Modules For Lie Algebras Research Articles

Extended Diffeomorphism Algebras¶and Trajectories in Jet Space

Let the DRO (Diffeomorphism, Reparametrization, Observer) algebra¶DRO(N) be the extension of diff(N)⊕ diff(1) by its four inequivalent Virasoro-like cocycles. Here diff(N) is the diffeomorphism algebra in N-dimensional spacetime and diff(1) describes reparametrizations of trajectories in the space of tensor-valued p-jets. DRO(N) has a Fock module for each p and each representation of gl(N). Analogous representations for gauge algebras (higher-dimensional Kac–Moody algebras) are also given. The reparametrization symmetry can be eliminated by a gauge fixing procedure, resulting in previously discovered modules. In this process, two DRO(N) cocycles transmute into anisotropic cocycles for diff(N). Thus the Fock modules of toroidal Lie algebras and their derivation algebras are geometrically explained.

Colimits in the Crossed Modules Category in Lie Algebras

Abstract In this paper a left adjoint is constructed for the restriction (or pullback) functor associated to a morphism of Lie algebras. Colimits in the category of crossed modules in Lie algebras and a new result on crossed modules induced by a morphism of Lie algebras ι : P → Q in the case where ι is the inclusion of an ideal, are obtained.

A Five-term Exact Sequence for Opext of Crossed Modules in Lie Algebras

Extensions of crossed modules in Lie algebras with abelian kernel are studied, particularly backward and forward induced extensions and related properties. The set Opext((U, Q, ɛ), (R, K, ∂)) of congruence classes of extensions of (R, K, ∂) by (U, Q, ɛ) is endowed with a K-vector space structure. This K-vector space appears in a five-term natural and exact sequence associated with an extension of crossed modules.

A Five-term Exact Sequence for Opext of Crossed Modules in Lie Algebras

Extensions of crossed modules in Lie algebras with abelian kernel are studied, particularly backward and forward induced extensions and related properties. The set Opext((U, Q, ɛ), (R, K, ∂)) of congruence classes of extensions of (R, K, ∂) by (U, Q, ɛ) is endowed with a K-vector space structure. This K-vector space appears in a five-term natural and exact sequence associated with an extension of crossed modules.

Constructions for Octonion and Exceptional Jordan Algebras

In this note we reverse the usual process of constructing the Lie algebras of types G_2 and F_4 as algebras of derivations of the split octonions or the exceptional Jordan algebra and instead begin with their Dynkin diagrams and then construct the algebras together with an action of the Lie algebras and associated Chevalley groups. This is shown to be a variation on a general construction of all standard modules for simple Lie algebras and it is well suited for use in computational algebra systems. All the structure constants which occur are integral and hence the construction specialises to all fields, without restriction on the characteristic, avoiding the usual problems with characteristics 2 and 3.

Capelli type identities on certain scalar generalized Verma modules

We give an analogue of the Capelli identity. The analogue is constructed on certain scalar generalized Verma modules of the complex simple Lie algebras of Hermitian symmetric type, and has stronger compatibility with group actions than the original Capelli identity. As in the original case, the analogue expresses certain invariant operators as operators from the center of the universal enveloping algebra. We also give examples for all the possible cases except for $E_{7}$.

A Crossed Module Representing the Godbillon–Vey Cocycle

We exhibit crossed modules of Lie algebras (i.e. certain four-term exact sequences) corresponding to the Godbillon–Vey generator for the Lie algebras of formal vector fields in one variable, of vector fields on the circle, of differentiable vector fields of holomorphic type and of holomorphic vector fields on a punctured Riemann surface.

Highest weight categories of Lie algebra modules

We study a certain class of categories of Lie algebra modules which include the well-known categories O and O S . We show that all these categories are highest weight categories.

Q -Identities and Affinized Projective Varieties¶II. Flag Varieties

In a previous paper we defined the concept of an affinized projective variety and its associated Hilbert series. We computed the Hilbert series for varieties associated to quadratic monomial ideals. In this paper we show how to apply these results to affinized flag varieties. We discuss various examples and conjecture a correspondence between the Hilbert series of an affinized flag variety and a modified Hall-Littlewood polynomial. We briefly discuss the application of these results to quasi-particle character formulas for affine Lie algebra modules.

Weight modules of direct limit Lie algebras

Journal Article Weight modules of direct limit Lie algebras Get access Ivan Dimitrov, Ivan Dimitrov Search for other works by this author on: Oxford Academic Google Scholar Ivan Penkov Ivan Penkov Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1999, Issue 5, 1999, Pages 223–249, https://doi.org/10.1155/S1073792899000124 Published: 01 January 1999 Article history Received: 21 October 1998 Published: 01 January 1999

Crystals for Demazure Modules of Classical Affine Lie Algebras

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of typesA(1)n,B(1)n,C(1)n,D(1)n,A(2)2n−1,A(2)2n, andD(2)n+1. We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show that if the highest weight islΛ0, the Demazure crystal has a remarkably simple structure.

