Lie Algebra Sl Research Articles

Virasoro type Lie algebras and deformations

The first and second cohomologies of Cartan Type Lie algebras with coefficients in irreducible tensor modules are calculated. The spaceH 1(L, U) is interpreted as a space of deformations of (L, U)-modules.H 2(L, L)≠0 ifL=S 2,S 2 + orL=H n ,H + . Lie algebra of divergenceless vector fieldsS 2 + has only one nontrivial local deformation. The two-sided simple hamiltonian algebraH n has 2n 2+n new local deformations in addition to Moyal cocycle. The Lie algebrasL=W n (n>3),S n−1(n>2),H n (n>1),K n+1(n>1) have 3, 1, 1, 3 nonisomorphic tensor modules with irreducible bases and nonzero 1-cohomologies; respectively, the corresponding numbers for 2-cohomologies are 9, 6, 7 and 9.

A Lie algebra attached to a projective variety

Each choice of a Kahler class on a compact complex manifold defines an action of the Lie algebra sl(2) on its total complex cohomology. If a nonempty set of such Kahler classes is given, then we prove that the corresponding sl(2)-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori, hyperkahler manifolds and flag varieties. We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra.

New quantum deformations ofD=4 conformal algebra

We consider new class of classicalr-matrices forD=4 conformal Lie algebra. These r-matrices do satisfy the classical Yang-Baxter equation and as two-tensors belong to the tensor product of Borel subalgebras. In such a way we generalize the lowest order of known nonstandard quantum deformation ofsl(2) to the Lie algebrasl(4)≅so(6). As an exercise we interpret nonstandard deformation ofsl(2) as describing quantumD=1 conformal algebra with fundamental mass parameter. Further we describeD=4 conformal bialgebras with deformation parameters equal to the inverse of fundamental masses. It appears that forD=4 the deformation of the Poincare algebra sector coincides with “null plane” quantum Poincare algebra.

The classification of global Lie wedges in sl(2)

In [2] Hofmann and Hilgert investigate semigroups in SL(2, ℝ) and its universal covering group. Their interest is focussed on semigroups that are generated by the exponential image of a so-called Lie wedgeW in the Lie algebra. For an arbitrary subsemigroupS of a Lie groupG the set L(S) of subtangent vectors at the group identity1 is always a Lie wedge. IfS is closed, then\(\overline {\left\langle {\exp L(S)} \right\rangle } \subseteq S\) is always true. If equality holds, thenS is called infinitesimally generated. Conversely, if we start with a Lie wedgeW, then it may happen that the inclusion\(W \subseteq L\overline {\left\langle {\exp W} \right\rangle } \) is proper. Those wedges for which equality holds are called global. The classification of all global Lie wedges in a given Lie algebra is equivalent to classifying all infinitesimally generated subsemigroups of the corresponding simply connected Lie group. We give the solution to this classification problem for the Lie algebra sl(2, ℝ) . Roughly speaking, a Lie wedgeW ⊑sl(2, ℝ) is global if and only if it is either not pointed or it is contained in the intersection of two distinct halfspaces whose boundaries are subalgebras.

On the structure of an α-stratified generalized Verma module over Lie algebrasl(n, ℂ))

We study α-stratified modules of Verma type for the Lie algebrasl(n, ℂ). Necessary and sufficient conditions are established for existence of a submodule in a generalized Verma module.

Finite-dimensional simple modules over certain iterated skew polynomial rings

We classify the finite-dimensional simple R-modules for a class of algebras which arise as iterated skew polynomial rings in two variables over commutative K-algebras for an algebraically closed field k of arbitrary characteristic. The class includes the enveloping algebra and the quantized enveloping algebra of the Lie algebra sl(2, k), the quantized Weyl algebra in two variables, various quantum groups, and the enveloping algebra of the dispin Lie superalgebra.

On Parasupersymmetries and Relativistic Descriptions for Spin one Particles: I. The Free Context

This series of two papers is devoted to a constructive review of the relativistic wave equations for vector mesons due to the recent impact of spin one developments in connection with parasupersymmetric quantum mechanics. The free case as well as the interacting context with an electromagnetic field will be successively visited and discussed. Their associated parasupersymmetric properties will be pointed out. In this first part, the free context is presented by studying systematically the (symmetric) forms of wave equations subtended by a 16-dimensional reducible representation of the Lie algebra sl (2, C) or, evidently, so (3, 1), this representation playing a well known role in p = 2-parastatistical developments. Their hamiltonian forms are also discussed and some second order descriptions are finally reviewed.

