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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.

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.

This article presents 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, M. 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 unicity 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. M. 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 20th century in various algebraic contexts., arXiv admin note: text overlap with arXiv:1801.00053

In this lecutre note, we consider infinite dimensional Lie algebras of generalized Jacobi matrices $\mathfrak{g}J(k)$ and $\mathfrak{gl}_\infty(k)$, which are important in soliton theory, and their orthogonal and symplectic subalgebras. In particular, we construct the homology ring of the Lie algebra $\mathfrak{g}J(k)$ and of the orthogonal and symplectic subalgebras., To appear in Banach Center Publications

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.

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.

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

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 $\cC_{\rm red}$.

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

In this note, we compute the homology with trivial coefficients of Lie algebras of generalized Jacobi matrices of type $B, C$ and $D$ over an associative unital $k$-algebra with $k$ being a field of characteristic $0$., To appear in Atti. Acad. Naz. Lincei Rend. Lincei Mat. Appl

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 .

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.

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.

Dans cet article, on classifie les modules de plus haut poids unitarisables sur la super-algebre de Virasoro N = 2 tordue.

In this note, we determine the structure of the pre-Verma modules \(N(z,\frac{1}{24}z)\) over the N = 1 Ramond algebra. The results of this note together with the results obtained in Iohara and Koga [Adv Math 178:1–65, 2003] completely determine the structure of Verma modules over the N = 1 super Virasoro algebra both in Neveu Schwarz and Ramond sectors.

The second homology of Lie superalgebras over a field of characteristic 0 extended over a supercommutative superalgebra A and their twisted version are obtained. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this article, we study the structure of Verma modules and Fock modules over the twisted sector of the N =2 super Virasoro algebras.

En generalisant la filtration de Jantzen, a partir des idees de B. Feigin et D. Fuchs dans Representations of the Virasoro algebra paru aux Adv. Stud. Contemp. Math., nous determinons completement la structure des modules de Fock pour les super-algebres de Virasoro N = 1. Comme application, nous demontrons l'existence de complexes d'un type de Felder pour les modeles super-minimaux N = 1 et leur descendants. Cet article est une version detaillee de l'article de Resolution de type Bechi-Rouet-Stora-Tyutin pour les super-algebres de Virasoro paru aux CRAS en 2000.

In this paper, we construct Wakimoto modules for basic affine Lie superalgebras of type A(m−1, n−1)(1) and D(2, 1, a)(1). As an application, we compute the characters of irreducible highest weight modules at the critical level.

For a commutative algebra A over a commutative ring k satisfying certain conditions, we construct the universal central extension of \( {\frak g}_k \otimes_k A \), regarded as a Lie superalgebra over k, where \( {\frak g}_k \) denotes a basic classical Lie superalgebra over k. To consider basic classical Lie superalgebras over an ring k, we also show the existence of their Chevalley basis. Our results contain not only the descriptions of the untwisted affine Lie superalgebras but also those of the toroidal Lie superalgebras.

Resume Nous construisons des resolutions de type Bechi–Rouet–Stora–Tyutin (abrege en BRST) pour les modeles super-minimaux N =1.

RésuméSoit g une algèbre de Kac-Moody symétrisable. Dans [KW1], Kac et Wakimoto ont démontré une formule des caractères pour une grande classe de g-modules simples de plus grand poids qui comprend des g-modules intégrables comme cas particulier. Nous obtenons une extension de la formule de Kac et Wakimoto.AbstractLet g be a symmetrizable Kac-Moody algebra. In [KW1], Kac and Wakimoto proved a character fomula for a large class of irreducible highest weight g-modules which includes integrable modules as a special case. We obtain an extension of Kac and Wakimoto's formula.

For the current realization of the quantum affine algebras, Drinfeld gave a simple comultiplication of the quantum current operators. With this comultiplication, we study the related vertex operators for the case of $U_q(\hgtsl_n)$ and give an explicit bosonization of these new vertex operators. We use these vertex operators to construct the quantum current operators of $U_q(\hgtsl_n)$ and discuss its connection with quantum boson-fermion correspondence., Comment: Amslatex 13 pages

Drinfeld gave a current realization of the quantum affine algebras as a Hopf algebra with a simple comultiplication for the quantum current operators. In this Letter, we will present a generalization of such a realization of quantum Hopf algebras. As a special case, we will choose the structure functions for this algebra to be elliptic functions to derive certain elliptic quantum groups as a Hopf algebra, which degenerates into quantum affine algebras if we take certain degeneration of the structure functions.

