arXiv subjects
Publications and source records attributed to Kenji Iohara.
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 $0$.
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.
We classify all simple strong Harish-Chandra modules for the Lie superalgebra $W(m,n)$. We show that every such module is either strongly cuspidal or a module of the highest weight type. We construct tensor modules for $W(m,n)$, which are parametrized by simple finite-dimensional $gl(m,n)$-modules and show that every simple strongly cuspidal $W(m,n)$-module is a quotient of a tensor module. Finally, we realize modules of the highest weight type as simple quotients of the generalized Verma modules induced from tensor modules for $W(m-1,n)$.
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.
Let $W_π$ be the lattice Lie algebra of Witt type associated with an additive inclusion $π: \mathbb{Z}^N \hookrightarrow \mathbb{C}^2$ with $N>1$. In this article, the classification of simple $\mathbb{Z}^N$-graded $W_π$-modules, whose multiplicities are uniformly bounded, is given.
In this note, we show the polynomiality of the ring of invariants with respect to the Weyl group of type $A_{2l}^{(2)}$.
This note surveys the historical background of the Gröbner basis theory for D-modules and linear rewriting theory. The objective is to present a deep interaction of these two fields largely developed in algebra throughout the twentieth century. We recall the work of M. Janet on the algebraic analysis on linear partial differential systems that leads to the notion of involutive division. We present some generalizations of the division introduced by M. Janet and their relations with Gröbner basis theory.
Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $\mathfrak{sl}_2$ at generic $q$. We show that for any $r\ge 1$, the endomorphism algebra of $M\otimes V^{\otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $\rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $\rm{TLB}_r(q,Q)$ as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.
The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a) $ - are usually considered for "non-singular" values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular values too, giving rise to "singular specializations" that are no longer simple: this extends the family of simple objects of type $D(2,1;a)$ in five different ways. The resulting five families coincide for general values of $a$, but are different at "singular" ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or "degenerations") at singular values of $a$. Although one may work with a single complex parameter $a$, in order to stress the overall $\mathfrak{S}_3$-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter $\boldsymbolσ = (σ_1,σ_2,σ_3)$ ranging in the complex affine plane $σ_1 + σ_2 + σ_3 = 0$.
Let X be a manifold. The classification of all equivariant bilinear maps between tensor density modules over X has been investigated by Yu Grozman, 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 Spec C[z,z^{-1}].
Let $Λ$ be a lattice of rank $n$. A Lie algebra on the lattice $Λ$ is a Lie algebra ${\cal L}=\oplus_{λ\inΛ}\,{\cal L}_λ$ such that $\dim\,{\cal L}_λ=1$ for all $λ$. In this article, we classify all simple graded Lie algebras on a lattice.
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 paper, 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.
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.
On the basis of `$RTT=TTR$' formalism, we introduce the quantum double of the Yangian $Y_{\hbar}(\gtg)$ for $\gtg=\gtgl_N,\gtsl_N$ with a central extension. The Gauss decomposition of 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 the 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.
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.
We explicitly write dowm integral formulas for solutions to Knizhnik-Zamolodchikov equations with coefficients in non-bounded -- neither highest nor lowest weight -- $\gtsl_{n+1}$-modules. The formulas are closely related to WZNW model at a rational level.
We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra $U_q(\slth)$. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin $1/2$, and that the $n$-particle space has an RSOS-type structure rather than a simple tensor product of the $1$-particle space. This agrees with the picture proposed earlier by Reshetikhin.