Surfaces on Lie groups,on Lie algebras,and their integrability

We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the twisted Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component.     

Lie algebras and representations of continuous groups

It is shown that every finitely generated continuous group has a subgroup generated by its infinitesimal transformations. This subgroup has a group algebra which is the Lie algebra of the group. By obtaining complete systems in the Lie algebra and complete rectangular arrays, it is shown that these can yield matrix representations of the continuous group. Illustrative examples are given for the rotation groups and for the full linear groups. It would seem that all the finite motion representations can be obtained by these methods, including spin representations of rotation groups. But the completeness of the method is not here demonstrated.     

Lie algebras of nonstandard relativity groups

It is shown that representations of Lie algebras of the possible nonstandard spacetime symmetry groups may be derived from the representations of the Poincaré group. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Lie algebras associated with topological nilpotent groups

A Lie algebra structure is defined on the set of all continuous one-parameter groups of nilpotent topological groups. Extensions are given to some inductive and projective limits.     

Representations of Lie algebras from representations of quantum groups

In paper [*] (P. Moylan: Czech. J. Phys., Vol. 47 (1997), p. 1251) we gave an explicit embedding of the three dimensional Euclidean algebra (2) into a quantum structure associated with U _q(so(2, 1)). We used this embedding to construct skew symmetric representations of (2) out of skew symmetric representations of U _q(so(2, 1)). Here we consider generalizations of the results in [*] to a more complicated quantum group, which is of importance to physics. We consider U _q(so(3, 2)), and we show that, for a particular representation, namely the Rac representation, many of the results in [*] carry over to this case. In particular, we construct representations of so(3, 2), P(2, 2), the Poincaré algebra in 2+2 dimensions, and the Poincaré algebra out of the Rac representation of U _q(so(3, 2)). These results may be of interest to those working on exploiting representations of U _q(so(3, 2)), like the Rac, as an example of kinematical confinement for particle constituents such as the quarks.     

Analytic vectors and irreducible representations of nilpotent Lie groups and algebras

Let U be a unitary irreducible locally faithful representation of a nilpotent Lie group G, U the universal enveloping algebra of G, M a simple module on U with kernel Ker dU, then there exists an automorphism of U keeping ker dU invariant such that, after transport of structure, M is isomorphic to a submodule of the space of analytic vectors for U.     

Generalized enveloping algebras and quantum kinematic coherent states of noncompact Lie groups

A new concept of generalized enveloping algebra is introduced by means of the generalized Heisenberg commutation relations of non-Abelian quantum kinematics. This concept is examined within the quantum-kinematic formalism of some noncompact Lie groups of a special kind. The well known Gel'fand theorem (which relates the center of the traditional enveloping algebra with the adjoint representation) is then extended to the generalized enveloping algebra of the group. In this way, the isomorphism of the generalized left-center and the traditional right-center of the corresponding enveloping algebras is proved within the left regular representation of noncompact Lie groups of the chosen kind. As an interesting application of generalized enveloping algebras, this paper contains a brief discussion of quantum-kinematic (boson) ladder operators for non-Abelian noncompact finite Lie groups and of their corresponding coherent states.     

Bicovariant quantum algebras and quantum Lie algebras

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun toU _q g, given by elements of the pure braid group. These operators—the reflection matrixYL ⁺ SL ^– being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO _q (N).This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139     

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.     

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namelyA ₂,B ₂, andG ₂. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.     

Lie point symmetry algebras and finite transformation groups of the general Broer--Kaup system

Using a new symmetry group theory, the transformation groups and symmetries of the general Broer--Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

Lie algebras and quasifield operators

It is shown that realisations of any Lie algebra by means of bilinear polynomials of quasifield operators exist. These realisations are used to find some class of representations of the algebra.     

Nilpotent locally convex Lie algebras and Lie field structures

The purpose of this work is to join Lie field structures with certain infinite-dimensional Lie algebras with locally convex topology. These topological Lie algebras allow topological groups which are a generalization of the connected nilpotent Lie groups. We showed the existence of the continuous unitary representations of the gained groups and then we proved the analogue of Gårding theorem. Using this theorem we established the existence of representations of Lie field structures into Lie algebras of skew-symmetric operators on Hilbert spaces.Work supported by National Science Foundation.On leave of absence from the Institute Rudjer Bokovi, Zagreb.     

Casimir invariants for quantized affine Lie algebras

Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest-weight representation.     

Affine Lie algebras, string junctions and 7-branes

We consider the realization of affine ADE Lie algebras as string junctions on mutually non-local 7-branes in Type 1113 string theory. The existence of the affine algebra is signaled by the presence of the imaginary root junction δ, which is realized as a string encircling the 7-brane configuration. The level k of an affine representation partially constrains the asymptotic (p, q) charges of string junctions departing the configuration. The junction intersection form reproduces the full affine inner product, plus terms in the asymptotic charges.     

Rational forms for twistings of enveloping algebras of simple Lie algebras

We give rational forms for twistings of classical enveloping algebras. We also remark a link with the generalized formalism of Gurevich, Manin, and Cartier.