Showing papers on "Cartan matrix published in 1999"
Book•
01 Jan 1999
TL;DR: The Weil Model and the Cartan Model were proposed by Cartan as discussed by the authors, who considered the Weil model as an extension of Cartan's formula and showed that it can be used in the context of equivariant cohomology in topology.
Abstract: 1 Equivariant Cohomology in Topology.- 3 The Weil Algebra.- 4 The Weil Model and the Cartan Model.- 5 Cartan's Formula.- 6 Spectral Sequences.- 7 Fermionic Integration.- 8 Characteristic Classes.- 9 Equivariant Symplectic Forms.- 10 The Thom Class and Localization.- 11 The Abstract Localization Theorem.- Notions d'algebre differentielle application aux groupes de Lie et aux varietes ou opere un groupe de Lie: Henri Cartan.- La transgression dans un groupe de Lie et dans un espace fibre principal: Henri Cartan.
513 citations
TL;DR: In this article, 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.
42 citations
TL;DR: In this article, a quasi-Hopf supergroup is constructed from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the twisting procedure to the supersymmetric case.
Abstract: We introduce the quasi-Hopf superalgebras which are Z2-graded versions of Drinfeld’s quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the supersymmetric case. Two types of elliptic quantum supergroups are defined, that is, the face type Bq,λ(G) and the vertex type Aq,p[sl(n|∧n)] (and Aq,p[gl(n|∧n)]), where G is any Kac–Moody superalgebra with symmetrizable generalized Cartan matrix. It appears that the vertex type twistor can be constructed only for Uq[sl(n|∧n) in a nonstandard system of simple roots, all of which are fermionic.
28 citations
TL;DR: In this paper, the authors provided a classification for volume-preserving Cartan actions of a connected semisimple Lie group with real rank at least 3. And they provided a C ∞ classification for multiplicity free, trellised, Anosov actions on compact manifolds.
Abstract: Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let ⊂ G be an irreducible lattice. We provide a C ∞ classification for volume-preserving Cartan actions of an d G. Also, if G has real rank at least 3, we provide a C ∞ classification for volume-preserving, multiplicity free, trellised, Anosov actions on compact manifolds.
24 citations
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TL;DR: In this article, it was shown that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra.
Abstract: We give proofs of the PBW and duality theorems for the quantum Kac-Moody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra; this quotient is trivial in all cases except the $A_1^{(1)}$ case.
20 citations
01 Jan 1999
TL;DR: In this paper, the main structural ingredients of finite dimensional associative algebras are described and polynomial time algorithms to find the main components: the Jacobson radical and the simple direct summands of the radical-free part.
Abstract: In this chapter we consider some basic algorithmic problems related to finite dimensional associative algebras. Our starting point is the structure theory of these algebras. This theory gives a description of the main structural ingredients of finite dimensional associative algebras, and specifies the way the algebra is constructed from these building blocks. We describe polynomial time algorithms to find the main components: the Jacobson radical and the simple direct summands of the radical-free part.
15 citations
TL;DR: In this paper, a relation between the number k(B) of ordinary irreducible characters in a p-block B of a finite group G and the Cartan invariants of B was established.
Abstract: In this article we exhibit a relation between the number k(B) of ordinary irreducible characters in a p-block B of a finite group G and the Cartan invariants c
ij
of B. Next, we give a lower bound of the Perron-Frobenius eigenvalue
$\rho (C_B)$
of the Cartan matrix C
B
of B in terms of B, that is
$k(B) \le \rho (C_B)l(B)$
, where l(B) is the number of irreducible Brauer characters in B. For p-solvable groups, we conjecture
$k(B) \le \rho (C_B)$
that is closely related to the Brauer's k(B) conjecture.
9 citations
TL;DR: In this article, a test for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert-series in the associated path incidence ring is presented.
Abstract: This article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup�. Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the�-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties. A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert�-series in the associated path incidence ring. Therationalityofthe�-Eulercharacteristic, theHilbert�-seriesandthePoincar´ e-Betti�-seriesisstudied whenis torsion-free commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.
8 citations
TL;DR: In this paper, the spectral radius of a Coxeter transformation of wild stars has been shown to be upper and lower bounds on the spectral radii of trees with a single branching point and are neither of Dynkin nor of Euclidean type.