The classification of the simple modular lie algebras: V. algebras with Hamiltonian two-sections

We investigate the structure of simple modular Lie algebrasL over an algebraically closed field of characteristic p > 7. Every optimal torusT in some p-envelope ofL defines uniquely a subalgebraQ(L, T) ofL. We classify all L, for whichQ(L, T) ≠ L and which have a two-section of typeH (2; 1;Ф(τ))(1)

The Lie algebra of the sl(2, ?)-valued automorphic functions on a torus

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, C) -valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2,C) -valued loop algebra, while the latter one - into the Lie algebra (sl(2)^)'/(centre) .

Ermakov systems: a group theoretic approach

Recently Leach [Phys. Lett. A 158 (1991) 102] reported that Ermakov systems have the Lie algebra sl (2, R). Leach also reported the Lie algebra of the Ermakov-Lewis invariant to be sl(2, R). This is misleading and we complete his results. We also determine the general equation invariant under this new symmetry. Finally we find all (four) first integrals for Hamiltonian Ermakov systems.

Graded contractions of representations of special linear Lie algebras with respect to their maximal parabolic subalgebras

Parabolic gradings of the classical simple Lie algebras sl(N,C) (N>or=3) are described for all maximal parabolic subalgebras. All contractions are found which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading (parabolic contractions of Lie algebras). Contractions of the irreducible representations of sl(N,C) for each parabolic contraction of the Lie algebra are the main results of the article.

Primitive ideals of the quantised enveloping algebra of a complex lie algebra

In this paper, we characterise all primitive ideals of the quantised enveloping algebra Uq[sl(2, ℂ)] of the complex Lie algebra sl(2, ℂ) and show how they are similar to those of U[sl(2, ℂ)], the enveloping algebra of sl(2, ℂ).

Conformal blocks and generalized theta functions

Let M(r) be the moduli space of rank r vector bundles with trivial determinant on a Riemann surface X . This space carries a natural line bundle, the determinant line bundle L . We describe a canonical isomorphism of the space of global sections of L^k with a space known in conformal field theory as the ``space of conformal blocks, which is defined in terms of representations of the Lie algebra sl(r, C((z))).

?-Stratified modules over the Lie algebra sl (n, ?)

Properties of the generalized Weyl group are studied for α-stratified modules over the Lie algebra sl (n, ℂ).

Lie and Noether Counting Theorems for One-Dimensional Systems

For a second-order equation E( t, q, q̇, q̈) = 0 defined on a domain in the plane, Lie geometrically proved that the maximum dimension of its point symmetry algebra is eight. He showed that the maximum is attained for the simplest equation q̈ = 0 and this was later shown to correspond to the Lie algebra sl(3, R ). We present an algebraic proof of Lie′s "counting" theorem. We also prove a conjecture of Lie′s, viz., that the full Lie algebra of point symmetries of any second-order equation is a subalgebra of sl(3, R ). Furthermore, we prove, the Noether "counting" theorem, that the maximum dimension of the Noether algebra of a particle Lagrangian is live and corresponds to A 5,40. Then we show that a particle Lagrangian cannot admit a maximal four-dimensional Noether point symmetry algebra. Consequently we show that a particle Lagrangian admits the maximal r ∈ {0, 1, 2, 3, 5}.dimensional Noether point symmetry algebra.

Adler–Kostant–Syms theorem and supersymmetric integrable systems

The Adler, Kostant, and Syms theorem has been used to construct integrable nonlinear dynamical systems involving both bosonic and fermionic variables. Explicit forms of the equations are displayed in case of two super Lie Algebra OSPU(1,1/1) and Sl(2/2). Integrability is proved by explicit construction of higher order conserved quantities.

The quantum Weyl group and the universal quantumR-matrix for affine lie algebraA 1 (1)

The universal quantumR-matrix is obtained in the case of the affine Kac-Moody Lie algebra sl(2).

Polynomial deformations of the Lie algebrasl(2) in problems of quantum optics

It is shown that specific (polynomial) deformations of Lie algebras arise naturally as dynamical symmetry algebrasgds of second-quantized models with nonquadratic HamiltoniansH invariant with respect to certain groupsGinv(H). Such deformationssld(2) of the Lie algebrasl(2) are found in a number of models of quantum optics (multiphoton processes, generalized Dicke model, and frequency conversion), and ways to apply thesl(2) formalism to the solution of physics problems are indicated.

A q-analogue of the dual space to the Lie algebras gl(2) and sl(2)

For the Lie group GL(n), the linear Poisson bracket on the dual space gl(n)* to the Lie algebra gl(n) is quadratically deformed to make the coadjoint action of GL(n) into a Poisson action, for arbitrary quadratic Poisson Lie structure on GL(n). For the case n=2, these formulae are quantized for both quantum groups GLq(2) and SLq(2).

Representations of real-valued forms of the graded analogue of a Lie algebra

Real-valued forms of the ℤ2n -graded analogue of the Lie algebra sl(2,C) are described and their irreducible representations are studied.