Showing papers on "Cartan matrix published in 1997"
TL;DR: In this paper, the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix was realized as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety, and a counterexample to the conjecture of Kazhdan--Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert cells of type A was given.
Abstract: We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample to the conjecture of Kazhdan--Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert cells of type A and also to the similar problem asked by Lusztig on the characteristic variety of the perverse sheaves corresponding to canonical bases.
282 citations
TL;DR: In this article, the authors consider the Hall algebra H whose basis is formed by isomorphism classes of coherent sheaves on a smooth projectibe curve over a finite field and study its structure similar to that of quantum affine algebras in their loop realization of Drinfeld.
Abstract: Let X be a smooth projectibe curve over a finite field. We consider the Hall algebra H whose basis is formed by isomorphism classes of coherent sheaves on X and whose typical structure constant is the number of subsheaves in a given sheaf belonging to a given isomorphism class and with given isomorphism class of quotient. We study this algebra in detail. It turns out that its structure is strikingly similar to that of quantum affine algebras in their loop realization of Drinfeld. In this analogy the set of simple roots of a semisimple Lie algebra g corresponds to the set of cusp forms on X, the entries of the Cartan matrix of g to the values of Rankin-Selberg L-functions associated to two cusp forms. For the simplest instances of both theories (the curve P^1, the Lie algebra g=sl_2) the analogy is in fact an identity.
113 citations
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TL;DR: In this paper, simple complex Lie superalgebras of vector fields on "supercircles" are defined and described, and three nontrivial cocycles on the N=4 extended Neveu-Schwarz and Ramond superalgebra are presented.
Abstract: We define and describe simple complex Lie superalgbras of vector fields on "supercircles" - simple stringy superalgebras. There are four series of such algebras and four exceptional stringy superalgebras. The 13 of the simple stringy Lie superalgebras are distinguished: only they have nontrivial central extensions; since two of the distinguish algebras have 3 nontrivial central extensions each, there are exactly 16 superizations of the Liouville action, Schroedinger equation, KdV hierarchy, etc. We also present the three nontrivial cocycles on the N=4 extended Neveu-Schwarz and Ramond superalgebras in terms of primary fields and describe the "classical" stringy superalgebras close to the simple ones. One of these stringy superalgebras is a Kac-Moody superalgebra G(A) with a nonsymmetrizable Cartan matrix A. Unlike the Kac-Moody superalgebras of polynomial growth with symmetrizable Cartan matrix, it can not be interpreted as a central extension of a twisted loop algebra.The stringy superalgebras are often referred to as superconformal ones. We discuss how superconformal stringy superalgebras really are.
109 citations
TL;DR: In this article, a finite-dimensional Hopf algebra with antipode-Sover a field k is shown to be a symmetric algebra if and only if its unimodular and semisimpleteness is inner.
98 citations
TL;DR: In this paper, the infinite-dimensional Lie superalgebras of Cartan type X(m, n) (X = W, S, H or K ) over field F of prime characteristic are constructed.
Abstract: IN this note the infinite-dimensional Lie superalgebras of Cartan type X(m, n ) ( X = W, S, H or K ) over field F of prime characteristic are constructed. Then the second class of finitedimensional Lie superalgebras of Cartan type over F is defined. Their simplicity and restrictability are discussed. Finally a conjecture about classification of the finite-dimensional simple Lie superalgebras over F is given. Let F be a field and charF = p >2. I3t n be a positive integer and n > 1. A ( n ) denotes the exterior algebra over F with generators E l , . . a , En. If u = ( i l , i 2 , ..., i r ), where a 1 < i l < i 2 < . . . < i , < n , thenweset Q = ~ i , [ i 2 . . . ~ i r , and / P I = r . LetDi=-, i = l , . . , n . a Ei Imitating the situation that the characteristic number of basic field is zero, we can get the
79 citations
TL;DR: In this paper, the authors give a concrete list of all isomorphy classes of real Cartan factors and give an explicit description of the full automorphism group as well as the group of all surjective ℝ-linear isometries for every non-exceptional real cartan factor and decide which of the real or complex Cartan factor are isometrically equivalent to each other as real Banach spaces.
