Representation Rings of Lie Superalgebras

Given a Lie superalgebra , we introduce several variants of the representation ring, built as subrings and quotients of the ring of virtual -supermodules, up to (even) isomorphisms. In particular, we consider the ideal of virtual -supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring on which the parity reversal operator takes the class of a virtual -supermodule to its negative. We also construct representation groups built from ungraded -modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring , including all degree shifts, is then a -graded ring in the complex case and a -graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings , and . We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring . In the real case, we obtain the expected 24-term version, as well as a surprising six-term version, of this periodic exact sequence. (Received: October 2004)     

Properties of Auto- and Antiautomorphisms of Maximal Chain Structures and their Relations to i-Perspectivities

Let E be a non empty set, let P : = E × E, := {x × E|x ∈ E}, := {E × x|x ∈ E}, and := {C ∈ 2 ^P |∀X ∈ : |C ∩ X| = 1} and let . Then the quadruple resp. is called chain structure resp. maximal chain structure. We consider the maximal chain structure as an envelope of the chain structure . Particular chain structures are webs, 2-structures, (coordinatized) affine planes, hyperbola structures or Minkowski planes. Here we study in detail the groups of automorphisms , , , related to a maximal chain structure . The set of all chains can be turned in a group such that the subgroup of generated by the left-, by the right-translations and by ι the inverse map of is isomorphic to (cf. (2.14)).     

{\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}

Let R be a unital associative ring and two classes of left R-modules. In this paper we introduce the notion of a In analogy to classical cotorsion pairs as defined by Salce [10], a pair of subclasses and is called a if it is maximal with respect to the classes and the condition for all and Basic properties of are stated and several examples in the category of abelian groups are studied. Received: 17 March 2005     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.     

New results on the peak algebra

The peak algebra is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of and to characterize the elements of in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals of , j = 0,..., , such that is the linear span of sums of permutations with a common set of interior peaks and is the peak algebra. We extend the above results to , generalizing results of Schocker (the case j = 0). Aguiar supported in part by NSF grant DMS-0302423 Orellana supported in part by the Wilson Foundation     

Crystal Base Elements of an Extremal Weight Module Fixed by a Diagram Automorphism

In our previous work, we introduced a bijection between the elements of the crystal base of the negative (resp. positive) part of the quantized universal enveloping algebra of a Kac–Moody algebra that are fixed by a diagram automorphism and the elements of the crystal base of the negative (resp. positive) part of the quantized universal enveloping algebra of the orbit Lie algebra of . In this paper, we prove that this bijection commutes with the *-operation. As an application of this result we show that there exists a canonical bijection between the elements ℬ⁰(λ) of the crystal base ℬ(λ) of an extremal weight module of extremal weight λ over that are fixed by a diagram automorphism and the elements of the crystal base of an extremal weight module of extremal weight over , if the crystal graph of is connected. Presented by P. Littelmann Mathematics Subject Classifications (2000) Primary: 17B37, 17B10; secondary: 81R50.     

A Remark on Support of the Principal Function for Class A Operators

Let be an invertible class A operator such that . Then we show that , where gT is the principal function of T. Moreover, we show that if T is pure, then .     

On the Matlis duals of local cohomology modules

Let ( ) be a commutative Noetherian local ring with non-zero identity, an ideal of R and M a finitely generated R-module with . Let D(–) := Hom _R (–, E) be the Matlis dual functor, where is the injective hull of the residue field . We show that, for a positive integer n, if there exists a regular sequence and the i-th local cohomology module H ⁱ _a (M) of M with respect to is zero for all i with i > n then The author was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 85130023). Received: 9 August 2006     

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Passive systems with and as an input and output space and as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system with is said to be quasi-selfadjoint if ran . The subclass of the Schur class is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass and the Q-function of T is given. Received: December 16, 2007., Accepted: March 4, 2008.     

Birkhoff coordinates for KdV on phase spaces of distributions

The purpose of this paper is to extend the construction of Birkhoff coordinates for the KdV equation from the phase space of square integrable 1-periodic functions with mean value zero to the phase space of mean value zero distributions from the Sobolev space endowed with the symplectic structure More precisely, we construct a globally defined real-analytic symplectomorphism where is a weighted Hilbert space of sequences supplied with the canonical Poisson structure so that the KdV Hamiltonian for potentials in is a function of the actions alone.     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

Extreme values of a portfolio of Gaussian processes and a trend

We consider the extreme values of a portfolio of independent continuous Gaussian processes ( ) which are asymptotically locally stationary, with expectations and variances , and a trend for some constants with . We derive the probability for , which may be interpreted as ruin probability. AMS 2000 Subject Classifications Primary—60G15, 62G32, 91B28     

Quantum duality in quantum deformations

In accordance with the quantum duality principle, the twisted algebra is equivalent to the quantum group and has two preferred bases: one inherited from the universal enveloping algebra and the other generated by coordinate functions of the dual Lie group . We show howthe transformation can be explicitly obtained for any simple Lie algebra and a factorable chain of extended Jordanian twists. In the algebra , we introduce a natural vector grading , compatible with the adjoint representation of the algebra. Passing to the dual-group coordinates allows essentially simplifying the costructure of the deformed Hopf algebra , considered as a quantum group . The transformation can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations and study it in terms of ; we find new realizations of the parabolic twist. Dedicated to the birthday of my teacher, Yurii Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 112–125, July, 2006.     

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005     

Dynamics of the Mapping Class Group on the Moduli of a Punctured Sphere with Rational Holonomy

Let M be a four-holed sphere and Γ the mapping class group of M fixing the boundary ∂M. The group Γ acts on which is the space of completely reducible SL (2, -gauge equivalence classes of flat SL -connections on M with fixed holonomy on ∂M. Let and be the compact component of the real points of . These points correspond to SU(2)-representations or SL(2, -representations. The Γ-action preserves and we study the topological dynamics of the Γ-action on and show that for a dense set of holonomy , the Γ-orbits are dense in . We also produce a class of representations such that the Γ-orbit of [ρ] is finite in the compact component of , but is dense in SL(2, .Mathematics Subject Classiffications (2000). 57M05, 54H20, 11D99     

Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization

In this note we continue the study of spectral properties of a self-adjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub–Matsaev condition on some interval Δ₀ and is boundedly invertible in the endpoints of Δ₀, then the ‘embedding’ of the original Hilbert space into the Hilbert space , where the linearization of A(z) acts, is in fact an isomorphism between a subspace of and . As a consequence, properties of the local spectral function of A(z) on Δ₀ and a so-called inner linearization of the operator function A(z) in the subspace are established.      

Spherical Monogenics: An Algebraic Approach

The space of spherical monogenics in can be regarded as a model for the irreducible representation of Spin(m) with weight . In this paper we construct an orthonormal basis for . To describe the symmetry behind this procedure, we define certain Spin(m − 2)-invariant representations of the Lie algebra (2) on . Received: October, 2007. Accepted: February, 2008.     

On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let be the Euclidean Dirac operator in the n-dimensional flat space the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials. Received: 27 October 2005