Semisimple Lie Research Articles

Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

We have recently proposed [1] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with SU(N) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots T[2,2k+1]. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank N and the representation R.

A generalized Noetherian condition for Lie algebras

A Lie algebra (over any field and of any dimension) is Noetherian if it satisfies the maximal condition on ideals. We introduce a new and more general class of quasi-Noetherian Lie algebras that possess several of the main properties of Noetherian Lie algebras. This class is shown to be closed under quotients and extensions. We obtain conditions under which a quasi-Noetherian Lie algebra is Noetherian. Next, we consider various questions about locally nilpotent and soluble radicals of quasi-Noetherian Lie algebras. We show that there exists a semisimple quasi-Noetherian Lie algebra that is not Noetherian. Finally, we consider some analogous results for groups and prove that a quasi-Noetherian group is countably recognizable.

Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups

Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S3 = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S3 of all geodesics on (G, d) with respect to p.

Constructing affine Lie algebras

ABSTRACTThe ℤ-grading determined by a long simple root of a rank n+1 affine Lie algebra over ℂ arises from a representation of a rank n semi-simple complex Lie algebra. Analysis of the relationship between the grading and the representation yields constructions that generalize the minuscule and adjoint algorithms as well as Kac’s construction of nontwisted affine Lie algebras.

A note on para-holomorphic Riemannian–Einstein manifolds

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. First, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler–Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler–Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits a natural para-Kählerian–Norden Einstein metric.

Stabilizers of actions of lattices in products of groups

We prove that any ergodic non-atomic probability-preserving action of an irreducible lattice in a semisimple group, with at least one factor being connected and of higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer [Stabilizers for ergodic actions of higher rank semisimple groups.Ann. of Math. (2)139(3) (1994), 723–747], who found that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is of higher-rank. We also prove a generalization of a result of Bader and Shalom [Factor and normal subgroup theorems for lattices in products of groups.Invent. Math.163(2) (2006), 415–454] by showing that any probability-preserving action of a product of simple groups, with at least one having property$(T)$, which is ergodic for each simple subgroup, is either essentially free or essentially transitive. Our method involves the study of relatively contractive maps and the Howe–Moore property, rather than relying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups, which is of independent interest.

Properties of Nilpotent Orbit Complexification

We consider aspects of the relationship between nilpotent orbits in a semisim-ple real Lie algebra g and those in its complexification giƒ. In particular, we prove that two distinct real nilpotent orbits lying in the same complex orbit are incomparable in the closure order. Secondly, we characterize those g having non-empty intersections with all nilpotent orbits in giƒ. Finally, for g quasi-split, we characterize those complex nilpotent orbits containing real ones.

Multidimensional gravity, flux and black brane solutions governed by polynomials

Two families of composite black brane solutions are overviewed, fluxbrane and black brane ones, in a model with scalar fields and fields of forms. The metric of any solution is defined on a manifold which contains a product of several Ricci-flat “internal” spaces. The solutions are governed by moduli functions $\mathcal{H}_s $ (for fluxbranes) and H s (for black branes), obeying nonlinear differential equations with certain boundary conditions. Themaster equations for $\mathcal{H}_s $ and H s are equivalent to Toda-like equations and depend on a nondegenerate matrix A related to brane intersection rules. The functions H s and $\mathcal{H}_s $ , as was conjectured and confirmed (at least partly) earlier, should be polynomials in proper variables if A is a Cartan matrix of some semisimple finite-dimensional Lie algebra. The fluxbrane polynomials $\mathcal{H}_s $ were shown to be used for the construction of black brane polynomials H s . This approach is illustrated by examples of nonextremal electric black p-brane solutions related to Lie algebras A 2, C 2, and G 2.

Control Affine Systems on Semisimple Three-Dimensional Lie Groups

Abstract We classify the full-rank left-invariant control affine systems evolving on (real) semisimple three-dimensional Lie groups. This is accomplished by reducing the problem to that of classifying the affine subspaces of the Lie algebras so (2; 1) and so (3).

