Existence,uniqueness and cohomology of the classical BRST charge with ghosts of ghosts

A complete canonical formulation of the BRST theory of systems with redundant gauge symmetries is presented. These systems includep-form gauge fields, the superparticle, and the superstring. We first define the Koszul-Tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic. The global existence of the BRST generator is then demonstrated, and the BRST charge is proved to be unique up to canonical transformations in the extended phase space, which includes the ghosts. Finally, the BRST cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits, as in the irreducible case. This is done by re-expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree, which are identified with the ghosts of ghosts. The quantum cohomology is not dealt with.Aspirant du Fonds National de la Recherche Scientifique (Belgium)Chercheur qualifié au Fonds National de la Recherche Scientifique (Belgium)     

Central extensions of quantum current groups

We describe Hopf algebras which are central extensions of quantum current groups. For a special value of the central charge, we describe Casimir elements in these algebras. New types of generators for quantum current algebra and its central extension for quantum simple Lie groups, are obtained. The application of our construction to the elliptic case is also discussed.     

BRST Formalism in Self-Dual Chern-Simons Theory with Matter Fields

We apply BRST method to the self-dual Chern-Simons gauge theory with matter fields and the generators of symmetries of the system from an elegant Lie algebra structure under the operation of Poisson bracket. We discuss four different cases: abelian, nonabelian, relativistic, and nonrelativistic situations and extend the system to the whole phase space including ghost fields. In addition, we obtain the BRST charge of the field system and check its nilpotence of the BRST transformation which plays an important role such as in topological quantum field theory and string theory.     

BRST cohomology for certain reducible topological symmetries

In this paper, two different definitions of the BRST complex are connected. We obtain the BRST complex of topological quantum field theories (leading to equivariant cohomology) from the standard definition of the classical BRST complex (leading to Lie algebra cohomology) provided that we include ghosts for ghosts. Hereby, we use a finite dimensional model with a semi-direct product action ofH DiffM on a configuration spaceM, whereH is a compact Lie group representing the gauge symmetry in this model.     

Standard complex for quantum Lie algebras

For a quantum Lie algebra Γ, let Γ^ be its exterior extension (the algebra Γ^ is canonically defined). We introduce a differential on the exterior extension algebra Γ^ which provides the structure of a complex on Γ^. In the situation when Γ is a usual Lie algebra, this complex coincides with the “standard complex.” The differential is realized as a commutator with a (BRST) operator Q in a larger algebra Γ^[Ω], with extra generators canonically conjugated to the exterior generators of Γ^. A recurrent relation which uniquely defines the operator Q is given.     

The algebraic structure of BRST quantization

It is shown that a system of first-class bosonic constraints obeying a Lie algebra has associated with it a natural superalgebra. BRST quantization arises as a non-linear representation of this superalgebra. Two distinct superalgebras are explicitly constructed and their associated BRST quantizations presented. The first BRST quantization is the canonical one with the BRST charge a grassmannian scalar. The second is new — the BRST charge is a grassmannian spinor transforming in the fundamental representation of the appropriate superalgebra. Generalizations are briefly discussed.     

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.     

On Centralizer Algebras for Spin Representations

We give a presentation of the centralizer algebras for tensor products of spinor representations of quantum groups via generators and relations. In the even-dimensional case, this can be described in terms of non-standard q-deformations of orthogonal Lie algebras; in the odd-dimensional case only a certain subalgebra will appear. In the classical case q?= 1 the relations boil down to Lie algebra relations.     

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     

Global Geometric Deformations of Current Algebras as Krichever-Novikov Type Algebras

We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. The algebras are explicitly given by generators and structure equations and yield new examples of infinite dimensional algebras of current and affine Lie algebra type.     

