Representation Theory Of Lie Algebras Research Articles

Smiley Theorem for Spectral Operators Whose Radical Part Is Locally Nilpotent

Generalizing bicommutant theorem to the higher-order commutator case is very useful for representation theory of Lie algebras, which plays an important role in symmetry analysis. In this paper, we mainly prove that for any spectral operator A on a complex Hilbert space whose radical part is locally nilpotent, if a bounded operator B lies in the k-centralizer of every bounded linear operator in the l-centralizer of A, where k and l are two arbitrary positive integers satisfying l⩾k, then B must belong to the von Neumann algebra generated by A and the identity operator. This result generalizes a matrix commutator theorem proved by M. F. Smiley. To this aim, Smiley operators are defined and an example of a non-spectral Smiley operator is given by the unilateral shift, indicating that Smiley-type theorems might also hold for general spectral operators.

Higher deformations of Lie algebra representations I

In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group.

Ensemble observability of Bloch equations with unknown population density

We introduce in the paper a novel observability problem for a large population (in the limit, a continuum ensemble) of nonholonomic control systems with unknown population density. We address the problem by focussing on a prototype, namely, the ensemble of Bloch equations which is known for its use of describing the evolution of the bulk magnetization of a collective of non-interacting nuclear spins in a static field modulated by a radio frequency (rf) field. The dynamics of the equations are structurally identical, but show variations in Larmor dispersion and rf inhomogeneity. We assume that the initial state of any individual system (i.e., individual Bloch equation) is unknown and, moreover, the population density of these individual systems is also unknown. Furthermore, we assume that at any time, there is only one scalar measurement output at our disposal. The measurement output integrates a certain observation function, common to all individual systems, over the continuum ensemble. The observability problem we pose in the paper is thus the following: Whether one is able to use the common control input (i.e., the rf field) and the single measurement output to estimate both the initial states of the individual systems and the population density? Amongst other things, we establish a sufficient condition for the ensemble system to be observable: We show that if the common observation function is any harmonic homogeneous polynomial of positive degree, then the ensemble system is observable. The proof of the result utilizes tools from representation theory of Lie algebras. Although the results established in the paper are for the specific ensemble of Bloch equations, the approach developed along the analysis can be generalized to investigate observability of other ensemble systems with single, integrated measurement outputs.

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.

Derived representation theory of Lie algebras and stable homotopy categorification of slk

We set up foundations of representation theory over S, the sphere spectrum, which is the “initial ring” of stable homotopy theory. In particular, we treat S-Lie algebras and their representations, characters, gln(S)-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, we construct a Khovanov slk-stable homotopy type with a large prime hypothesis, which is a new link invariant, using a stable homotopy analogue of the method of J. Sussan.

Orthogonal Stochastic Duality Functions from Lie Algebra Representations

We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between *-representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the Heisenberg algebra and mathfrak {su}(1,1). Both cases lead to orthogonal (self-)duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes.

Self-Duality of Markov Processes and Intertwining Functions

We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of examples, including Markov processes with a discrete state space (e.g. interacting particle systems) and Markov processes with continuous state space (e.g. diffusion processes). In the examples we use explicit representations of Lie algebras that are unitary equivalent. As a consequence, in the discrete setting self-duality functions are given by orthogonal polynomials whereas in the continuous context they are Bessel functions.

Marked chain-order polytopes

We introduce in this paper the marked chain-order polytopes associated to a marked poset, generalizing the marked chain polytopes and marked order polytopes by putting them as extremal cases in an Ehrhart equivalent family. Some combinatorial properties of these polytopes are studied. This work is motivated by the framework of PBW degenerations in representation theory of Lie algebras.

Lie Algebras of Derivations and Resolvent Algebras

This paper analyzes the action {\delta} of a Lie algebra X by derivations on a C*-algebra A. This action satisfies an "almost inner" property which ensures affiliation of the generators of the derivations {\delta} with A, and is expressed in terms of corresponding pseudo-resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*-algebra A, it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo- resolvents. If the Lie action {\delta} is ergodic, i.e. the only elements of A on which all the derivations in {\delta}_x vanish are multiples of the identity, then this extension is given by a (non-degenerate) symplectic form {\sigma} on X. Moreover, the algebra generated by the pseudo-resolvents coincides with the resolvent algebra based on the symplectic space (X, {\sigma}). Thus the resolvent algebra of the canonical commutation relations, which was recently introduced in physically motivated analyses of quantum systems, appears also naturally in the representation theory of Lie algebras of derivations acting on C*-algebras.

On the hyperdeterminant for 2×2×3 arrays

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2 × 2 × 3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x ijk where i, j = 1, 2 and k = 1, 2, 3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2 × 2 × 2 arrays.

Valued Graphs and the Representation Theory of Lie Algebras

Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

In this paper we present Affine.m—a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Program summaryProgram title: Affine.mCatalogue identifier: AENA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, UKLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 24844No. of bytes in distributed program, including test data, etc.: 1045908Distribution format: tar.gzProgramming language: Mathematica.Computer: i386–i686, x86_64.Operating system: Linux, Windows, Mac OS, Solaris.RAM: 5–500 MbClassification: 4.2, 5.Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems.Solution method:We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases.Restrictions:Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical.Unusual features:We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface.Running time:From seconds to days depending on the rank of the algebra and the complexity of the representation.

