Representations Of Lie Algebras Research Articles

The geometry of special symplectic representations

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions. The main algebraic result is that suitably generic elements of these representation spaces can be uniquely written as the sum of two elements of a naturally defined Lagrangian subvariety. We give universal explicit formulae for the summands and show how they lead to the existence of geometric structure on appropriate subsets of the representation space. Over the real numbers this structure reduces to either a conic, special pseudo-Kähler metric or a conic, special para-Kähler metric.

Applications of the Capelli identities in physics and representation theory

Capelli identities are shown to facilitate the construction of representations of various Heisenberg algebras that arise in many-particle quantum mechanics and the construction of holomorphic representations of many Lie algebras by vector coherent state methods. We consider the original Capelli identity and its generalizations by Turnbull and by Howe and Umeda.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

This paper is the first part of a two part paper which introduces a method for determining the vanishing topology of nonisolated matrix singularities. A foundation for this is the introduction in this first part of a method for obtaining new classes of free divisors from representations V of connected solvable linear algebraic groups G. For equidimensional representations where dimG = dimV , with V having an open orbit, we give sufficient conditions that the complement E of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor). We do so by introducing the notion of a “block representation”which is especially suited for both solvable groups and extensions of reductive groups by them. This is a representation for which the matrix representing a basis of associated vector fields on V defined by the representation can be expressed using a basis for V as a block triangular matrix, with the blocks satisfying certain nonsingularity conditions. We use the Lie algebra structure of G to identify the blocks, the singular structure, and a defining equation for E. This construction naturally fits within the framework of towers of Lie groups and representations yielding a tower of free divisors which allows us to inductively represent the variety of singular matrices as fitting between two free divisors. We specifically apply this to spaces of matrices including m×m symmetric, skew-symmetric or general matrices, where we prove that both the classical Cholesky factorization of matrices and a further “modified Cholesky factorization”which we introduce are given by block representations of solvable group actions. For skew-symmetric matrices, we further introduce an extension of the method valid for a representation of a nonlinear infinite dimensional solvable Lie algebras. In part II, we shall use these geometric decompositions and results for computing the vanishing topology for nonlinear sections of free divisors to compute the vanishing topology for matrix singularities for all of the classes.

Essential signatures and canonical bases of irreducible representations of the group G 2

We consider representations of simple Lie algebras and the problem of constructing a “canonical” weight basis in an arbitrary irreducible finite-dimensional highest-weight module. Vinberg suggested a method for constructing such bases by applying the lowering operators corresponding to all negative roots to the highest-weight vector and put forward a number of conjectures on the parametrization and structure of such bases. It follows from papers by Feigin, Fourier, and Littelmann that these conjectures are true for the cases of A n and C n . In the present paper, we prove these conjectures for the case of G 2 by using a different approach suggested by Vinberg.

Some remarks on representations of Yang-Mills algebras

In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043–1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.

Characters of the Feigin-Stoyanovsky subspaces and Brion’s theorem

We give an alternative proof of the main result of [1]; the proof relies on Brion’s theorem about convex polyhedra. The result itself can be viewed as a formula for the character of the Feigin-Stoyanovsky subspace of an integrable irreducible representation of the affine Lie algebra \(widehat {s{l_n}}(\mathbb{C})\). Our approach is to assign integer points of a certain polytope to vectors comprising a monomial basis of the subspace and then compute the character by using (a variation of) Brion’s theorem.

A uniform realization of the combinatorial $R$-matrix

Kirillov-Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape KR crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger’s sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial $R$-matrix in type $A$. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves Les cristaux de Kirillov–Reshetikhin (KR) sont des graphes orientés avec des arêtes étiquetées qui encodent certaines représentations de dimension finie des algèbres de Lie affines. Les produits tensoriels des cristaux KR de type colonne ont été récemment réalisés de manière uniforme, pour tous les types affines symétriques, en termes du modèle des alcôves quantique. Nous enrichissons ce modèle en l’utilisant pour donner une réalisation uniforme de la $R$-matrice combinatoire, c’est à dire, l’isomorphisme des cristaux affines unique quit permute les facteurs dans un produit tensoriel des cristaux KR. En d’autres termes, nous généralisons pour tous les types de Lie le jeu de taquin de Schützenberger sur les tableaux de Young, qui réalise la $R$-matrice combinatoire dans le type $A$. Nous montrons aussi que le modèle des alcôves quantique ne dépend pas du choix d’une suite d’alcôves.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

We introduce a method for obtaining new classes of free divisors from representations $V$ of connected linear algebraic groups $G$ where $\dim(G)=\dim(V)$, with $V$ having an open orbit. We give sufficient conditions that the complement of this open orbit, the "exceptional orbit variety", is a free divisor (or a slightly weaker free* divisor) for "block representations" of both solvable groups and extensions of reductive groups by them. These are representations for which the matrix defined from a basis of associated "representation vector fields" on $V$ has block triangular form, with blocks satisfying certain nonsingularity conditions. For towers of Lie groups and representations this yields a tower of free divisors, successively obtained by adjoining varieties of singular matrices. This applies to solvable groups which give classical Cholesky-type factorization, and a modified form of it, on spaces of $m \times m$ symmetric, skew-symmetric or general matrices. For skew-symmetric matrices, it further extends to representations of nonlinear infinite dimensional solvable Lie algebras.