Abstract: JBW*-triples can be described (modulo W*-algebras, compare [13]) by those of type I. Among these the (complex) Cartan factors are the building blocks. We determine for every complex Cartan factorU all conjugations of the underlying complex Banach space and hence all real forms (in the sense of [15]) ofU, calledreal Cartan factors. We also give a concrete list of all isomorphy classes of real Cartan factors which generalizes the classification of LOOS [23] to infinite dimensions. Furthermore, we give an explicit description of the full automorphism group as well as the group of all surjective ℝ-linear isometries for every non-exceptional real Cartan factor and decide which of the real or complex Cartan factors are isometrically equivalent to each other as real Banach spaces.
70 citations
TL;DR: In this paper, the Cartan matrices of Lie superalgebras with Cartan matrix can be shown to be neither integer nor symmetrizable nor non-Serre relations encountered.
Abstract: We completely describe presentations of Lie superalgebras with Cartan matrix if they are simple Z-graded of polynomial growth. Such matrices can be neither integer nor symmetrizable. There are non-Serre relations encountered. In certain cases there are infinitely many relations. Our results are applicable to the Lie algebras with the same Cartan matrices as the Lie superalgebras considered.
60 citations
TL;DR: In this article, the authors introduced a new class of simple Lie subalgebras of generalized Cartan type W, referred as the algebra of W. The Lie algebra W is A-graded with homogeneous components W s t T, x g A.
57 citations
TL;DR: In this article, the authors give a mild generalization of Cartan's theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes, which is used to complete the proof of the following theorem claimed in an earlier paper by the author: given hyperplanes in general position, there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union (even linearly degenerate curves) satisfy the inequality, except for the ramification term.
Abstract: We give a mild generalization of Cartan's theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes. This generalization is used to complete the proof of the following theorem claimed in an earlier paper by the author: Given hyperplanes in projective space in general position, there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union (even linearly degenerate curves) satisfy the inequality of Cartan's theorem, except for the ramification term. In addition, it is shown how these methods can lead to a shorter proof of Nochka's theorem on Cartan's conjecture and (in the number field case) how Nochka's theorem gives a short proof of Wirsing's theorem on approximation of algebraic numbers by algebraic numbers of bounded degree.
54 citations
TL;DR: In this paper, the authors studied the Bethe ansatz equations for a generalized XXZ model on a one-dimensional lattice and proved the combinatorial completeness of Bethe's states.
Abstract: We study the Bethe ansatz equations for a generalized XXZ model on a one-dimensional lattice. Assuming the string conjecture we propose an integer version for vacancy numbers and prove a combinatorial completeness of Bethe's states for a generalized XXZ model. We find an exact form for the inverse matrix related with vacancy numbers and compute its determinant. This inverse matrix has a tridiagonal form, generalizing the Cartan matrix of type A.
41 citations
01 Oct 1997
TL;DR: In this paper, dual pairs acting on some infinite dimensional Fock representations between a finite dimensional classical Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix are studied.
Abstract: We construct and study in detail various dual pairs acting on some infinite dimensional Fock representations between a finite dimensional classical Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to the action of a dual pair, present explicit formulas for the highest weight vectors and calculate the corresponding highest weights. We further outline applications of these dual pairs to the study of tensor products of modules of such an infinite dimensional Lie algebra.
TL;DR: In this article, it was shown that the finite dimensional half-quantum group u + (G) is of wild representation type, except for G = sl2, where G is a simple Lie algebra of type A, D or E and q a primitive root of unity of order n ≥ 5.
