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16 results on '"Kenji Iohara"'

1. Equivalence of a tangle category and a category of infinite dimensional 𝑈_{𝑞}(𝔰𝔩₂)-modules

Author

Ruibin Zhang, Gustav I. Lehrer, and Kenji Iohara

Subjects
Pure mathematics, Mathematics (miscellaneous), Verma module, 010102 general mathematics, 02 engineering and technology, 0101 mathematics, 021001 nanoscience & nanotechnology, 0210 nano-technology, 01 natural sciences, Equivalence (measure theory), Tangle, Mathematics
Abstract

It is very well known that if V V is the simple 2 2 -dimensional representation of U q ( s l 2 ) \mathrm {U}_q(\mathfrak {sl}_2) , the category of representations V ⊗ r V^{\otimes r} , r = 0 , 1 , 2 , … r=0,1,2,\dots , is equivalent to the Temperley-Lieb category T L ( q ) \mathrm {TL}(q) . Such categorical equivalences between tangle categories and categories of representations are rare. In this work we give a family of new equivalences by extending the above equivalence to one between the category of representations M ⊗ V ⊗ r M\otimes V^{\otimes r} , where M M is a projective Verma module of U q ( s l 2 ) \mathrm {U}_q(\mathfrak {sl}_2) and the type B B Temperley-Lieb category T L B ( q , Q ) \mathbb {TLB}(q,Q) , realised as a subquotient of the tangle category of Freyd, Yetter, Reshetikhin, Turaev and others.

Published
2021
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2. Maurice Janet’s algorithms on systems of linear partial differential equations

Author

Philippe Malbos, Kenji Iohara, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Monomial, Partial differential equation, 010102 general mathematics, Multiplicative function, 06 humanities and the arts, Formal methods, 01 natural sciences, Algebra, Mathematics (miscellaneous), 060105 history of science, technology & medicine, History and Philosophy of Science, [MATH.MATH-HO]Mathematics [math]/History and Overview [math.HO], Compatibility (mechanics), Partition (number theory), 0601 history and archaeology, Uniqueness, 0101 mathematics, Algebraic number, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

This article describes the emergence of formal methods in theory of partial differential equations (PDE) in the French school of mathematics through Janet’s work in the period 1913–1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the uniqueness of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the twentieth-century in various algebraic contexts.

Published
2021
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3. From analytical mechanical problems to rewriting theory through M. Janet's work

Author

Philippe Malbos, Kenji Iohara, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Polynomial ring, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], 010102 general mathematics, Elimination theory, 01-08, 13P10, 12H05, 35A25, 58A15, 68Q42, 010103 numerical & computational mathematics, 01 natural sciences, Algebra, Algebraic equation, symbols.namesake, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, symbols, Homological algebra, Ideal (order theory), Rewriting, 0101 mathematics, Computational problem, Mathematics, Hilbert–Poincaré series
Abstract

This chapter is devoted to a survey of the historical background of Grobner bases for D-modules and linear rewriting theory largely developed in algebra throughout the twentieth century and to present deep relationships between them. Completion methods are the main streams for these computational theories. In the theory of Grobner bases, they were motivated by algorithmic problems in elimination theory such as computations in quotient polynomial rings modulo an ideal, manipulating algebraic equations, and computing Hilbert series. In rewriting theory, they were motivated by computation of normal forms and linear bases for algebras and computational problems in homological algebra.

Published
2020
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4. Introduction to Representations of Quivers

Author

Kenji Iohara, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
010102 general mathematics, 0103 physical sciences, 010307 mathematical physics, [MATH]Mathematics [math], 0101 mathematics, 01 natural sciences, ComputingMilieux_MISCELLANEOUS
Abstract

The main purpose of this lecture note is to provide a quick introduction to quivers and their representations. In particular, as there already exists several introductory and complete texts on quivers, the author tries motivating the reader to develop the theory by showing several concrete examples.