Morse linear functionals on orbits of adjoint action of semisimple Lie groups

Linear functionals on the Lie algebra of an arbitrary semisimple compact Lie group with restrictions of these functionals onto an arbitrary orbit of the adjoint action are considered. Criteria for the criticality and non-degenerate criticality of a point on the orbit are formulated and proved for a given functional, a necessary and sufficient condition for a linear functional to be a Morse function on the orbit is also proved. A method calculating the indices of critical points and its applications in the study of topological properties of orbits are indicated.

Nearly Kähler and Hermitian f-structures on homogeneous Φ-spaces of order k with special metrics

We consider arbitrary homogeneous Φ-spaces of order k ≥ 3 of semisimple compact Lie groups G in the case of a series of special metrics. We give formulas for the Nomizu function of the Levi-Civita connection of these metrics. Using these formulas and other relations for Φ-spaces of order k, we prove necessary and sufficient conditions for the canonical f-structures on these spaces to lie in some generalized Hermitian geometry classes of f-structures: nearly Kahler (NKf-structures) and Hermitian (Hf-structures).

Embedding types and canonical affine maps between Bruhat-Tits buildings of classical groups

P. Broussous and S. Stevens studied maps between enlarged Bruhat-Tits buildings to construct types for p-adic unitary groups. They needed maps which respect the Moy-Prasad filtrations. That property is called (CLF), i.e. compatibility with the Lie algebra filtrations. In the first part of this thesis we generalise their results on such maps to the Quaternion-algebra case. Let k0 be a p-adic field of residue characteristic not two. We consider a semisimple k0-rational Lie algebra element beta of a unitary group G:=U(h) defined over k0 with a signed hermitian form h. Let H be the centraliser of beta in G. We prove the existence of an affine H(k0)-equivariant CLF-map j from the enlarged Bruhat-Tits building B^1(H,k0) to B^1(G,k0). As conjectured by Broussous the CLF-property determines j, if none of the factors of H is k0-isomorphic to the isotropic orthogonal group of k0-rank one and all factors are unitary groups. Under the weaker assumption that the affine CLF-map j is only equivariant under the center of H^0(k0) it is uniquely determined up to a translation of B^1(H,k0). The second part is devoted to the decoding of embedding types by the geometry of a CLF-map. Embedding types have been studied by Broussous and M. Grabitz. We consider a division algebra D of finite index with a p-adic center F. The construction of simple types for GLn(D) in the Budhnell-Kutzko framework required an investigation of strata which had to fulfil a rigidity property. Giving a stratum especially means to fix a pair (E,a) consisting of a field extension E|F in Mn(D) and a hereditary order a which is stable under conjugation by E^x, in other words we fix an embedding of E^x into the normalizer of a. Broussous and Grabitz classified these pairs with invariants. We describe and prove a way to decode these invariants using the geometry of a CLF-map.

The work of Vladimir Morozov on Lie algebras

We survey the works of V. V. Morozov on Lie algebras concentrating on the following three results on subalgebras of a semisimple complex Lie algebra: the theorem on a nilpotent element, the triangulizability theorem for solvable subalgebras, and the regularity theorem for nonsemisimple maximal subalgebras.

Homogeneous spaces with invariant flat Cartan structures.

Let L/L' be a homogeneous space associated with a semi-simple graded Lie algebra [ = l-1 ⊕ l 0 ⊕ l 1 . On a homogeneous space G/H, with H connected, the existence of a G-invariant flat Cartan structure of graded type L/L' is equivalent to the existence of a homomorphism f: g → l of the Lie algebra g of G into l satisfying natural conditions.

ON QUASI-TORAL RESTRICTED LIE ALGEBRAS

This paper gives some sufficient conditions for the commutativity of quasi-toral restricted Lie algebras and characterizes some properties on semisimple quasi-toral restricted Lie algebras.

New Examples of Almost Homogeneous Kahler-Einstein Manifolds

We deal with compact Kahler manifolds M which are acted on by a semisimple compact Lie group G of isometries with codimension one regular orbits. We provide an explicit description of the standard blow-ups of such manifolds along complex singular orbits, in case b 1 (M) = 0 and the regular orbits are Levi nondegenerate. Up to very few exceptions, all the nonhomogeneous manifolds in this class are shown to admit a G-invariant Kahler-Einstein metric, giving completely new examples of compact Kahler-Einstein manifolds.