On 2 - 3 Matrix Chevalley Eilenberg Cohomology

The main objective of this paper is to provide the tool rather than the classical adjoint representation of Lie algebra; which is essential in the conception of the Chevalley Eilenberg Cohomology. We introduce the notion of representation induced by a 2 - 3 matrix. We construct the corresponding Chevalley Eilenberg differential and we compute all its cohomological groups.

A lower bound for faithful representations of nilpotent Lie algebras

Given a finite-dimensional nilpotent Lie algebra , let (resp. ) be the minimal dimension of such that admits a faithful representation (resp. nilrepresentation) on . In this paper, we present a lower bound for for a -step nilpotent Lie algebra over a field of characteristic zero. Our bound is given as the minimum of a quadratically constrained linear optimization problem, it works for arbitrary and takes into account a given filtration of . We present some estimates of this minimum which leads to a very explicit lower bound for that involves the dimensions of and its centre. This bound allows us to obtain for some families of nilpotent Lie algebras.

The Serpentine Representation of the Infinite Symmetric Group and the Basic Representation of the Affine Lie Algebra $${\widehat{\mathfrak{sl}_2}}$$ sl 2 ^

We introduce and study the so-called serpentine representations of the infinite symmetric group $\sinf$, which turn out to be closely related to the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and representations of the Virasoro algebra. This is a new version of the manuscript of the same authors "On a relation between the basic representation of the affine Lie algebra $\widehat\sl$ and a Schur--Weyl representation of the infinite symmetric group" (arXiv:1403.1558)

Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections

Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Frechet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γ_c(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γ_c(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C^∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗A, where A is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)_0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SU_n(A)_0 with A a complete involutive commutative continuous inverse algebra.

Spin Matrix theory: a quantum mechanical model of the AdS/CFT correspondence

We introduce a new quantum mechanical theory called Spin Matrix theory (SMT). The theory is interacting with a single coupling constant g and is based on a Hilbert space of harmonic oscillators with a spin index taking values in a Lie (super)algebra representation as well as matrix indices for the adjoint representation of U(N). We show that SMT describes N=4 super-Yang-Mills theory (SYM) near zero-temperature critical points in the grand canonical phase diagram. Equivalently, SMT arises from non-relativistic limits of N=4 SYM. Even though SMT is a non-relativistic quantum mechanical theory it contains a variety of phases mimicking the AdS/CFT correspondence. Moreover, the infinite g limit of SMT can be mapped to the supersymmetric sector of string theory on AdS_5 x S^5. We study SU(2) SMT in detail. At large N and low temperatures it is a theory of spin chains that for small g resembles planar gauge theory and for large g a non-relativistic string theory. When raising the temperature a partial deconfinement transition occurs due to finite-N effects. For sufficiently high temperatures the partially deconfined phase has a classical regime. We find a matrix model description of this regime at any coupling g. Setting g=0 it is a theory of N^2+1 harmonic oscillators while for large g it becomes 2N harmonic oscillators.

Conjugation properties of tensor product multiplicities

It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products and , are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for the fusion product of the affine algebra of SU(3) at finite levels, but this is not investigated in the present paper.

Representations of Leibniz Algebras

In this paper we prove that every irredicuble representation of a Leibniz algebra can be obtained from irreducible representations of the semisimple Lie algebra from the Levi decomposition. We also prove that - in general - for (semi)simple Leibniz algebras it is not true that a representation can be decomposed to a direct sum of irreducible ones.

Jacobi–Trudi determinants and characters of minimal affinizations

In their study of characters of minimal affinizations of representations of orthogonal and symplectic Lie algebras, Chari and Greenstein conjectured that certain Jacobi-Trudi determinants satisfy an alternating sum formula. In this note, we prove their conjecture and slightly more. The proof relies on some symmetries of the ring of symmetric functions discovered by Koike and Terada. Using results of Hernandez, Mukhin-Young, and Naoi, this implies that the characters of minimal affinizations in types B, C, and D are given by a Jacobi-Trudi determinant.

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature (0,n) the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function (EGF) carrying the continuum Dirac operator D=∑j=1nej∂xj together with the Lie-algebraic representation of raising and lowering operators acting on the lattice hZn is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson–Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.

Classification of irreducible representations of Lie algebra of vector fields on a torus

Abstract We solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on n-dimensional torus for any n. This generalizes the classical result of O.Mathieu on simple weight modules for the Virasoro algebra ( n = 1 ${n=1}$ ). Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao.

Quantum Torus Symmetry of the KP, KdV and BKP Hierarchies

In this paper, we construct the quantum Torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum Torus Lie algebra in the KP system by acting on its tau function. Comparing to the $W_{\infty}$ symmetry, this quantum Torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum Torus symmetries of the KdV and BKP hierarchies and further derive the quantum Torus constraints on their tau functions. These quantum Torus constraints might have applications in the quantum field theory, supersymmetric gauge theory and so on.

The quantum Rabi model and Lie algebra representations of

The aim of the present paper is to understand the spectral problem of the quantum Rabi model in terms of Lie algebra representations of . We define a second order element of the universal enveloping algebra of , which, through the image of a principal series representation of , provides a picture equivalent to the quantum Rabi model drawn by confluent Heun differential equations. By this description, in particular, we give a representation theoretic interpretation of the degenerate part of the spectrum (i.e., Juddʼs eigenstates) of the Rabi Hamiltonian due to Kuś in 1985, which is a part of the exceptional spectrum parameterized by integers. We also discuss the non-degenerate part of the exceptional spectrum of the model, in addition to the Judd eigenstates, from a viewpoint of infinite dimensional irreducible submodules (or subquotients) of the non-unitary principal series such as holomorphic discrete series representations of .