Showing papers on "Cartan matrix published in 1987"
TL;DR: In this paper, it was shown that Cartan subalgebras with maximal toral part have dimension equal to the rank of the algebra, where is the nilspace of the endomorphism.
Abstract: It is shown, using the technique of switching toral subalgebras, that in finitedimensional Lie -algebras every Cartan subalgebra with maximal toral part has dimension equal to the rank of the algebra. As is known, every Cartan subalgebra of a?Lie -algebra? is of the form , where is the nilspace of the endomorphism?, . It is proved that there exists a?Zariski-open subset? such that for every? the subspace is a?Cartan subalgebra with maximal toral part. A?further result is the proof that the class of Cartan subalgebras with maximal toral part is the same as the class of Cartan subalgebras with minimal nilpotent part. The results are used to settle a?question concerning anisotropic forms of Lie algebras over finite fields. Bibliography: 12 titles.
31 citations
01 Jan 1987
TL;DR: In this article, the authors posed the question of what the best way is to define integrability, and the most promising definition seems to be one based on an associated overdetermined linear system which is "allowable".
Abstract: I began this lecture by posing the question of what the best way is to define integrability. The most promising definition seems to be one based on an associated overdetermined linear system which is “allowable”. The linear systems described in sections 3 and 4 are certainly allowable, and most known integrable equations can be obtained in this way. But there are exceptions, such as the KP equation, so the question is not yet settled. With this sort of definition, it would be difficult to establish whether or not a given equation was integrable. But one could try to classify all the integrable equations which arose from a certain type of linear system.
30 citations
TL;DR: In this article, it is shown that the classical Cartan form θ 1L is not adequate for the determination of all the natural symmetries and conservation laws for a Lagrangian L.
Abstract: This paper demonstrates that the classical Cartan form θ1L is not adequate for the determination of all the natural symmetries and conservation laws for a Lagrangian L. It is shown that the various extensions θ2L,..., θrL of the classical Cartan form, introduced in recent papers, give larger symmetry groups: G1⊆G2⊆⋅⋅⋅⊆Gr. This paper also introduces the notion of contact equivalent Lagrangians, which serves to clarify the idea that different Lagrangians can give rise to the same variational and symmetry theories.
26 citations
TL;DR: Naruki and Tomari as mentioned in this paper gave a proof of the equality in Section 4 using the self-intersection number of a Weil divisor on a rational surface, which is a version of that for the strange duality on the 14 exceptional unimodular singularities of Arnold due to Pinkham.
20 citations
01 Oct 1987
TL;DR: The Cartan matrix C of a left artinian ring A, with indecomposable projectives P 1,…, P n and corresponding simples S i = P i / JP i, is an n × n integral matrix with entries C ij, the number of copies of the simple s j which appear as composition factors of P i as discussed by the authors.
Abstract: The Cartan matrix C of a left artinian ring A , with indecomposable projectives P 1 ,…, P n and corresponding simples S i = P i / JP i , is an n × n integral matrix with entries C ij , the number of copies of the simple s j which appear as composition factors of P i . A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A C = ± 1 (Eilenberg [ 3 ]). More recently Zacharia [ 9 ] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A 1 ) or [ 3 ]). However if A is left serial, gl dim A C = l [ 1 ]. If A = ⊕ n ≧ 0 A n is ℤ-graded and the radical J = ⊕ n ≧ 0 A n , Wilson [ 8 ] calls such rings positively graded . Here there is a graded Cartan matrix with entries from ℤ[ X ] and gl dim A = 1 and, hence, det C = l [ 8 , Prop. 2.2].
9 citations
TL;DR: In this paper, the existence of an imbedding of Verma modules over a Kac-Moody algebra, where the two highest weights both lie in the Weyl group orbit of a dominant integral weight, was shown.
5 citations
TL;DR: In this paper, the authors apply the scalar L-A pair to finding the simplest soliton solutions associated with the vector fundamental representations of the unitary series A/sub k/, for which, as usual, the calculations can be carried through to the final expressions without resource to the invariant root technique, only the explicit form of the Cartan matrix being needed.
Abstract: This paper is devoted to the specific question of constructing soliton solutions to the equations of the periodic Toda chain on the basis of the formalism of the scalar L-A pair, i.e., the system of linear equations of higher dimensions for one unique scalar function psi, the condition of compatibility of the system being the equations of the generalized periodic Toda chain. With each representation of a semisimple algebra there is associated a scalar L-A pair. The number of equations in it exceeds by unity the dimension of the corresponding representation, and the maximal degree of the derivative in the system is equal to it. The author limits ourselves to applying the technique of the scalar L-A pair to finding the simplest soliton solutions associated with the vector fundamental representations of the unitary series A/sub k/, for which, as usual, the calculations can be carried through to the final expressions without resource to the invariant root technique, only the explicit form of the Cartan matrix being needed
5 citations
TL;DR: For p ≥ 3 the author computes the indices of p-algebras of Cartan type of the series W, S, H, K, K as discussed by the authors.
Abstract: For p ≥ 3 the author computes the indices of p-algebras of Cartan type of the series W, S, H, K. Bibliography: 11 titles.
3 citations
3 citations
01 Jan 1987
TL;DR: In this article, a complex semisimple Lie algebra and a Cartan subalgebra of this algebra are defined, and the authors show how to construct Cartan subsets of these subalgebras.
Abstract: Throughout this chapter, \(\mathfrak{g}\)denotes a complex semisimple Lie algebra, and \(\mathfrak{h}\) a Cartan subalgebra of \(\mathfrak{g}\) (cf. Chap. III).
2 citations