Representations Of Semisimple Lie Algebras Research Articles

Dimensions of modular irreducible representations of semisimple Lie algebras

In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished p p -characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category O \mathcal {O} . For this, we prove an equivalence between two categorifications of a parabolically induced module over the affine Hecke algebra conjectured by the first named author. For the general nilpotent p p -character, we get character formulas by explicitly computing the duality operator on a suitable equivariant K-group.

Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians

Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighboring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians.

Classifications of $\Gamma$-Colored $d$-Complete Posets and Upper $P$-Minuscule Borel Representations

The $\Gamma$-colored $d$-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify $\Gamma$-colored $d$-complete posets which specifies the structure of the associated representations. We show that finite $\Gamma$-colored $d$-complete posets are precisely the dominant minuscule heaps of J.R. Stembridge. These heaps are reformulations and extensions of the colored $d$-complete posets of R.A. Proctor. We also show that connected infinite $\Gamma$-colored $d$-complete posets are precisely order filters of the connected full heaps of R.M. Green.

Radon Transforms of Twisted D-Modules on Partial Flag Varieties

In this paper we study intertwining functors (Radon transforms) for twisted D-modules on partial flag varieties and their relation to the representations of semisimple Lie algebras. We show that certain intertwining functors give equivalences of derived categories of twisted D-modules. This is a generalization of a result by Marastoni. We also show that these intertwining functors from dominant to antidominant direction are compatible with taking global sections.

A Cohomological Proof that Real Representations of Semisimple Lie Algebras Have Q-Forms

A Lie algebra $\mathfrak{g}_\mathbb{Q}$ over $\mathbb{Q}$ is said to be $\mathbb{R}$-universal if every homomorphism from $\mathfrak{g}_\mathbb{Q}$ to $\mathfrak{gl}(n,\mathbb{R})$ is conjugate to a homomorphism into $\mathfrak{gl}(n,\mathbb{Q})$ (for every $n$). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an $\mathbb{R}$-universal $\mathbb{Q}$-form. We also provide a classification of the $\mathbb{R}$-universal Lie algebras that are semisimple.

Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution

We prove most of Lusztig’s conjectures on the canonical basis in homology of a Springer ber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic. We check this for almost all characteristics. To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the Langlands dual group. This equivalence established by Arkhipov and the rst author ts the framework of local geometric Langlands duality. The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality.

Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis

This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of $\mathfrak{ sl}$$_n$, the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a new interpretation and a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity (all the known proofs use more sophisticated algebraic tools). A constructive approach to the Gelfand-Tsetlin formulas is then given, based on a simple algorithm for solving certain equations on the lattice of semistandard Young tableaux. This algorithm also implies certain extremal properties of the Gelfand-Tsetlin basis.

Book Review: Representations of semisimple Lie algebras in the BGG category $\mathcal O$

Book Review: Representations of semisimple Lie algebras in the BGG category $\mathcal O$

Exploring Representation Theory of Unitary Groups via Linear Optical Passive Devices

In this paper, we investigate some mathematical structures underlying the physics of linear optical passive (LOP) devices. We show, in particular, that with the class of LOP transformations on N optical modes one can associate a unitary representation of U (N) in the N-mode Fock space, representation which can be decomposed into irreducible sub-representations living in the subspaces characterized by a fixed number of photons. These (sub-)representations can be classified using the theory of representations of semi-simple Lie algebras. The remarkable case where N = 3 is studied in detail.

On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior

This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53–64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119–171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173–177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91–117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have Q -forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17–22, 2001, TIFR, Mumbai, 2001, pp. 469–490] of D. Morris (Witte) on a completely different topic. We will answer that question too.

Quantum channels and representation theory

In the study of d-dimensional quantum channels (d⩾2), an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su(n) is a depolarizing channel for all n, but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su(2), we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g2 and the Clifford algebra.

