L1 Representations Research Articles

Cohomology of the infinite-dimensional Lie algebra 𝐿₁ with nontrivial coefficients

with the bracket [ei, ej ] = (j − i)ei+j . The Lie algebra L is naturally graded, the degree of ei being i. The most natural modules over L are the so-called tensor field modules. A tensor field on the circle is of the form g(φ)(d/dφ). A vector field acts on this by infinitesimally changing the coordinate φ, where g(φ) is a section of some line bundle on the circle S with a flat connection. In the space of tensor fields we choose a basis fi, i ∈ Z, such that ei(fj) = (−λ(i + 1) + μ + j) · fi+j . Here λ, μ ∈ C are the invariants characterizing the module, i.e., the power of d/dφ and the logarithm of the monodromy of the flat connection. We denote such a module by Fλ,μ (see [3]). Denote by L1 the subalgebra of L with basis (e1, e2, e3, . . . ). It is easy to see that L1 is isomorphic to the Lie algebra of vector fields on the line, with polynomial coefficients, having a two-fold zero at the origin. The strategy of the cohomology computation for L1 with coefficients in the adjoint module is the following: we first compute the cohomology of L1 with coefficients in Fλ,μ, and then remark that the adjoint representation of L1 is a submodule of such an Fλ,μ. After this the spaces H(L1, L1) can easily be determined. The computations of H(L1, L1) and H (L1, L1) are contained in [3]. Deformations of L1 are studied in [5]. In this paper we shall describe a more general method for the computation of H∗(L1,Fλ,μ). It will be more convenient for us to deal with homology instead of cohomology. It is easy to see that H ∗ (L1,Fλ,μ) is dual to H∗(L1,F−1−λ,−μ). Then, using the fact that L∗1 is the factor of some Fλ,μ, we can compute Hi(L1, L ∗ 1). Notice that for almost every (λ, μ) the module Fλ,μ is an irreducible representation of L, and L1 is the maximal nilpotent subalgebra in L. That means that the problem of determining H∗(L1,Fλ,μ) is analogous to that of determining the cohomology of the maximal nilpotent subalgebra of a complex semisimple Lie

Primitive ideals with bounded approximate units in L1-algebras of exponential lie groups

AbstractLet G be an exponential Lie group. We study primitive ideals (i.e. kernels of irreducible *-representations of L1(G)), with bounded approximate units (b.a.u.). We prove a result relating the existence of b.a.u. in certain primitive ideals with the geometry of the corresponding Kirillov orbits. This yields for a solvable group of class 2, a characterization of the primitive ideals with b.a.u.

Orbital angular momentum eigenfunctions for many-particle systems

Methods are presented for constructing eigenfunctions of the total orbital angular momentum operator of a many-particle system without the use of the Clebsch–Gordan coefficients. One of the equations derived in this paper is analogous to Dirac's identity for total spin; and through this equation, a connection is established between eigenfunctions of L2 and irreducible representations of the symmetric group Sn.

A note on separating families of representations

Conditions are given that imply that a separating family of unitary representations of a locally compact group G G lifts to a separating family of ∗ ^ * -representations of L 1 ( G ) {L^1}(G) .

Representations of the ℓ1-algebra of an inverse semigroup having the separation property

Let S be a semigroup, and let ℓ1(S) denote the l1-semigroup algebra of S. Beginning with the fundamental paper of E. Hewitt and H. Zuckerman [5], there has been a considerable amount of research done concerning the Banach algebra ℓ1(S) in the case when S is abelian; see the bibliography [7]. However, until recently, there was very little information known concerning ℓ1(S) when S was nonabelian and infinite. Now for certain classes of infinite nonabelian semigroups with involution, recent progress has been made in the study of the Banach *-algebra ℓ1(S) and the *-representations of l1(S). In [2], B. Barnes and J. Duncan prove that ℓ1(S) is Jacobson semisimple, study the spectrum of elements in ℓ1(S), and construct and study *-representations of ℓ1(S) when S is the free semigroup with a finite or countably infinite set of generators (and also in some cases where the generators satisfy certain relations). In [1], the present author considered the representation theory of ℓ1(S) where S is an inverse semigroup. This paper is a sequel to [1].

Localizability of relativistic particles in fuzzy phase space

The concept of fuzzy phase space Gamma s, recently introduced in non-relativistic quantum mechanics, is extended to the relativistic case. The L2( Gamma s) representation of the wave packet is used as a basis for discussing localizability in Gamma s for non-zero mass particles as well as for the photon.

Unitary and Uniformly Bounded Representations of the Two by Two Unimodular Group Over Local Fields

Introduction. In this paper, we study unitary and uniformly bounded representations of the group G= SL(2,K) of two by two matrices of determinant 1 over a locally compact, totally disconnected, non-discrete field K. We obtain results similar to those obtained by Kunze and Stein in [8], [9] for SL(2,R) and SL(n,C) and by the author in [13] for the universal covering group of SL(2,R). The unitary representations with which we are primarily concerned are the principal series and complementary series of irreducible unitary representations of G. We also deal with the so-called special which has no direct analogue in the classical situation. These representations have recently received the attention of several authors ([2], [3], [4], [11], [12], [16]). We also note the occurrence of the special in the paper of Ihara [6] (pp. 229-230). In ? 1, we summarize the basic results of harmonic analysis on K which we require for the rest of the paper. Some of these results are well-known and others are taken from [15] where a more detailed treatment may be found. In ? 2, we discuss the principal series of representations of G. These representations are indexed by the unitary characters on K*. After summarizing the standard construction of these representations in L2 (K), we show that the principal series can also be realized as operators in L2(CU), where CT is the analogue of the unit circle in the complex plane. More precisely, if L = K ( Vr) is a ramified quadratic extension of K, then CT is the kernel of the norm from L to K. Here v denotes a generator of the valuation ideal in K. This is similar to the original construction of Bargmann [1] for SL(2,R) which was also utilized by the author in [13]. In this picture, it turns out that, for any character on L* (not necessarily unitary), we obtain a representation of G by bounded operators on L2(CT). The group CT is isomorphic to the subgroup of G defined by