Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A_n⁽¹⁾, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary 16G10, 17B37. Alistair Savage: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and was partially conducted by the author for the Clay Mathematics Institute.     

Cm Elliptic Curves and Bicolorings of Steiner Triple Systems

It is shown that a necessary condition for the existence ofa bicolored Steiner triple system of order n is that n can bewritten in the form A²+3B² for integers A and B. In the casewhen n=q is either a prime congruent to 1 mod 3, or the squareof a prime congruent to 2 mod 3, it is shown that the numbersof colored vertices in the triple system would be unique, andare given by the number of points on specific twists of theCM elliptic curve y²=x³–1 over the finite field F_q. 2000Mathematics Subject Classification 05B07, 11G20, 14G15 (primary);11G15, 14K22 (secondary).     

Crystals of type and Young walls

We give a new realization of arbitrary level perfect crystals and arbitrary level irreducible highest weight crystals of type , in the language of Young walls. We refine the notions of splitting of blocks and slices that have appeared in our previous works, and these play crucial roles in the construction of crystals. The perfect crystals are realized as the set of equivalence classes of slices, and the irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls which, in turn, are concatenations of slices.

     

Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantumgeneralized Kac–Moody algebras. For a quantum generalizedKac–Moody algebra U_q(g), we first introduce the categoryO_int of U_q(g)-modules and prove its semisimplicity. Next, wedefine the notion of crystal bases for U_q(g)-modules in thecategory O_int and for the subalgebra . We then prove the tensor product rule and the existence theoremfor crystal bases. Finally, we construct the global bases forU_q(g)-modules in the category O_int and for the subalgebra . 2000 Mathematics Subject Classification17B37, 17B67.     

Extensions of Isometric Dual Representations of Semigroups

Let T be a dual representation of a suitable subsemigroup Sof a locally compact abelian group G by isometries on a dualBanach space X=(X_*)^*. It is shown that (X, T) can be extendedto a dual representation of G on a dual Banach space Y containingX, and that this extension can be done in a canonical way. Inthe case of a representation by *-monomorphisms of a von Neumannalgebra, the extension is a representation of G by *-automorphismsof a von Neumann algebra.     

Homogeneous Orthogonally Additive Polynomials on Banach Lattices

The main result in this paper is a representation theorem forhomogeneous orthogonally additive polynomials on Banach lattices.The representation theorem is used to study the linear spanof the set of zeros of homogeneous real-valued orthogonallyadditive polynomials. It is shown that in certain lattices everyelement can be represented as the sum of two or three zerosor, at least, can be approximated by such sums. It is also indicatedhow these results can be used to study weak topologies inducedby orthogonally additive polynomials on Banach lattices. 2000Mathematics Subject Classification 46G25, 46B42, 47B38.     

A Basic Analogue of Hermite's Equation

Using basic number and the analogues of differentiation andintegration, a q-analogue of Hermite's equation is introduced.Series solutions are given, and it is shown that polynomialforms of these solutions are orthogonal with respect to basicintegration. By reversing the series representation of thesesolutions, a basic analogue of the Hermite polynomial is obtainedfor which a generating function and a three-term recurrencerelation are deduced. Finally, an orthogonality relation isgiven.     

Small Maximal Spaces of Non-Invertible Matrices

The rank of a vector space A of n x n-matrices is by definitionthe maximal rank of an element of A. The space A is called rank-criticalif any matrix space that properly contains A has a strictlyhigher rank. This paper exhibits a sufficient condition forrank-criticality, which is then used to prove that the imagesof certain Lie algebra representations are rank-critical. Arather counter-intuitive consequence, and the main novelty inthis paper, is that for infinitely many n, there exists an eight-dimensionalrank-critical space of n x n-matrices having generic rank n– 1, or, in other words: an eight-dimensional maximalspace of non-invertible matrices. This settles the question,posed by Fillmore, Laurie, and Radjavi in 1985, of whether sucha maximal space can have dimension smaller than n. Another consequenceis that the image of the adjoint representation of any semisimpleLie algebra is rank-critical; in both results, the ground fieldis assumed to have characteristic zero. 2000 Mathematics SubjectClassification 15A30, 17B10, 20G05.     

Categories with Projective Functors

We introduce a notion of a category with full projective functors.It encodes certain common properties of categories appearingin representation theory of Lie groups, Lie algebras and quantumgroups. We describe the left or right exact functors which naturallycommute with projective functors and provide a unified approachto the verification of relations between such functors. 2000Mathematics Subject Classification 17B10.     