Abstract: Let G be a simple Lie algebra of type A, D or E and q a primitive root of unity of order n ≥ 5. We show that the finite dimensional half-quantum group u + (G) is of wild representation type, except for G = sl2. Moreover, the algebra u + (G) is an admissible quotient of the path algebra of the Cayley graph of the abelian group (Z/nZ) t with respect to the columns of the t × t Cartan matrix of G.
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TL;DR: In this article, it was shown that half-quantum groups at roots of unity corresponding to simple Lie algebras having symmetric Cartan matrix are of wild representation type, except for sl_2.
Abstract: We show that finite dimensional half-quantum groups at roots of unity corresponding to simple Lie algebras having symmetric Cartan matrix are of wild representation type, except for sl_2. Moreover, the underlying associative algebra is isomorphic to an admissible quotient of the path algebra of the Cayley graph of an abelian group. A quantum type Fourier transform enables to describe an explicit isomorphism.
TL;DR: In this article, the authors used the Bose-Fermi polynomial identities for the minimal models M (p, p ) to demonstrate the existence of a Bailey flow from M p, p to the coset models.
TL;DR: The Lie algebra of Cartan type K which occurs as a subalgebra of the Lie algebra for derivations of the polynomial algebra F[x 0, x 1, x 2, x 3, x 4, x 5,x 6,x 7,x 8,x 9,x 10,x 11,x 12,x 13,x 14,x 15,x 16,x 17,x 18,x 19,x 20,x 21,x 22,x 23,x 24,x 25,x 26,x 28,x
Abstract: The Lie algebra of Cartan type K which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra F[x0, x1,…, xn,xn−1,…,x−n], where F is a field of characteristic 0, was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials F[x0,x1,…, xn,x−1,…,x−n,X0 −1x1 -1,…,xn −1,…,x−1 −1…,x−n −1]A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, determine all possible
TL;DR: In this article, the concepts of generalized restricted Lie algebras and their generalized restricted representations over F are introduced, and the irreducible generalized restricted representation of L is determined, which are in a one-to-one correspondence with p sd weight functions; here d is the dimension of the standard torus.
TL;DR: In this paper, the structure of a general two-dimensional Toda field theory involving bosons and fetmions is given in terms of a set of simple roots for a Lie superalgebra.
01 Jan 1997
TL;DR: In this paper, the semisimple Lie algebras among the quadratic Lie algesbras are characterized using Casimir elements, which is a generalization of Cartan's second criterion.
Abstract: Using Casimir elements, we characterize the semisimple Lie algebras among the quadratic Lie algebras. This characterization gives, in particular, a generalization of a consequence of Cartan's second criterion.
TL;DR: In this paper, the representation theory of Hopf algebras of Cartan type was studied in terms of Frobenius kernels of reductive algebraic groups in positive characteristics.
TL;DR: In this article, the Cartan matrix of a simple Lie algebra is replaced with that of an affine Lie algebra, and then a system of functional equations different from the T-system is obtained.
Abstract: It is known that a family of transfer matrix functional equations, the T-system, can be compactly written in terms of the Cartan matrix of a simple Lie algebra. We formally replace this Cartan matrix of a simple Lie algebra with that of an affine Lie algebra, and then we obtain a system of functional equations different from the T-system. It may be viewed as an X n ( a ) type affine Toda field equation on discrete space time. We present, for A n (1) , B n (1) , C n (1) , D n (1) , A n (2) and D n +1 (2) , its solutions in terms of determinants or Pfaffians.
TL;DR: In this article, Avramidi et al. extended the covariant Taylor expansion method to the heat kernel with respect to a massless fermion of spin coupled to a vector gauge field in six-dimensional Riemann-Cartan spacetime.
Abstract: In order to evaluate the asymptotic expansion coefficients of the heat kernel associated with a fermion of spin in an arbitrary dimensional Riemann - Cartan spacetime, the limit of application of the covariant Taylor expansion method (Avramidi I G 1989 Teor. Mat. Fiz. 79 219) to the heat kernel is extended. The first three coefficients in the approach are manifestly computed. In the case of a massless fermion of spin coupled to a vector gauge field in six-dimensional Riemann - Cartan spacetime, the third coefficient which is intimately connected with some anomalies is expressed in the form of arranging it with respect to an irreducible -matrix basis.
