LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART

For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

A natural extension of the conformal Lorentz group in a field theory context

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field theory at fixed momentum for a spin 1/2 particle along with its antiparticle. It is shown that the essential part of the corresponding automorphism group can naturally be related to the conformal Lorentz group. In addition, the non-semisimple part of the automorphism group can be understood as “dressing” of the pure one-particle states. The studied mathematical structure may help in constructing quantum field theories in a non-perturbative manner. In addition, it provides a simple example of circumventing Coleman–Mandula theorem using non-semisimple groups, without SUSY.

Frobenius Conjugacy Classes

This chapter analyzes Frobenius conjugacy classes. It shows that in either the split or nonsplit case, when χ‎ is good for N, the conjugacy class FrobE,X has unitary eigenvalues in every representation of the reductive group Garith,N. Now fix a maximal compact subgroup K of the complex reductive group Garith,N (ℂ). The semisimple part (in the sense of Jordan decomposition) of FrobE,X gives rise to a well-defined conjugacy class θE,X in K.

On the continuity of Arthur’s trace formula: the semisimple terms

We show that the semisimple part of the trace formula converges for a wide class of test functions.

A derived equivalence for a degree 6 del Pezzo surface over an arbitrary field

Let S be a degree six del Pezzo surface over an arbitrary field F. Motivated by the first author's classification of all such S up to isomorphism [3] in terms of a separable F-algebra B×Q×F, and by his K-theory isomorphism Kn(S) ≅ Kn(B×Q×F) for n ≥ 0, we prove an equivalence of derived categorieswhere A is an explicitly given finite dimensional F-algebra whose semisimple part is B×Q×F.

Rationality and Orbit Closures

Suppose we are given a finite-dimensional vector space V equipped with an F-rational action of a linearly algebraic group G, with F a characteristic zero field. We conjecture the following: to each vector v ∈ V(F) there corresponds a canonical G(F)-orbit of semisimple vectors of V. In the case of the adjoint action, this orbit is the G(F)-orbit of the semisimple part of v, so this conjecture can be considered a generalization of the Jordan decomposition. We prove some cases of the conjecture.