Young wall realization of crystal bases for classical Lie algebras

In this paper, we give a new realization of crystal bases for finite-dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.

     

Cabling procedure for the colored HOMFLY polynomials

We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and $\mathcal{R}$ -matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and $\mathcal{R}$ -matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦Q¦m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental $\mathcal{R}$ -matrices and clarifying some conjectures formulated in previous papers.     

Specht Filtrations for Hecke Algebras of Type A

Let H_q(d) be the Iwahori–Hecke algebra of the symmetricgroup, where q is a primitive 1th root of unity. Using resultsfrom the cohomology of quantum groups and recent results aboutthe Schur functor and adjoint Schur functor, it is proved that,contrary to expectations, for l 4 the multiplicities in a Spechtor dual Specht module filtration of an H_q(d)-module are welldefined. A cohomological criterion is given for when an H_q(d)-modulehas such a filtration. Finally, these results are used to givea new construction of Young modules that is analogous to theDonkin–Ringel construction of tilting modules. As a corollary,certain decomposition numbers can be equated with extensionsbetween Specht modules. Setting q = 1, results are obtainedfor the symmetric group in characteristic p 5. These resultsare false in general for p = 2 or 3.     

Continuity of Lie Isomorphisms of Banach Algebras

We prove that if A and B are semisimple Banach algebras, thenthe separating subspace of every Lie isomorphism from A ontoB is contained in the centre of B. 1991 Mathematics SubjectClassification 17B40, 17B60, 46H40.     

Characteristic Zero Loop Space Homology of Two-Cones

Given a principal ideal domain R of characteristic zero, containing1/2, and a two-cone X of appropriate connectedness and dimension,we present sufficient algebraic conditions for the Hopf algebraFH(X; R) to be isomorphic with the universal enveloping algebraof an R-free graded Lie algebra; here, F stands for free part(that is, quotient by the R-torsion), H for homology, and forthe Moore loop space functor. 2000 Mathematics Subject Classification55P35, 55P62, 57T05, 17B70, 17B35.     

Arens Regularity of the Algebra of Operators on a Banach Space

A short proof is given that if E is a super-reflexive Banachspace, then B(E), the Banach algebra of operators on E withcomposition product, is Arens regular. Some remarks are madeon necessary conditions on E for B(E) to be Arens regular. 2000Mathematics Subject Classification 47L10 (primary), 46H05 (secondary).     

Quantum Deformations of Imaginary Verma Modules

We construct quantum deformations of imaginary Verma modulesover and show that, for generic q, imaginary Verma modules over can be deformed to those over the quantum group in such a way that the dimensions of the weightspaces are invariant under the deformation. We also prove thePBW theorem for with respect to the triangular decomposition induced from the root partitioncorresponding to the imaginary Verma modules. 1991 MathematicsSubject Classification: 17B67, 17B65, 17B10.     

Ubiquity of Free Subgroups

Necessary and sufficient conditions are given for a Polish topologicalgroup to be ‘almost free’. It is deduced that theexistence of one free subgroup of a Polish group can lead tothe existence of many free subgroups of maximal rank. Applicationsare given to permutation groups, profinite groups, Lie groupsand unitary groups. 2000 Mathematics Subject Classification22F50, 20E05, 54H05 (primary), 12F10, 20B27, 20B30, 20E18 (secondary).     

The Location of the Control Points in the Case of Box Splines

In this paper a theorem of Greville (1967) for univariate splinesis carried over to multivariate box splines; namely, it is shownhow the vector-valued function s(x)=x can expressed in termsof some translates of a box spline. The result means that thegraph of a spline function s(x), which is given as a linearcombination of box splines, can be viewed as a parametric splinesurface. The control points of the graph are explicitly given.     

The Steinberg Lattice of a Finite Chevalley Group and its Modular Reduction

Let p be a prime and let q = p^a, where a is a positive integer.Let G 7equals; G(F_q) be a Chevalley group over F_q, with associatedsystem of roots and Weyl group W. Steinberg showed in 1957that G has an irreducible complex representation whose degreeequals the p-part of |G| [11]. This representation, now knownas the Steinberg representation, has remarkable properties,which reflect the structure of G, and there have been many researchpapers devoted to its study. The module constructed in [11]is in fact a right ideal in the integral group ring ZG of G,and is thus a ZG-lattice, which we propose to call the Steinberglattice of G. It should be noted that lattices not integrallyisomorphic to the Steinberg lattice may also afford the Steinbergrepresentation, and such lattices may differ considerably intheir properties compared with the Steinberg lattice.