Composition of Lie Group Elements from Basis Lie Algebra Elements

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.     

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras . On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra an abstract quantum Lie algebra independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra . In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same are isomorphic, 2) the quantum Lie product of any is q-antisymmetric. We also describe a construction of which establishes their existence. Received: 23 May 1996 / Accepted: 17 October 1996     

Noncommutative symmetric functions and laplace operators for classical Lie algebras

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N).     

Reduction Groups and Automorphic Lie Algebras

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts.On leave from, L.D. Landau Institute for Theoretical Physics Chernogolovka, Russia     

UniversalR-matrix for quantized (super)algebras

For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universalR-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties ofq-exponential functions. We don't use quantum Weyl group.     

Modular Classes of Regular Twisted Poisson Structures on Lie Algebroids

We derive a formula for the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.      

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations ofZ-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras continuum Lie algebras. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.     

Fine Gradings on Non-simple Lie Algebras: Example of o(4,C)

On any Lie algebra L, it is of significant convenience to have at one's disposal all the possible fine gradings of L, since they reflect the basic structural properties of the Lie algebra. They also provide useful bases of the representations of the algebra -- namely such bases that are preserved by the commutator.We list all the six fine gradings on the non-simple Lie algebra o(4,C) and we explain their relation to the fine gradings of the Lie algebra sl(2,C) where relevant. The existence of such relation is not surprising, since o(4,C) is in fact a product of two specimen of sl(2,C). The example of o(4,C) is especially important due to the fact that one of its fine gradings is not generated by any MAD-group. This proves that, unlike in the case of classical simple Lie algebras over C, on the non-simple classical Lie algebras over C there can exist a fine grading that is not generated by any MAD-group on the Lie algebra.     

Structural parameters of semi-simple lie algebras

Based on the exploitations of properties of the Killing forms of semi-simple Lie algebras, we set out in a readily programmable form, the structural analysis and the Iwasawa-type decompositions of semi-simple Lie algebras. As an example, the case ofSO(3,1) and its covering groupSL(2,C) is worked out in some detail.     

Casimir Elements for Some Graded Lie Algebras and Superalgebras

We consider a class of Lie algebras L such that L admits a grading by a finite Abelian group so that each nontrivial homogeneous component is one-dimensional. In particular, this class contains simple Lie algebras of types A, C and D where in C and D cases the rank of L is a power of 2. We give a simple construction of a family of central elements of the universal enveloping algebra U(L). We show that for the A-type Lie algebras the elements coincide with the Gelfand invariants and thus generate the center of U(L). The construction can be extended to Lie superalgebras with the additional assumption that the group grading is compatible with the parity grading.     

A star-product approach to noncompact quantum groups

Using the duality and the topological theory of well-behaved Hopf algebras, we construct star-product models of noncompact quantum groups from Drinfeld and Reshetikhin standard deformations of enveloping Hopf algebras of simple Lie algebras. Our star-products act not only on coefficient functions of finite-dimensional representations, but actually on allC functions, and they even exist for nonlinear (semi-simple) Lie groups.     

Conformal Covariance Subalgebras

We give a direct Lie algebraic characterisation of conformal inclusions of chiral current algebras associated with compact, reductive Lie algebras. We use quantum field theoretic arguments and prove a longstanding conjecture of Schellekens and Warner on grounds of unitarity and positivity of energy. We explore the structures found to characterise conformal covariance subalgebras and coset current algebras.     

The A-D-E classification of minimal andA 1 (1) conformal invariant theories

We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant sesquilinear forms, with positive integral coefficients, in the characters of the Virasoro or of theA ₁ ⁽¹⁾ Kac-Moody algebras, which describe the corresponding partition functions on a torus. A remarkable correspondence emerges with simply laced Lie algebras.     

Deformation theory and quantization. I. Deformations of symplectic structures

We present a mathematical study of the differentiable deformations of the algebras associated with phase space. Deformations of the Lie algebra of C^∞ functions, defined by the Poisson bracket, generalize the well-known Moyal bracket. Deformations of the algebra of C^∞ functions, defined by ordinary multiplication, give rise to noncommutative, associative algebras, isomorphic to the operator algebras of quantum theory. In particular, we study deformations invariant under any Lie algebra of “distinguished observables”, thus generalizing the usual quantization scheme based on the Heisenberg algebra.     

Kerr–Schild Symmetries

We study continuous groups of generalized Kerr–Schild transformations and the vector fields that generate them in any n-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixed. The properties of these Lie algebras are briefly analyzed and we show that they are generically finite-dimensional but that they may have infinite dimension in some relevant situations. The most general vector fields of the above type are explicitly constructed for the following cases: any two-dimensional metric, the general spherically symmetric metric and deformation direction, and the flat metric with parallel or cylindrical deformation directions.     

On the (3, N) Maurer-Cartan equation

Abstract

Deformations of the 3-differential of 3-differential graded algebras are controlled by the (3, N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of N-differential graded algebras, and use these results to study N Lie algebroids.     

Quantization and representation theory of finiteW algebras

In this paper we study the finitely generated algebras underlyingW algebras. These so called finiteW algebras are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings ofsl ₂ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finiteW algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finiteW symmetry. In the second part we BRST quantize the finiteW algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finiteW algebras in one stroke. Examples are given. In the last part of the paper we study the representation theory of finiteW algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finiteW algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finiteW algebras.     

Lie Algebras and Superalgebras Defined by a Finite Number of Relations: Computer Analysis

Abstract

The presentation of Lie (super)algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. It is very important, for instance, for investigation of the particular Lie (super)algebras arising in different (super)symmetric physical models. Generally, one can put the following question: what is the most general Lie algebra or superalgebra satisfying to the given set of Lie polynomial equations? To solve this problem, one has to perform a large volume of algebraic transformations which sharply increases with growth of the number of generators and relations. By this reason, in practice, one needs to use a computer algebra tool. We describe here an algorithm and its implementation in C for constructing the bases of finitely presented Lie (super)algebras and their commutator tables.     

On the calculation of group characters

It is known that characters of irreducible representations of finite Lie algebras can be obtained using the Weyl character formula including Weyl group summations which make actual calculations almost impossible except for a few Lie algebras of lower rank. By starting from the Weyl character formula, we show that these characters can be re-expressed without referring to Weyl group summations. Some useful technical points are given in detail for the instructive example of G₂ Lie algebra.