Published
2020
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5. Invariants of the Weyl Group of Type $A_{2l}^{(2)}$

Author

Kenji Iohara, Yoshihisa Saito, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0012,GéoLie,Méthodes géométriques en théorie de Lie(2015)

Subjects
Condensed Matter::Quantum Gases, Pure mathematics, Weyl group, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], General Mathematics, High Energy Physics::Lattice, 010102 general mathematics, Type (model theory), 01 natural sciences, symbols.namesake, FOS: Mathematics, symbols, Representation Theory (math.RT), 0101 mathematics, [MATH]Mathematics [math], Mathematics - Representation Theory, Mathematics
Abstract

In this note, we show the polynomiality of the ring of invariants with respect to the Weyl group of type $A_{2l}^{(2)}$., 23 pages, some typos corrected and details added. Accepted version for publication

Published
2019
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6. Temperley-Lieb Algebras At Roots of Unity, A Fusion category and the Jones Quotient

Author

Kenji Iohara, Gus Lehrer, Ruibin Zhang, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), School of Mathematics and statistics [Sydney], and The University of Sydney

Subjects
Endomorphism, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Symmetric bilinear form, Root of unity, General Mathematics, 010102 general mathematics, Dimension (graph theory), Order (ring theory), 01 natural sciences, Combinatorics, Tensor product, Product (mathematics), [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], 0101 mathematics, Quotient, MSC2010: 81R15, 46L37, 16W22, Mathematics
Abstract

When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $\TL_n(q)$ is non-semisimple for almost all $n$. In this work, using cellular methods, we give explicit generating functions for the dimensions of all the simple $\TL_n(q)$-modules. Jones showed that if the order $|q^2|=\ell$ there is a canonical symmetric bilinear form on $\TL_n(q)$, whose radical $R_n(q)$ is generated by a certain idempotent $E_\ell\in\TL_{\ell-1}(q)\subseteq\TL_n(q)$, which is now referred to as the Jones-Wenzl idempotent, for which an explicit formula was subsequently given by Graham and Lehrer. Although the algebras $Q_n(\ell):=\TL_n(q)/R_n(q)$, which we refer to as the Jones algebras (or quotients), are not the largest semisimple quotients of the $\TL_n(q)$, our results include dimension formulae for all the simple $Q_n(\ell)$-modules. This work could therefore be thought of as generalising that of Jones {\it et al.} on the algebras $Q_n$. We also treat a fusion category $\cC_{\rm red}$ introduced by Reshitikhin, Turaev and Andersen, whose objects are the quantum $\fsl_2$-tilting modules with non-zero quantum dimension, and which has an associative truncated tensor product (the fusion product). We show $Q_n(\ell)$ is the endomorphism algebra of a certain module in $\cC_{\rm red}$ and use this fact to recover a dimension formula for $Q_n(\ell)$. We also show how a fusion rule for $K(Q_\infty):=\bigoplus_{n\geq 1}K_0(Q_n(\ell))$ is determined from the structure of $\cC_{\rm red}$.

Published
2019
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7. Classification of Simple Cuspidal Modules over a Lattice Lie Algebra of Witt type

Author

Yuly Billig, Kenji Iohara, Carleton University, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0012,GéoLie,Méthodes géométriques en théorie de Lie(2015), and ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010)

Subjects
Pure mathematics, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], General Mathematics, 010102 general mathematics, Lattice Lie Algebra, Lattice (group), AV-module, 0102 computer and information sciences, Type (model theory), 01 natural sciences, 010201 computation theory & mathematics, Simple (abstract algebra), Lie algebra, FOS: Mathematics, cuspidal module, Uniform boundedness, 0101 mathematics, Representation Theory (math.RT), [MATH]Mathematics [math], Mathematics - Representation Theory, Mathematics
Abstract

Let $W_��$ be the lattice Lie algebra of Witt type associated with an additive inclusion $��: \mathbb{Z}^N \hookrightarrow \mathbb{C}^2$ with $N>1$. In this article, the classification of simple $\mathbb{Z}^N$-graded $W_��$-modules, whose multiplicities are uniformly bounded, is given., 29 pages

Published
2018
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8. SINGULAR DEGENERATIONS OF LIE SUPERGROUPS OF TYPE D(2, 1; a)

