras can have more than one irrep of the same dimension and different irreps with the same dimension are usually distinguished by one or more primes (e.g. 175 and 175 ′ of A 4 ). To determine the need for one or more primes of an irrep a brute-force loop over other irreps must be performed to search for irreps with the same dimensionality. Since Lie algebras have an infinite number of irreps, this loop must be cut off, which is done by limiting the maximum Dynkin digit in the loop. In rare cases for irreps of high dimensionality in high-rank algebras the used cutoff is too low and the assignment of primes is incorrect. However, this only affects the display of the irrep. All computations involving this irrep are correct, since the internal unique Representation of Dynkin labels is used.
Matthew B Young - One of the best experts on this subject based on the ideXlab platform.
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burnside rings for real 2 Representation Theory the linear Theory
2020Co-Authors: Dmitriy Rumynin, Matthew B YoungAbstract:This paper is a fundamental study of the Real 2-Representation Theory of 2-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a 2-equivariant Morita ...
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burnside rings for real 2 Representation Theory the linear Theory
2019Co-Authors: Dmitriy Rumynin, Matthew B YoungAbstract:This paper is a fundamental study of the Real $2$-Representation Theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real $2$-Representations as a Real variant of the Burnside ring of the fundamental group of the $2$-group and study the Real categorical character Theory. This paper unifies two previous lines of inquiry, the approach to $2$-Representation Theory via Morita Theory and Burnside rings, initiated by the first author and Wendland, and the Real $2$-Representation Theory of $2$-groups, as studied by the second author.