Unitary Highest Weight Representations in Hilbert Spaces of Holomorphic Functions on Infinite Dimensional Domains

Automorphism groups of symmetric domains in Hilbert spaces form a natural class of infinite dimensional Lie algebras and corresponding Banach Lie groups. We give a classification of the algebraic category of unitary highest weight modules for such Lie algebras and show that infinite dimensional versions of the Lie algebras so(2,n) have no unitary highest weight representations and thus do not meet the physical requirement of having positive energy. Highest weight modules correspond to unitary representations of global Banach Lie groups realized in Hilbert spaces of vector valued holomorphic functions on the relevant domains in Hilbert spaces. The construction of such holomorphic representations of certain Banach Lie groups, followed by the application of the general framework of Harish-Chandra type groups in an appropriate Banach setting, leads to the integration of the Lie algebra representation to a group representation. The extension of this theory to infinite dimensional settings is explored.

The actor of a crossed module in lie algebras

In the category D,M of crossed modules in Lie algebras we introduce the actor of a crossed module, an object that plays a similar role to derivations of a Lie algebra in the Lie algebras category. From this concept we derive fundamental notions in D,M such as the centre of a crossed module and the action between crossed modules. Using this concept of action we obtain a description of a crossed square (crossed module in the crossed modules category).

Vertex operator algebras and irreducibility of certain modules for affine Lie algebras

We find the connection between the representation theory of vertex operator algebra $L(k Lambda_0)$ and the irreducibility of tensor products $overline V(mu) otimes L(k Lambda_0)$. In the case of affine Lie algebra $A_1 ^{(1)}$, on every admissible rational level we construct a family of irreducible modules having infinite-dimensional weight spaces.

Generalization of the Knizhnik–Zamolodchikov Equations

In this Letter, we introduce a generalization of the Knizhnik–Zamolodchikov equations from affine Lie algebras to a wide class of conformal field theories (not necessarily rational). The new equations describe correlations functions of primary fields and of a finite number of their descendents. Our proposal is based on Nahm's concept of small spaces which provide adequate substitutes for the lowest energy subspaces in modules of affine Lie algebras. We explain how to construct the first order differential equations and investigate properties of the associated connections, thereby preparing the grounds for an analysis of quantum symmetries. The general considerations are illustrated in examples of Virasoro minimal models.

Combinatorial constructions of modules for infinite-dimensional Lie algebras, I. Principal subspace

This is the first of a series of papers studying combinatorial (with no “subtractions”) bases and characters of standard modules for affine Lie algebras, as well as various subspaces and “coset spaces” of these modules. In part I we consider certain standard modules for the affine Lie algebra g ̂ , g: = sl(n + 1, C) , n ≥ 1, at any positive integral level k and construct bases for their principal subspaces (introduced and studied recently by Feigin and Stoyanovsky (1994)). The bases are given in terms of partitions: a color i, 1 ≤ i ≤ n, and a charge s, 1 ≤ s ≤ k, are assigned to each part of a partition, so that the parts of the same color and charge comply with certain difference conditions. The parts represent “Fourier coefficients” of vertex operators and can be interpreted as “quasi-particles” enjoying (two-particle) statistical interaction related to the Cartan matrix of g. In the particular case of vacuum modules, the character formula associated with our basis is the one announced in Feigin and Stoyanovsky (1994). New combinatorial characters are proposed for the whole standard vacuum ĝ-modules at level one.

Structure of α-Stratified Modules for Finite-Dimensional Lie Algebras, I

For complex simple finite-dimensional Lie algebras we study the structure of generalized Verma modules which are torsion free with respect to some subalgebra isomorphic tosl(2). We obtain a criterion of irreducibility and establish necessary and sufficient conditions for the existence of a non-trivial homomorphism between such modules generalizing the BGG theorem for Verma modules.

Polynomial Analogues of the Character Formulas in Vertex Models

New polynomial identities are presented in a context of statistical mechanics. They are associated with the one-dimensional configuration sums of the corner transfer matrix calculation of the solvable vertex models. In the infinite order limit they produce well-known principally specialized character formulas of level 1 modules of affine Lie algebras A n (1) , B n (1) and D n (1) .

Representations of Blocks Associated to Induced Modules of Restricted Lie Algebras

AbstractLet (L, [p]) be a finite dimensional restricted Lie algebra over an algebraically closed field F of characteristic p ≧ 3, and x ε L* be a linear form. In this article we investigate the structure and representation theory of those blocks of the reduced enveloping algebra u(L,x) that are associated to irreducible modules of the form u(L, x) ⊗ u(K,x|k)Fλ.

New irreducible modules for affine Lie algebras at the critical level

Journal Article New irreducible modules for affine Lie algebras at the critical level Get access Drazen Adamović Drazen Adamović Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1996, Issue 6, 1996, Pages 253–261, https://doi.org/10.1155/S1073792896000189 Published: 01 January 1996 Article history Published: 01 January 1996 Received: 05 February 1996 Revision received: 16 February 1996