Jordan-Chevalley decomposition in finite dimensional Lie algebras

Let g \mathfrak {g} be a finite dimensional Lie algebra over a field k k of characteristic zero. An element x x of g \mathfrak {g} is said to have an abstract Jordan-Chevalley decomposition if there exist unique s , n ∈ g s,n\in \mathfrak {g} such that x = s + n x=s+n , [ s , n ] = 0 [s,n]=0 and given any finite dimensional representation π : g → g l ( V ) \pi :\mathfrak {g}\to \mathfrak {gl}(V) the Jordan-Chevalley decomposition of π ( x ) \pi (x) in g l ( V ) \mathfrak {gl}(V) is π ( x ) = π ( s ) + π ( n ) \pi (x)=\pi (s)+\pi (n) . In this paper we prove that x ∈ g x\in \mathfrak {g} has an abstract Jordan-Chevalley decomposition if and only if x ∈ [ g , g ] x\in [\mathfrak {g},\mathfrak {g}] , in which case its semisimple and nilpotent parts are also in [ g , g ] [\mathfrak {g},\mathfrak {g}] and are explicitly determined. We derive two immediate consequences: (1) every element of g \mathfrak {g} has an abstract Jordan-Chevalley decomposition if and only if g \mathfrak {g} is perfect; (2) if g \mathfrak {g} is a Lie subalgebra of g l ( n , k ) \mathfrak {gl}(n,k) , then [ g , g ] [\mathfrak {g},\mathfrak {g}] contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods. Our proof uses only elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lie’s Theorem, in addition to the fundamental theorems of Ado and Levi.

The BQP-hardness of approximating the Jones polynomial

A celebrated important result due to Freedman et al (2002 Commun. Math. Phys.227 605–22) states that providing additive approximations of the Jones polynomial at the kth root of unity, for constant k=5 and k⩾7, is BQP-hard. Together with the algorithmic results of Aharonov et al (2005) and Freedman et al (2002 Commun. Math. Phys.227 587–603), this gives perhaps the most natural BQP-complete problem known today and motivates further study of the topic. In this paper, we focus on the universality proof; we extend the result of Freedman et al (2002) to ks that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values to which the AJL algorithm applies (Aharonov et al 2005), proving that for all those values, the problems are BQP-complete. As a side benefit, we derive a fairly elementary proof of the Freedman et al density result, without referring to advanced results from Lie algebra representation theory, making this important result accessible to a wider audience in the computer science research community. We make use of two general lemmas we prove, the bridge lemma and the decoupling lemma, which provide tools for establishing the density of subgroups in SU(n). Those tools seem to be of independent interest in more general contexts of proving the quantum universality. Our result also implies a completely classical statement, that the multiplicative approximations of the Jones polynomial, at exactly the same values, are #P-hard, via a recent result due to Kuperberg (2009 arXiv:0908.0512). Since the first publication of those results in their preliminary form (Aharonov and Arad 2006 arXiv:quant-ph/0605181), the methods we present here have been used in several other contexts (Aharonov and Arad 2007 arXiv:quant-ph/0702008; Peter and Stephen 2008 Quantum Inf. Comput.8 681). The present paper is an improved and extended version of the results presented by Aharonov and Arad (2006) and includes discussions of the developments since then.

There is No “Theory of Everything” Inside E8

We analyze certain subgroups of real and complex forms of the Lie group E8, and deduce that any "Theory of Everything" obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E8 lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin's classic papers and are written for mathematicians.

On Representations of sl(n, C) Compatible with a Z2-grading

This paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a representation compatible with a given grading is applied to finite-dimensional representations of sl(n,C) in relation to its Z2-gradings. For representation theory of sl(n,C) the Gel’fand-Tseitlin method turned out very efficient. We show that it is not generally true that every irreducible representation can be compatibly graded.

Superpotentials for superconformal Chern–Simons theories from representation theory

These notes provide a detailed account of the universal structure of superpotentials defining a large class of superconformal Chern–Simons theories with matter, many of which appear as the low-energy descriptions of multiple M2-brane configurations. The amount of superconformal symmetry in the Chern–Simons matter theory determines the minimum amount of global symmetry that the associated quartic superpotential must realize, which in turn restricts the matter superfield representations. Our analysis clarifies the necessary representation-theoretic data which guarantees a particular amount of superconformal symmetry. Thereby we shall recover all the examples of M2-brane effective field theories that have appeared in the recent literature. The results are based on a refinement of the unitary representation theory of Lie algebras to the case when the Lie algebra admits an ad-invariant inner product. The types of representation singled out by the superconformal symmetry turn out to be intimately associated with triple systems admitting embedding Lie (super)algebras and we obtain a number of new results about these triple systems which might be of independent interest. In particular, we prove that any metric 3-Lie algebra embeds into a real metric 3-graded Lie superalgebra in such a way that the 3-bracket is given by a nested Lie bracket.

Continuous crystal and Duistermaat–Heckman measure for Coxeter groups

We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat–Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm–Liouville equations.

Representation Theory of Finite Dimensional Algebras

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

THE EIGENFUNCTION EXPANSION METHOD IN MULTI‐FACTOR QUADRATIC TERM STRUCTURE MODELS

We propose the eigenfunction expansion method for pricing options in quadratic term structure models. The eigenvalues, eigenfunctions, and adjoint functions are calculated using elements of the representation theory of Lie algebras not only in the self‐adjoint case, but in non‐self‐adjoint case as well; the eigenfunctions and adjoint functions are expressed in terms of Hermite polynomials. We demonstrate that the method is efficient for pricing caps, floors, and swaptions, if time to maturity is 1 year or more. We also consider subordination of the same class of models, and show that in the framework of the eigenfunction expansion approach, the subordinated models are (almost) as simple as pure Gaussian models. We study the dependence of Black implied volatilities and option prices on the type of non‐Gaussian innovations.