8 citations
TL;DR: In this paper, it was shown that the colength of API-variety of Lie algebras grows polynomially, and gave a number of examples in which colength grows more rapidly than any polynomial function does.
Abstract: It is proved that the colength of every API-variety of Lie algebras grows polynomially, and we give a number of examples in which the colength grows more rapidly than any polynomial function does. These indicate that for many of the important varieties of Lie algebras, such as varieties of solvable algebras of derived length 3, varieties generated by some infinite-dimensional simple algebras of Cartan type, or by certain Katz-Mudi algebras, the growth of colength will be superpolynomial.
5 citations
TL;DR: In this article, it was shown that H2(L, F) is isomorphic to the space of skew-symmetric bilinear forms T × T → F.
TL;DR: The well-known interlacing inequalities are examined in the framework of classical real simple Lie algebras.
Abstract: The well-known interlacing inequalities are examined in the framework of classical real simple Lie algebras.
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TL;DR: In this paper, the integrability of the Calogero-Moser-Sutherland models is studied for simple Lie algebras with Cartan subalgebra.
Abstract: We develop a new, systematic approach towards studying the integrability of the ordinary Calogero-Moser-Sutherland models as well as the elliptic Calogero models associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras. It is based on the introduction of a function F, defined on the relevant root system and with values in the respective Cartan subalgebra, satisfying a certain set of combinatoric identities that ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, given by completely explicit formulas. It is shown that among the simple Lie algebras, only those belonging to the A-series admit such a function F, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p,q)/S(U(p) x U(q)), allows non-trivial solutions when |p-q| <= 1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our approach provides new ones, namely for the ordinary models when |p-q| = 1 and for the elliptic models when |p-q| = 1 or p = q.
TL;DR: In this paper, a semi-classical version of quantum groups is used to reconstruct Kac-Moody algebras when the Cartan matrix is symmetrisable.
TL;DR: In this article, the authors investigated the structure of a non-semisimple Lie algebra of characteristic 0 by using the action of a Cartan subalgebra and gave an algorithm for calculating the nilradical and an algorithm to find a Levi subalgorithm.
TL;DR: In this paper, the S-matrix/conserved charge identities in affine Toda field theories were extended to nonsimply-laced Lie algebras.
TL;DR: In this article, the authors characterize the three and four dimensional commutative non-hermitian fusion algebras and construct some new examples of these objects by restricting to a special subclass of orthogonal matrices obtained by the above characterization.
TL;DR: In this paper, the S-matrix/conserved charge identities in affine Toda field theories of the type recently noted by Khastgir can be put on a more systematic footing.
Abstract: We note that S-matrix/conserved charge identities in affine Toda field theories of the type recently noted by Khastgir can be put on a more systematic footing. This makes use of a result first found by Ravanini, Tateo and Valleriani for theories based on the simply-laced Lie algebras (A,D and E) which we extend to the nonsimply-laced case. We also present the generalisation to nonsimply-laced cases of the observation - for simply-laced situations - that the conserved charges form components of the eigenvectors of the Cartan matrix.
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TL;DR: A sequence of generalizations of Cartan's conservation of torsion theorem for n-dimensional differentiable manifolds having a general linear connection is given in this article, where the authors also give a generalization of the Cartan conservation theorem for manifolds with general linear connections.
Abstract: A sequence of generalizations of Cartan's conservation of torsion theorem is given for n-dimensional differentiable manifolds having a general linear connection.
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TL;DR: The families of Cayley-Klein algebras associated to antihermitian matrices over R, C and H are described in a unified framework in this article.
Abstract: The families of quasi-simple or Cayley–Klein algebras associated to antihermitian matrices over R, C and H are described in a unified framework These three families include simple and non-simple real Lie algebras which can be obtained by contracting the pseudo-orthogonal algebras so(p, q) of the Cartan series Bl and Dl, the special pseudo-unitary algebras su(p, q) in the series Al, and the quaternionic pseudo-unitary algebras sq(p, q) in the series Cl This approach allows to study many properties for all these Lie algebras simultaneously In particular their non-trivial central extensions are completely determined in arbitrary dimension