On the Quantum KPRV Determinants for Semisimple and Affine Lie Algebras

Let g be a semisimple or affine Lie algebra and U q (g) its quantized enveloping algebra. Extending earlier work, the KPRV determinant for an admissible integrable U q (g) module V relative to a parabolic subalgebra p⊂g is defined and shown to be nonzero. These determinants had previously been evaluated for g semisimple and p a Borel subalgebra. The present results can be used to extend this to g affine as will be shown in a subsequent publication. For a parabolic subalgebra the evaluation of these determinants is much more difficult. For appropriate overalgebras of the primitive quotients of the enveloping algebra U(g) defined by one-dimensional representations of p, these determinants had been calculated for g semisimple. However the quantum case is interesting because it is unnecessary to pass to overalgebras and besides for U(g):g affine, it is not even clear how these determinants should be defined. Here for g semisimple, the degrees of the determinants are computed and shown to depend on being the same type of functions as in the enveloping algebra case; yet in a different fashion. Some special cases (in type A 4) are computed explicity. Here, as in the Borel case, the determinants take a remarkably simple form and notably can be expressed as a product of ‘linear’ factors. However compared to the enveloping algebra case one finds additional factors corresponding to what are called quantum zeros and whose origin remains unknown.

Theory of Finite Pseudoalgebras

Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional'' analogues of conformal algebras. They are defined as Lie algebras in a certain ``pseudotensor'' category instead of the category of vector spaces. A pseudotensor category (as introduced by Lambek, and by Beilinson and Drinfeld) is a category equipped with ``polylinear maps'' and a way to compose them. This allows for the definition of Lie algebras, representations, cohomology, etc. An instance of such a category can be constructed starting from any cocommutative (or more generally, quasitriangular) Hopf algebra $H$. The Lie algebras in this category are called Lie $H$-pseudoalgebras. The main result of this paper is the classification of all simple and all semisimple Lie $H$-pseudoalgebras which are finitely generated as $H$-modules. We also start developing the representation theory of Lie pseudoalgebras; in particular, we prove analogues of the Lie, Engel, and Cartan-Jacobson Theorems. We show that the cohomology theory of Lie pseudoalgebras describes extensions and deformations and is closely related to Gelfand-Fuchs cohomology. Lie pseudoalgebras are closely related to solutions of the classical Yang-Baxter equation, to differential Lie algebras (introduced by Ritt), and to Hamiltonian formalism in the theory of nonlinear evolution equations. As an application of our results, we derive a classification of simple and semisimple linear Poisson brackets in any finite number of indeterminates.

Invariant Lie Algebras and Lie Algebras with a Small Centroid

A subalgebra of a Lie algebra is said to be invariant if it is invariant under the action of some Cartan subalgebra of that algebra. A known theorem of Melville says that a nilpotent invariant subalgebra of a finite-dimensional semisimple complex Lie algebra has a small centroid. The notion of a Lie algebra with small centroid extends to a class of all finite-dimensional algebras. For finite-dimensional algebras of zero characteristic with semisimple derivations in a sufficiently broad class, their centroid is proved small. As a consequence, it turns out that every invariant subalgebra of a finite-dimensional reductive Lie algebra over an arbitrary definition field of zero characteristic has a small centroid.

ON HOLOMORPHIC EFFECTIVE ACTIONS OF HYPERMULTIPLETS COUPLED TO EXTERNAL GAUGE SUPERFIELDS

We study the structure of holomorphic effective action generated by hypermultiplet models interacting with background super-Yang–Mills fields. A general form of holomorphic effective action depending on background fields is found for hypermultiplet belonging to arbitrary representation of any semisimple compact Lie group spontaneously broken to its maximal Abelian subgroup. Resulting effective action is defined by the superfield strengths lying in Cartan subalgebra of the gauge algebra. The applications of the obtained results to hypermultiplets in fundamental and adjoint representations of the SU (n), SO (n), Sp (n) groups are considered.