TL;DR: In this article, the conjugacy classes of tori in special Lie p-algebra of Cartan type based on the properties of eigenfunctions are shown to be stable.
Abstract: The new proof of the Theorem on conjugacy classes of tori in special Lie p-algebra of Cartan type based on the properties of eigenfunctions is given
TL;DR: In this paper, the homological support variety of a generalized restricted L-module M is defined to be the primitive p-envelope P{L] for a graded Lie algebra of Cartan type over an algebraically closed field of characteristic p ≥ 3, which has been proved to be generalized restricted in the sense of Shul, Shu and Shu2.
Abstract: Let L be a graded Lie algebra of Cartan type over an algebraically closed field of characteristic p≧3, which has been proved to be generalized restricted in the sense of [Shul, Shu2]. For a generalized restricted L-module M, the homological support variety ‖L‖M is defined to be that of the primitive p-envelope P{L). A realization L of P(L) is given in Der(&(m : n)). Furthermore, a class of generalized restricted highest weight L-modules lift to Dist(Tx)V(p)-module structures and their support varieties can be computed by using algebraic group techniques developed in [LN].
TL;DR: It is conjecture and proved that a criterion due to Bell for primeness of the universal enveloping algebra of a Lie superalgebra appl ies to the Cartan type Liesuperalgebras W(n) for n = 3 but does not apply for odd n ≥ 5.
Abstract: On the basis of experimental work involving matrix computations, we conjecture and prove that a criterion due to Bell for primeness of the universal enveloping algebra of a Lie superalgebra appl ies to the Cartan type Lie superalgebras W(n) for n = 3 but does not apply for odd n ≥ 5.
TL;DR: This work considers the problem of decomposing a semisimple Lie algebra defined over a field of characteristic zero as a direct sum of its simple ideals as a result of the decomposition of the action of a Cartan subalgebra.
TL;DR: A description of maximal tori in the -closure of a general Lie algebra of Cartan type over an algebraically closed field of characteristics of tori is given in this article.
Abstract: A description is given of all maximal tori in the -closure of a general Lie algebra of Cartan type over an algebraically closed field of characteristics of tori are found (dimension, algebra of invariants, weight functions root subspaces, the centralizer of a maximal torus) in terms of a defining system of functions of a maximal torus, and criteria are obtained for standartness and optimality of a maximal torus. A connection is established between the maximal tori in the -closure and the Cartan subalgebras of .
01 Jan 1997
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TL;DR: In this article, dual pairs acting on some Fock representations between a finite dimensional Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix are studied.
Abstract: We construct and study in detail various dual pairs acting on some Fock representations between a finite dimensional Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to the action of a dual pair, present explicit formulas for the common highest weight vectors and calculate the corresponding highest weights. We further outline applications of these dual pairs to the study of tensor products of modules of such an infinite dimensional Lie algebra.
Journal Article•
TL;DR: In this article, the structure of certain pairs of Lie algebras is investigated and some inter- esting information on the relationship between representations, between coadjoint orbits and between completely prime primitive ideals in corre- sponding enveloping algesbras are obtained.
Abstract: In this paper some author's results on the structure of certain pairs of Lie algebras are presented. For such pairs some inter- esting information on the relationship between representations, between coadjoint orbits and between completely prime primitive ideals in corre- sponding enveloping algebras can be obtained.
TL;DR: In this article, it was shown that any automorphism of an infinite dimensional Lie algebra of Cartan type over an algebraically closed field of characteristic p>2 can be induced from an automomorphism of the associated divided power algebra.
Abstract: In this paper it is proved that any automorphism of an infinite dimensional Lie algebra of Cartan type over an algebraically closed field of characteristicp>2 can be induced from an automorphism of the associated divided power algebra.