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.

On the basis of the `RTT=TTR' formalism, we introduce the quantum double of the Yangian for with a central extension. The Gauss decomposition of the T-matrices gives us the so-called Drinfel'd generators. Using these generators, we present some examples of both finite- and infinite-dimensional representations that are quite natural deformations of their corresponding affine counterpart.

A central extension of $\cD Y_{\hbar}(\gtgl_2)$ is proposed. The bosonization of level $1$ module and vertex operators are also given., Comment: 10 pages, AmsLatex, to appear in Lett. in Math. Phys

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.

An elliptic deformation of $$\widehat{s1}_2 $$ is proposed. Our presentation of the algebra is based on the relationRLL = LLR *, whereR andR * are eight-vertexR-matrices with the elliptic moduli chosen differently. In the trigonometric limit, this algebra reduces to a quotient of that proposed by Reshetikhin and Semenov-Tian-Shansky. Conjectures concerning highest-weight modules and vertex operators are formulated, and the physical interpretation ofR * is discussed.

We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a ``q-deformation'' of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certain non-commutaive rings of quotients coming from complex powers of quantum group generators; this is applied to explicit calculation of singular vectors in Verma modules over $U_{q}(\gtsl_{n+1})$. We finally give a definition of a $q-$connection with coefficients in a ring of skew polynomials and study the structure of quantum group modules twisted by a $q-$connection., Comment: 25 pages

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.

The rationality of the vertex operator algebras associated to minimal series representations is given as an application of the results obtained in Chapters 5 and 6. The fusion algebra associated to such a vertex operator algebra is also given. One of the beautiful and important characterisations of the BPZ series representations is presented with two appendices which provide some necessary background.

We show that certain irreducible highest weight Vir-modules are unitarisable with the aid of unitarisability of integrable highest weight \(\hat {\mathfrak {sl}}_{2}\)-modules. The complete classification of the unitarisable Harish-Chandra modules is given as the goal of these two chapters.

We show that certain irreducible highest weight Vir-modules are unitarisable with the aid of unitarisability of integrable highest weight \(\hat {\mathfrak {sl}}_{2}\)-modules. The complete classification of the unitarisable Harish-Chandra modules is given as the goal of these two chapters.

We explain the proof of the conjecture of V. Kac which says that any simple ℤ-graded Vir-module with finite multiplicities is either a highest weight module, a lowest weight module, or the module of type t λ ℂ[t,t −1](dt) μ . The proof is given by the reduction to positive characteristic cases. Hence, for the reader who is not familiar with Lie p-algebras and their representations, we make a brief survey of the general theory of Lie p-algebras in Appendix B that are used in the proof. This chapter can be read independently of other chapters.

The structure of Verma modules is analyzed in detail. Starting from the classification of the highest weights, the structure of the Jantzen filtration of Verma modules is completely determined from which the Bernstein−Gelfand−Gelfand type resolution follows. As a simple corollary, the characters of the all irreducible highest weight modules over Vir are given. In particular, the characters of minimal series representations, with fixed central charge, forms a vector-valued SL(2,ℤ)-modular form and its modular transformations are also calculated.

The tilting equivalence for a certain class of ℤ-graded Lie algebras is explained. In particular, for the Virasoro algebra, this explains the structural duality between Verma modules with highest weights (c,h) and (26−c,1−h) which can be observed by analyzing Verma modules. This chapter can be read independently of other chapters.

The structure of Verma modules is analyzed in detail. Starting from the classification of the highest weights, the structure of the Jantzen filtration of Verma modules is completely determined from which the Bernstein−Gelfand−Gelfand type resolution follows. As a simple corollary, the characters of the all irreducible highest weight modules over Vir are given. In particular, the characters of minimal series representations, with fixed central charge, forms a vector-valued SL(2,ℤ)-modular form and its modular transformations are also calculated.

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).

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