Lie algebraic approach to the time-dependent quantum general harmonic oscillator and the bi-dimensional charged particle in time-dependent electromagnetic fields

We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express a quantum Hamiltonian and therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the Hamiltonian of a charged particle in electromagnetic fields exploiting the similarities between the terms of these two Hamiltonians. These results are applied to the solution of five different examples: the linear potential which is used to introduce the Lie algebraic method, a radio frequency ion trap, a Kanai–Caldirola-like forced harmonic oscillator, a charged particle in a time dependent magnetic field, and a charged particle in constant magnetic field and oscillating electric field. In particular we present exact analytical expressions that are fitting for the study of a rotating quadrupole field ion trap and magneto-transport in two-dimensional semiconductor heterostructures illuminated by microwave radiation. In these examples we show that this powerful method is suitable to treat quadratic Hamiltonians with time dependent coefficients quite efficiently yielding closed analytical expressions for the propagator and the Heisenberg picture position and momentum operators.     

Duality on the quantum space(3) with two parameters

In this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.     

Differential geometry of the quantum supergroupGL q (1/1)

We construct a right-invariant differential calculus on the quantum supergroupGL _q (1/1) and we show that the quantum Lie algebra generators satisfy the undeformed Lie superalgebra. The deformation becomes apparent when one studies the comultiplication for these generators. We bring the algebra into the standard Drinfeld-Jimbo form by performing a suitable change of variables, and we check the consistency of the map with the induced comultiplication.     

The Cohomology Algebra of the Semi-Infinite Weil Complex

In 1993, Lian-Zuckerman constructed two cohomology operations on the BRST complex of a conformal vertex algebra with central charge 26. They gave explicit generators and relations for the cohomology algebra equipped with these operations in the case of the c = 1 model. In this paper, we describe another such example, namely, the semi-infinite Weil complex of the Virasoro algebra. The semi-infinite Weil complex of a tame -graded Lie algebra was defined in 1991 by Feigin-Frenkel, and they computed the linear structure of its cohomology in the case of the Virasoro algebra. We build on this result by giving an explicit generator for each non-zero cohomology class, and describing all algebraic relations in the sense of Lian-Zuckerman, among these generators.     

A remark on quasi-triangular Lie algebras

Quasi-triangular quantum Lie algebras are algebras of quantum Lie derivatives. We show how to associate a quasi-triangular quantum Lie algebra to any quantum group defined by exchange relations. This provides a systematical way for constructing bi-covariant differential calculus on exchange quantum groups.     

Higher-Order Simple Lie Algebras

It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order “structure constants”) which characterise the Lie algebra cocycles. This analysis allows us to present a classification of the higher-order simple Lie algebras as well as a constructive procedure for them. Our results are synthesised by the introduction of a single, complete BRST operator associated with each simple algebra. Received: 3 June 1996 / Accepted: 8 November 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).     

Contractions, Deformations and Curvature

The role of curvature in relation with Lie algebra contractions of the pseudo-orthogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley–Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such “quantum” spaces.     

Superalgebras,symplectic bosons and the Sugawara construction

New representations of affine Lie algebras are constructed using symplectic bosons of the sort that occur naturally in the BRST treatment of fermionic string theories. These representations are shown to have analogous properties to the current algebra representations in terms of free fermion fields, though they do not act in a positive space. In particular, the condition for the Sugawara construction of the Virasoro algebra to equal the free one is the existence of a superalgebra with a quadratic Casimir operator, paralleling the symmetric space theorem for fermionic field constructions. Both results are seen to be particular cases of a more general super-symmetric space theorem, which arises from considering an affinisation of the superalgebras. These algebras are realised in terms of free fermions and symplectic bosons and lead to a super-Sugawara construction of the Virasoro algebra. The conditions for this to equal a Virasoro algebra obtained from the free fields are provided by the super-symmetric space theorem.     

Reducible gauge algebra of BRST-invariant constraints

We show that it is possible to formulate the most general first-class gauge algebra of the operator formalism by only using BRST-invariant constraints. In particular, we extend a previous construction for irreducible gauge algebras to the reducible case. The gauge algebra induces two nilpotent, Grassmann-odd, mutually anti-commuting BRST operators that bear structural similarities with BRST/anti-BRST theories but with shifted ghost number assignments. In both cases we show how the extended BRST algebra can be encoded into an operator master equation. A unitarizing Hamiltonian that respects the two BRST symmetries is constructed with the help of a gauge-fixing boson. Abelian reducible theories are shown explicitly in full detail, while non-Abelian theories are worked out for the lowest reducibility stages and ghost momentum ranks.