Decomposition of symmetric powers of irreducible representations of semisimple Lie algebras and the Brion polytope

Decomposition of symmetric powers of irreducible representations of semisimple Lie algebras and the Brion polytope

Extremal Properties of Bases for Representations of Semisimple Lie Algebras

Let \cal L be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of \cal L with weight basis \cal B. The supporting graph P of \cal B is defined to be the directed graph whose vertices are the elements of \cal B and whose colored edges describe the supports of the actions of the Chevalley generators on V. Four properties of weight bases are introduced in this setting, and several families of representations are shown to have weight bases which have or are conjectured to have each of the four properties. The basis \cal B can be determined to be edge-minimizing (respectively, edge-minimal) by comparing P to the supporting graphs of other weight bases of V. The basis \cal B is solitary if it is the only basis (up to scalar changes) which has P as its supporting graph. The basis \cal B is a modular lattice basis if P is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of sl(n, \Bbb C) serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of sp(2n, \Bbb C) and the irreducible “one-dimensional weight space” representations of any semisimple Lie algebra, are shown to be solitary and edge-minimal and to have modular lattice supports. Tools developed here are used to construct uniformly the irreducible one-dimensional weight space representations. Similar results for certain irreducible representations of the odd orthogonal Lie algebra o(2n + 1, \Bbb C), the exceptional Lie algebra G2, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.

Group Theory in Physics: An Introduction

The Basic Framework. The Structure of Groups. Lie Groups. Representation of Groups--Principal Ideas. Representation of Groups--Developments. Group Theory in Quantum Mechanical Calculations. Crystallographic Space Groups. The Role of Lie Algebras. The Relationships Between Lie Groups and Lie Algebras Explored. The Three-Dimensional Rotation Groups. The Structure of Semi-Simple Lie Algebras. Representations of Semi-Simple Lie Algebras. Symmetry Schemes for the Elementary Particles. Appendices. References. Subject Index.

Tensor products and fusion rules

Methods of decomposing tensor products of integrable representations of semisimple Lie algebras are described. They include a formula due to Zelobenko, the descendant of a conjecture made by Parthasarathy et al., and identities found by Feingold. The methods are adapted to the calculation of fusion rules in Wess–Zumino–Novikov–Witten models.

Kronecker products, minuscule representations, and polynomial identities

The special role of the highest long and highest short roots in the derivation of extremum properties of the weights of irreducible representations of semisimple Lie algebras is pointed out. These properties are used to give an intrinsic and unifying reformulation, as well as a new proof of Klimyk’s theorem on Kronecker products [Ukrain. Mat. Z. 18, 19 (1966)]. This new form of Klimyk’s theorem reveals the special position of the minuscule representations in Kronecker products; as an immediate consequence, the explicit formula for the Kronecker product of an arbitrary representation with a minuscule representation is obtained. Explicit expressions for the weights of the minuscule representations are given. New derivations of theorems of Dynkin [Trudy Mosk. Obsch. 1, 39 (1952)] and Feingold [Proc. Am. Math. Soc. 70, 109 (1978)] on Kronecker products, as well as the necessary condition for a Kronecker product to decompose in two irreducible components are obtained. The proofs are based on a theorem of Parthasarathy, Ranga Rao, and Varadarajan [Ann. Math. 85, 383 (1967)].

A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA 1 (1) .

New Weyl groups forA 1 (1) and characters of singular highest weight modules

We consider singular Verma modules overA 1 (1) , i.e., Verma modules for which the central charge is equal to minus the dual Coxeter number. We calculate the characters of certain factor modules of these Verma modules. In one class of cases we are able to prove that these factor modules are actually the irreducible highest modules for those highest weights. We introduce new Weyl groups which are infinitely generated abelian groups and are proper subgroups or isomorphic between themselves. Using these Weyl groups we can rewrite the character formulae obtained in the paper in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semisimple Lie algebras (respectively Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras) so that the new Weyl groups play the role of the usual Weyl group (respectively affine Weyl group).

Symmetry adaptation of wavefunctions and matrix elements

The authors describe a method for the symmetrisation of basis states of representations of semisimple Lie algebras according to semisimple symmetry chains. This method also yields the matrix elements for the generators of all algebras of the symmetry chain, and thus also for the elements of their enveloping algebras. The symmetry adaptation coefficients, and the matrix elements, are obtained in the explicit form p/q, p, q in z. The present paper describes the method for the case of su(l+1), 1 being the rank, as the leading algebra of the chain, and for completely symmetric representations (N) of su(l+1). The semisimple symmetry chain with leading algebra su(l+1) is arbitrary, and can be specified according to individual requirements. The symmetry adaptation of states and the calculation of matrix elements, as outlined in this paper, has been implemented for computer evaluation. As a consequence of the method of evaluation all results are obtained in an exact manner.

One construction of irreducible representations of semisimple Lie algebras

One construction of irreducible representations of semisimple Lie algebras