Representations Of Lie Algebras Research Articles

A Minimal Representation of the Orthosymplectic Lie Supergroup

Abstract We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$ for $p+q$ even, generalizing the Schrödinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions on the minimal orbit inside the Jordan superalgebra associated with $\mathfrak{osp}(p,q|2n)$, so that our construction is in line with the orbit philosophy. Its annihilator is given by a Joseph-like ideal for $\mathfrak{osp}(p,q|2n)$, and therefore the representation is a natural generalization of a minimal representation to the context of Lie superalgebras. We also calculate its Gelfand–Kirillov dimension and construct a nondegenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an $L^2$-inner product in the supercase.

Representations for three-point Lie algebras of genus zero

In this paper, we study representations for three-point Lie algebras of genus zero based on the Cox–Jurisich’s presentations. We construct two functors which transform simple restricted modules with nonzero levels over the standard affine algebras into simple modules over the three-point affine algebras of genus zero. As a corollary, vertex representations are constructed for the three-point affine algebra of genus zero using vertex operators. Moreover, we construct a Fock module for certain quotient of three-point Virasoro algebra of genus zero.

On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

In this paper we study the asymptotic of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotic of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.

Automorphic equivalence in the varieties of representations of Lie algebras

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.

Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions

Meta-conformal transformations are constructed as sets of time-space transformations which are not angle-preserving but contain time- and space translations, time-space dilatations with dynamical exponent and whose Lie algebras contain conformal Lie algebras as sub-algebras. They act as dynamical symmetries of the linear transport equation in d spatial dimensions. For d = 1 spatial dimensions, meta-conformal transformations constitute new representations of the conformal Lie algebras, while for their algebraic structure is different. Infinite-dimensional Lie algebras of meta-conformal transformations are explicitly constructed for d = 1 and d = 2 and they are shown to be isomorphic to the direct sum of either two or three centre-less Virasoro algebras, respectively. The form of co-variant two-point correlators is derived. An application to the directed Glauber–Ising chain with spatially long-ranged initial conditions is described.

Representations of Lie algebras of vector fields on affine varieties

For an irreducible affine variety X over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on X—gauge modules and Rudakov modules, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov [23]. We prove general simplicity theorems for these two types of modules and establish a pairing between them.

Representations of nine-dimensional Levi decomposition Lie algebras

We obtain a matrix representation for each of the indecomposable 9-dimensional real Lie algebras that have a non-trivial Levi decomposition.

Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras

We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over sl ^ 2 . In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights.

On irreducible representations of a certain subalgebra of extended Schrödinger-Virasoro Lie algebra

We study certain representations of Lie algebra , which is a subalgebra of extended Schrödinger-Virasoro Lie algebra. We prove the irreducibility of its certain representations.

Modular Representation Theory of Affine and Cyclotomic Yokonuma-Hecke Algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type A (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic p. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type A.

Minimal representations of filiform Lie algebras and their application for construction of Leibniz algebras

In this paper we find minimal faithful representations of several classes of filiform Lie algebras by means of strictly upper-triangular matrices. We investigate Leibniz algebras whose corresponding Lie algebras L∕I are filiform Lie algebras such that the action I×L∕I→I gives rise to a minimal faithful representation of a filiform Lie algebra, where I is the ideal generated by the squares of the elements of the algebra L. The classification up to isomorphism of such Leibniz algebras is given for low-dimensional cases.

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous n-tuples of operators in the Cowen-Douglas class of the Euclidean unit ball in Cn.

Algorithm to compute minimal matrix representation of nilpotent lie algebras

ABSTRACTAs it is well-known there exist matrix representations of any given finite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional nilpotent Lie algebra from its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representative for each nilpotent Lie algebra of dimensions 6 and 7 and we have also obtained the representation of some families with an arbitrary dimension.

Faithful irreducible representations of modular Lie algebras

Let L be a finite-dimensional Lie algebra over a field of characteristic . By a theorem of Jacobson, L has a finite-dimensional faithful completely reducible module. We show that if F is not algebraically closed, then L has an irreducible such module.

Orthogonal and Symplectic Yangians and Lie Algebra Representations

Orthogonal or symplectic Yangians are defined by the Yang–Baxter RLL relation involving the fundamental R-matrix with so(n) or sp(2m) symmetry. We investigate the conditions on the first- and second-order evaluations as restrictions imposed on the representation weights.

On the Hochschild Cohomologies of Associative Conformal Algebras with a Finite Faithful Representation

Associative conformal algebras of conformal endomorphisms are of essential importance for the study of finite representations of conformal Lie algebras (Lie vertex algebras). We describe all semisimple algebras of conformal endomorphisms which have the trivial second Hochschild cohomology group with coefficients in every conformal bimodule. As a consequence, we state a complete solution of the radical splitting problem in the class of associative conformal algebras with a finite faithful representation.

Vertex Algebras and Coordinate Rings of Semi-infinite Flags

The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type $ADE$ carries a structure of $P/Q$-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of $P/Q$-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Pl\"ucker-type relations in the homogeneous coordinate ring.

Models of Lie algebra sl(2,C) and special matrix functions by means of a matrix integral transformation

In this paper, we present four models of irreducible representations of special complex Lie algebra s l ( 2 , C ) from the special matrix functions point of view. These models, which involve differential operators, are transformed into matrix difference–differential operators using an integral transformation motivated by the integral representation of beta matrix function. We also obtain the matrix identities involving one or two variable special matrix functions.

Jet modules for the centerless Virasoro-like algebra

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a [Formula: see text]-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over [Formula: see text]. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188–231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23–42]. Consequently, indecomposable jet modules are described using modules over the algebra [Formula: see text], which is the “positive part” of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic [Formula: see text] related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153–165] and recently by [A [Formula: see text]-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508–564]).

Efficient Pose Tracking based on Line Segment Matching

Pose tracking is a crucial issue for many applications such as robotic tasks and facility operations. Vision-based approaches with non-contact properties are appropriate choices for these tasks. However, vision-based approaches are not sufficiently robust and fast. In this work, we propose a vision-based pose tracking to deal with these problems. We estimate poses using Lie group and Lie algebra representation theory. Such operation is performed in a linearized space, therefore it is convenient for pose estimation. To provide reliable visual information for our pose estimation, we detect line segments. Our detection of line segment depends on semi-global image information. We describe all line segments and match those detected in consecutive frames. Our line segment detector and matching descriptor are good at discarding ambiguous line segments and finding real ones in noisy situations. The integration of group theory and line segment detection and matching plays an important role for developing a robust vision-based pose tracking system. Our system proves to be efficient and robust.