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Showing papers in "Duke Mathematical Journal in 1991"


Journal ArticleDOI
TL;DR: In this article, the existence and uniqueness of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra I was proved for the case when g is one of the classical Lie algebras A, B, C, and D,. K.
Abstract: 0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the statistical mechanics. This algebra Uq(g) contains a parameter q, and, when q 1, this coincides with the universal enveloping algebra. In the context of exactly solvable models, the parameter q is that of temperature, and q 0 corresponds to the absolute temperature zero. For that reason, we can expect that the q-analogue has a simple structure at q 0. In [K1] we named crystallization the study at q 0, and we introduced the notion of crystal bases. Roughly speaking, crystal bases are bases of Uq(9)-modules at q 0 that satisfy certain axioms. There, we proved the existence and the uniqueness of crystal bases of finite-dimensional representations of U(g) when g is one of the classical Lie algebras A,, B,, C, and D,. K. Misra and T. Miwa ([M]) proved the existence of a crystal base of the basic representation of U,(A1)) and gave its combinatorial description. The aim of this article is to give the proof of the existence and uniqueness theorem of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra I. Moreover, we globalize this notion. Namely, with the aid of a crystal base we construct a base named the global crystal base of any highest weight irreducible integrable

1,420 citations



Journal ArticleDOI
TL;DR: In this article, the authors investigated certain group actions on manifolds, which have a simple convex polytope as orbit space, and showed that these actions can be locally isomorphic to the standard representation of Z.
Abstract: 0. Introduction. An n-dimensional convex polytope is simple if the number of codimension-one faces meeting at each vertex is n. In this paper we investigate certain group actions on manifolds, which have a simple convex polytope as orbit space. Let P\" denote such a simple polytope. We have two situations in mind. (1) The group is Z, M\" is n-dimensional and (2) The group is T\", M is 2n-dimensional and M2\"/T\" P\". Up to an automorphism of the group, the action is required to be locally isomorphic to the standard representation of Z. on in the second case. In the first case, we call M a \"small cover\" of P\"; in the second, it is a \"toric manifold\" over P. First examples are provided by the natural actions of Z. and T\" on RP\" and CP\", respectively. In both cases the orbit space is an n-simplex. Associated to a small cover of P\", there is a homomorphism 2\" Z’ Z, where m is the number of codimension-one faces of P\". The homomorphism 2 specifies an isotropy subgroup for each codimension-one face. We call it a \"characteristic function\" of the small cover. Similarly, the characteristic function ofa toric manifold over pn is a map Z --) 7/\". A basic result is that small covers and toric manifolds over P\" are classified by their characteristic functions (see Propositions 1.7 and 1.8). The algebraic topology of these manifolds is very beautiful. The calculation of their homology and cohomology groups is closely related to some well-known constructions in commutative algebra and the combinatorial theory of convex polytopes. We discuss some of these constructions below. Let f denote the number of/-faces of P\" and let h denote the coefficient of \"in f(t 1). Then (fo, f,) is called the f-vector and (ho, h,) the h-vector of P\". The f-vector and the h-vector obviously determine one another. The Upper Bound Theorem, due to McMullen, asserts that the inequality h < (,-,{-x), holds for all n-dimensional convex polytopes with m faces of codimension one. In 1971 McMullen conjectured simple combinatorial conditions on a sequence (h0, h,) of integers necessary and sufficient for it to be the h-vector of a simple convex polytope. The sufficiency of these conditions was proved by Billera and Lee and necessity by Stanley (see [Bronsted] for more details and references). Research on

821 citations








Journal Article
TL;DR: In this paper, a tensor structure on a category of representations of affine Lie algebras is defined, and the tensor category of finite-dimensional representations of a quantized enveloping algebra is identified.
Abstract: Let g be a finite dimensional simple Lie algebra of simply laced type. Drinfeld has shown that the tensor category of finite-dimensional representations of the corresponding quantized enveloping algebra over formal power series is equivalent to a tensor category whose objects are the finite-dimensional representations of g and whose tensor structure is obtained from the Knizhnik-Zamolodchikov equations. Our paper can be considered as an extension of Drinfeld's work. Following ideas from conformal field theory we define a tensor structure on a category of representations of an affine Lie algebra, and we identify it with the tensor category of finite-dimensional representations of a quantized enveloping algebra

127 citations












Journal ArticleDOI
TL;DR: The boundary Harnack principle holds in all Holder domains as discussed by the authors for the range α ∈ (1/2, 1] by Bass and Burdzy (1990a), Theorems 3.5 and 4.5.
Abstract: The constant c depends on L only through the constant cL defined in (1) below. Of course, c also depends on D, V and K. The above result shows that the boundary Harnack principle holds in all Holder domains. It was first proven for the range α ∈ (1/2, 1] by Bass and Burdzy (1990a), Theorems 3.5 and 4.5. It was subsequently extended by Banuelos (unpublished) to the full range of α provided the domain also satisfies a uniform capacity condition on the boundary. In Banuelos (1990) a class of domains called uniformly Holder domains of order β was introduced. A boundary Harnack principle may be obtained for such domains (for all β ∈ (0, 1)) by a variation of the proof of Theorem 1. In order to make this note compact, we refer









Journal ArticleDOI
TL;DR: In this paper, it was shown that for any linear space of real algebraic curves, the number of integral lattice points on the graph of a function can be bounded by local conditions on the function f(x) that control the multiplicity of the intersection of Γ with any algebraic curve.
Abstract: This paper is devoted to giving refinements and extensions of some of the results of Bombieri and the author [1] obtaining upper bounds for the number of integral lattice points on the graphs of functions Consider a sufficiently smooth function f(x) with graph Γ, and a positive integer d The main device of that paper was to consider integral points on Γ that do not lie on any real algebraic curve of degree d The Main Lemma of [1] shows that such points cannot be too close together relative to certain norms of the function We pursue here two different goals relative to the Main Lemma The first is to obtain local conditions on the function f(x) that control the multiplicity of the intersection of Γ with any algebraic curve of degree d This is essentially an investigation into the hypotheses of the Main Lemma, and constitutes the Geometric Postulation of the title Indeed, in sections 2 and 3 we will obtain such conditions for any linear space of real algebraic curves The applications to integral points are given in section 5 For example, we show that if f(x) ∈ C on [0, 1] and W (f, 2) = f ′′ ∣∣∣∣∣ f ′′′ 3f ′′ 0 f iv 4f ′′′ 6f ′′ f 5f iv 20f ′′′ ∣∣∣∣∣ is nowhere zero then, for every e > 0,