Corrected automatic continuity conditions for finite-dimensional representations of connected Lie groups

We continue the study of automatic continuity conditions for finite-dimensional representations of connected Lie groups. In particular, we claim that every locally bounded finite-dimensional representation of a connected Lie group is continuous on the commutator subgroup in the intrinsic Lie topology of the subgroup and continuous on the intersection of the commutator subgroup with the radical of the group in the original topology of the Lie group, thus correcting one of our previous results.     

Matrix elements of tensor operators

The Wigner-Eckart theorem for matrix elements of tensor operators transforming according to a finite-dimensional representation of a non-compact semisimple Lie group is proved. The applications of this theorem are discussed.     

Continuity conditions for finite-dimensional representations of connected locally compact groups

We obtain simple necessary and sufficient conditions for the continuity of a locally bounded finite-dimensional representation of a connected locally compact group. We also prove that the discontinuity group of a locally bounded finite-dimensional representation of a connected locally compact group is a central subgroup in the closure of the image.     

The Poincare-Dulac theorem for nonlinear representations of nilpotent lie algebras

The aim of this Letter is to show that the Poincare-Dulac theorem for holomorphic finite-dimensional representation, is valid for any nilpotent Lie algebrag. We reduce the classification problem of representations with a semisimple linear part satisfying the Poincaré condition to an algebraic problem. We develop a complete computation in a particular case.     

Fourier-Stieltjes localization in neighborhoods of finite-dimensional irreducible representations of locally compact groups

An analog of the localization principle for the almost convergence for any element of the Fourier-Stieltjes algebra (Fourier-Stieltjes transforms of bounded measures) is established for any unimodular amenable locally compact groups in neighborhoods of finite-dimensional representations of these groups. To the blessed memory of Yuri Solov’ev Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by the INTAS grant.     

Connected Lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations

In the paper, it is proved that a connected Lie group admits a (possibly discontinuous) faithful finite-dimensional representation if and only if it admits a continuous faithful finite-dimensional representation, i.e., if and only if it is a linear Lie group.     

Continuity Conditions for Finite-Dimensional Locally Bounded Representations of Connected Locally Compact Groups

A criterion for the continuity of the restriction of a finite-dimensional locally bounded representation of a connected locally compact group to the commutator subgroup of the group is given.     

Continuity conditions for finite-dimensional representations of some locally bounded groups

Necessary and sufficient conditions are obtained for the continuity of finite-dimensional representations of “almost divisible” locally pseudocompact groups in terms of the oscillations of these representations at a point. Applications to the continuity problem for finite-dimensional representations of almost connected locally compact groups and to the theory of quasirepresentations are given. To the blessed memory of Eckehart Hotzel Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by an INTAS grant.     

The Muhly–Renault–Williams Theorem for Lie Groupoids and its Classical Counterpart

A theorem of Muhly–Renault–Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C*-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply connected, then their associated Poisson manifolds (viz. the dual bundles to their Lie algebroids) are Morita equivalent in the sense of P. Xu.     

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z₊-graded vertex algebra is construsted on the vacuum representation V_k(\hat{g}[ θ]) of \hat{g}[θ], which is a one-dimentional central extension of θ-invariant subspace on the loop algebra Lg=g\otimes C((t^1/p)).     

On sufficient and necessary conditions for linearity of the transverse Poisson structure

We study the possibility of bringing the transverse Poisson structure to a coadjoint orbit (on the dual of a real Lie algebra) to a normal linear form. We study the relation between two sufficient conditions for linearity of such structures (P. Molino’s condition and our own). We then use these conditions to conclude that, if the isotropy subgroup of the (singular) point in question is compact, or if the isotropy subalgebra is semisimple, then there is a linear transverse Poisson structure to the corresponding coadjoint orbit.     

An invitation to higher gauge theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity.     

A New Recursion Relation of Fractional Parent age Coefficients in L-S Coupling Case

A system of identical fermions with single orbital angular momentum l and spin 1/2 is under consideration. A new recursion relation with explicit seniority of fractional parentage coefficients in L-S coupling case is derived by using the generalized Wigner-EcJcart theorem of semisimple compact Lie groups.     

Poisson Geometry of Discrete Series Orbits,and Momentum Convexity for Noncompact Group Actions

The main result of this paper is a convexity theorem for momentum mappings of certain Hamiltonian actions of noncompact semisimple Lie groups. The image is required to fall within a certain open subset D of the (dual of the) Lie algebra, and the momentum map itself is required to be proper as a map to D. The set D corresponds roughly, via the orbit method, to the discrete series of representations of the group, Much of the paper is devoted to the study of D itself, which consists of the Lie algebra elements which have compact centralizer. When the group is Sp(2n), these elements are the ones which are called 'strongly stable' in the theory of linear Hamiltonian dynamical systems, and our results may be seen as a generalization of some of that theory to arbitrary semisimple Lie groups. As an application, we prove a new convexity theorem for the frequency sets of sums of positive definite Hamiltonians with prescribed frequencies.     

Symmetries of Fractional Parentage Coefficients with Isospin

A system of identical fermions with single angular momentum j and isospin 1/2 is under consideration. The classification of its wave functions, the factorization and the symmetry of fractional parentage coefficients are discussed by using the generalized Wigner-Eckart theorem of semisimple compact Lie groups.     

A unification of boson expansion theories: (II). Boson expansions as provided by the functional representation method

It is shown that the boson expansions hitherto known, as well as an infinite number of the new ones, can be derived in a unified way in terms of the functional representation technique. Each boson expansion (boson representation) is valid for the carrier space of an irreducible representation of a semisimple compact Lie subgroup of SO(2N + 1) and can be presented as Dyson-type, Holstein-Primakoff-type or Garbaczewski (Marumori)-type expansion with the corresponding properties concerning finiteness, convergence and hermitian conjugation. The physical boson space can be determined for every expansion and the projection operator is explicitly found. The methods for solving the Schrödinger equation are discussed and the generator coordinate method is shown to be equivalent to the boson expansion approach.     

On the geometrical representation of the Jacobian in the path integral reduction problem

By using the formula for the scalar curvature of the manifold with the Kaluza-Klein metric we obtain the geometrical representation of the Jacobian resulted from the path integral reduction problem in Wiener path integrals for a scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group.