Author

Kenji Iohara, Fabio Gavarini, Algèbre, géométrie, logique ( AGL ), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ) -École Centrale de Lyon ( ECL ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), Dipartimento di Matematica [Rome], Università degli Studi di Roma Tor Vergata [Roma], Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Algèbre, géométrie, logique (AGL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and ANR-15-CE40-0012,GéoLie,Méthodes géométriques en théorie de Lie(2015)

Subjects
Mathematics - Differential Geometry, contractions, [ MATH ] Mathematics [math], Pure mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Structure (category theory), Type (model theory), 01 natural sciences, 17B20 (Primary), Simple (abstract algebra), 0103 physical sciences, FOS: Mathematics, Lie superalgebras, [MATH]Mathematics [math], 0101 mathematics, singular degenerations, Mathematical Physics, Mathematics, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Lie supergroups, 010102 general mathematics, Order (ring theory), Sigma, Mathematics - Rings and Algebras, 16. Peace & justice, 14A22, 13D10 (Secondary), 14A22, 17B20 (Primary), 13D10 (Secondary), [ MATH.MATH-RT ] Mathematics [math]/Representation Theory [math.RT], Singular value, Settore MAT/02 - Algebra, Differential Geometry (math.DG), Rings and Algebras (math.RA), Affine plane (incidence geometry), 010307 mathematical physics, Geometry and Topology, Settore MAT/03 - Geometria, Analysis
Abstract

International audience; The complex Lie superalgebras g of type D(2, 1; a) are usually defined for " non-singular " values of the parameter a , for which they are simple. In this paper we introduce five suitable integral forms of g , that are well-defined at those singular values too, giving rise to " singular specializations " that are no longer simple. This extends (in five different ways) the classically known D(2, 1; a) family. Basing on this construction, we perform the parallel one for complex Lie supergroups and describe their singular specializations (or " degenerations ") at singular values of the parameter. This is done via a general construction based on suitably chosen super Harish-Chandra pairs, which suits the Lie group theoretical framework; nevertheless, it might also be realized by means of a straightforward extension of the method introduced in [FG] and [Ga1] to construct " Chevalley supergroups " , which is fit for the context of algebraic supergeometry. Although one may adopt Kac' presentation for the Lie superalgebras of type D(2, 1; a) , in order to stress the overall S 3 –symmetry of the whole situation, we shall work (like Scheunert does, for instance, see [Sc]) with a two-dimensional parameter σ = (σ 1 , σ 2 , σ 3) ranging in the complex affine plane σ 1 + σ 2 + σ 3 = 0 instead of the single parameter a ∈ C .

Published
2017
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9. Double loop algebras and elliptic root systems

Author

Kenji Iohara, Hiroshi Yamada, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Weyl group, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Group (mathematics), Applied Mathematics, 010102 general mathematics, Extension (predicate logic), Space (mathematics), 01 natural sciences, Principal bundle, Action (physics), Invariant theory, invariant theory, Algebra, Elliptic curve, symbols.namesake, elliptic root system, 2-Toroidal Lie algebras, 0103 physical sciences, symbols, 010307 mathematical physics, MSC2010: 14J17, 14K25, 17B65, 22E65, 0101 mathematics, Mathematics
Abstract

International audience; In this note, we describe an elliptic root system and elliptic Weyl group, due to K. Saito (Publ. RIMS 21 (1985), 75--179), from view point of double loop algebra and its group. A natural action of the double loop group will be introduced on a trivial $\C^\ast$-bundle over the space of $\overline{\partial}$-connections on a $C^\infty$-trivial principal bundle over an elliptic curve that would be constructed from $2$-dimensional central extension of a double loop algebra. The invariant theory of the elliptic Weyl group will be also discussed.

Published
2017
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10. Enright functors for Kac-Moody superalgebras

Author

Kenji Iohara, Yoshiyuki Koga, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Verma module, Singular vector formula, Kac-Moody superalgebra, General Mathematics, Scalar (mathematics), Enright functor, 01 natural sciences, High Energy Physics::Theory, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Mathematics::Category Theory, Mathematics::Quantum Algebra, 0103 physical sciences, MSC 17B10, 17B67, Uniqueness, 0101 mathematics, Mathematics::Representation Theory, Mathematics, Functor, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, Superalgebra, Algebra, Number theory, Differential geometry, Homomorphism, 010307 mathematical physics
Abstract

International audience; A simple generalization of the Enright functor associated with a non-isotropic simple root of Kac-Moody superalgebras is introduced. Two applications for a Kac-Moody superalgebra without isotropic simple root are given: the uniqueness (up to scalar) of homomorphisms between Verma modules and the Malikov-Feigin-Fuks type singular vector formula. The braid relations of the Enright functors are also discussed.

Published
2012
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11. A Global version of Grozman's theorem

Author

Olivier Mathieu, Kenji Iohara, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Algèbre, géométrie, logique (AGL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)

Subjects
Pure mathematics, Geometric context, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], General Mathematics, 010102 general mathematics, Bilinear interpolation, Algebraic geometry, Differential operator, 01 natural sciences, Manifold, Mathematics - Quantum Algebra, 0103 physical sciences, FOS: Mathematics, Equivariant map, Quantum Algebra (math.QA), 010307 mathematical physics, 0101 mathematics, Representation Theory (math.RT), Tensor density, Mathematics - Representation Theory, Mathematics
Abstract

Let \(X\) be a manifold. The classification of all equivariant bilinear maps between tensor density modules over \(X\) has been investigated by Grozman (Funct Anal Appl 14(2):58–59, 1980), who has provided a full classification for those which are differential operators. Here we investigate the same question without the hypothesis that the maps are differential operators. In our paper, the geometric context is algebraic geometry and the manifold \(X\) is the circle \(\text{ Spec}\, \mathbb{C }[z,z^{-1}]\). Our main motivation comes from the fact that such a classification is required to complete the proof of the main result of Iohara and Mathieu (Proc Lond Math Soc, 2012, in press). Indeed it requires to also include the case of deformations of tensor density modules.

Published
2013
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12. Symmetries, Integrable Systems and Representations

Author

Kenji Iohara, Sophie Morier-Genoud, Bertrand Rémy, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Global COE programme 'The research and training center for new development in mathematics' (Graduate School of Mathematical Science, University of Tokyo), Institut Universitaire de France, GDR 3395 'Théorie de Lie algébrique et géométrique', GDRE 571 'Representation theory', Université Lyon 1, Université Paris 6., Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Pure mathematics, Structure constants, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, Lie superalgebra, Category O, Dynkin index, 01 natural sciences, Cluster algebra, Representations, Symmetry, Tensor product, Infinite Analysis, Vertex operator algebra, Theoretical, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Integrable systems, 010307 mathematical physics, Lie theory, 0101 mathematics, Mathematics::Representation Theory, Mathematical & Computational Physics, Mathematics
Abstract

A presentation of the deformed W1+1 algebra.- Generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes.- PBW filtration over Z and compatible bases for VZ(_) in type An and Cn.- On the subgeneric restricted blocks of affine category O at the critical level.- Slavnov determinants, Yang-Mills structure constants, and discrete KP.- Monodromy of partial KZ functors for rational Cherednik algebras.- Category of finite dimensional modules over an orthosymplectic Lie superalgebra: small rank examples.- Monoidal categorifications of cluster algebras of type A and D.- A classification of roots of symmetric Kac-Moody root systems and its application.- Fermions acting on quasi-local operators in the XXZ model.- The Romance of the Ising Model.- A(1) n -Geometric Crystal corresponding to Dynkin index i = 2 and its ultra-discretization.- A Z3-orbifold theory of lattice vertex operator algebra and Z3-orbifold constructions.- Words, automata and Lie theory for tilings.- Toward Berenstein-Zelevinsky data in affine type A, part III: Proof of the connectedness.- Quiver varieties and tensor products, II.- Derivatives of Schur, Tau and Sigma Functions on Abel-Jacobi Images.- Pade interpolation for elliptic Painleve equation.- Non-commutative harmonic oscillators.- The inversion formula of polylogarithms and the Riemann-Hilbert problem.- Some remarks on the Quantum Hall Effect.- Ordinary differential equations on rational elliptic surfaces.- On the spectral gap of the Kac walk and other binary collision processes on d-dimensional lattice.- A restricted sum formula for a q-analogue of multiple zeta values.- A trinity of the Borcherds _-function.- Sum rule for the eight-vertex model on its combinatorial line.

Published
2012
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13. Classification of Simple Lie Algebras on a Lattice

Author

Olivier Mathieu, Kenji Iohara, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Algèbre, géométrie, logique (AGL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)

Subjects
Pure mathematics, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Mathematics::Commutative Algebra, General Mathematics, High Energy Physics::Lattice, Mathematics::Rings and Algebras, 010102 general mathematics, Open set, FOS: Physical sciences, Mathematical Physics (math-ph), 01 natural sciences, Lattice (order), 0103 physical sciences, Lie algebra, FOS: Mathematics, 010307 mathematical physics, Affine transformation, Representation Theory (math.RT), 0101 mathematics, Mathematics - Representation Theory, Mathematical Physics, Mathematics
Abstract

Let Λ = Z for some n ≥ 1. The aim of the paper is to classify all simple Λ-graded Lie algebras L = ⊕ λ∈Λ Lλ, such that dimLλ = 1 for all λ. The classification involves two affine Lie algebras, namely A (1) 1 and A (2) 2 , and a familly (Wπ), parametrized by a dense open set of the space of all embeddings π : Λ→ C. The family (Wl) of generalized Witt algebras, indexed by all embeddings l : Λ → C, appears as a sub-family. In general, the algebras Wπ are described as Lie algebras of symbols of twisted pseudo-differential operators.

Published
2011
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14. Unitarizable Highest Weight Modules of the N=2 Super-Virasoro Algebras: Untwisted sector

Author

Kenji Iohara, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010102 general mathematics, N = 2 super Virasoro algebras, Statistical and Nonlinear Physics, 01 natural sciences, Algebra, unitarizable modules, High Energy Physics::Theory, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Mathematics::Quantum Algebra, 0103 physical sciences, Classification theorem, Virasoro algebra, 0101 mathematics, Mathematics::Representation Theory, 010306 general physics, Mathematical Physics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

In this article, we prove the classification theorem of the unitarizable highest weight modules over the N = 2 super Virasoro algebras for the untwisted sector, stated by Boucher et al. (Phys Lett B 172:316–322, 1986).

Published
2010
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15. Fusion Algebras for Superconformal Field Theories through Coinvariants II : Ramond Sector

Author

Kenji Iohara and Yoshiyuki Koga

Subjects
Fusion, Series (mathematics), Field (physics), 010308 nuclear & particles physics, General Mathematics, 010102 general mathematics, 01 natural sciences, High Energy Physics::Theory, Theoretical physics, Mathematics::K-Theory and Homology, 0103 physical sciences, Fusion rules, 0101 mathematics, Mathematics
Abstract

We determine the fusion rules for the minimal series representations over the super-Virasoro algebras including the Ramond sector.

Published
2009
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16. Note on spin modules associated to Z-graded Lie superalgebras

Author

Kenji Iohara, Yoshiyuki Koga, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Verma module, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], 010308 nuclear & particles physics, 010102 general mathematics, Statistical and Nonlinear Physics, Universal enveloping algebra, Lie superalgebra, 01 natural sciences, Affine Lie algebra, Super-Poincaré algebra, Lie conformal algebra, Graded Lie algebra, Algebra, High Energy Physics::Theory, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Mathematics::Quantum Algebra, 0103 physical sciences, Virasoro algebra, 0101 mathematics, Mathematics::Representation Theory, Mathematical Physics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

The aim of this note is to clarify what can be a correct analog of modules of ‘semi-infinite wedges’ over the Virasoro algebra, considered by Feigin and Fuchs [“Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra,” Funct. Anal. Appl. 16, 114 (1982)], for a certain class of Lie superalgebras.

Published
2009
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