We formulate quantum mechanics in the two-dimensional torus without using position operators. We define an algebra with only momentum operators and shift operators and construct an irreducible representation of the algebra. We show that it realizes quantum mechanics of a charged particle in a uniform magnetic field. We prove that any irreducible representation of the algebra is unitarily equiva...

In this review we show that a Clifford algebra possesses a unique irreducible representation ; the spinor representation. We discuss what types of spinors can exist in Minkowski space-times and we explain how to construct all the supersymmetry algebras that contain a given space-time Lie algebra. After deriving the irreducible representations of the superymmetry algebras, we explain how to use ...

We introduce 3-irreducible modules, even roots and odd roots for Leibniz algebras, produce a basis for a root space of a Leibniz algebra with a semisimple Lie factor, and classify finite dimensional simple Leibniz algebras with Lie factor sl2. In this paper, all vector spaces are vector spaces over an algebraically closed field k of characteristic 0. This paper consists of three sections. In Se...

The relation between facets of Rees cones of clutters and irreducible b-vertex covers is examined. Let G be a simple graph and let Ic(G) be its ideal of vertex covers. We give a graph theoretical description of the irreducible bvertex covers of G, i.e., we describe the minimal generators of the symbolic Rees algebra of Ic(G). As an application we recover an explicit description of the edge cone...

For each connected real semisimple matrix group, one obtains a constructive list of the irreducible tempered unitary representations and their characters. These irreducible representations all turn out to be instances of a more general kind of representation, here called basic. The result completes Langland's classification of all irreducible admissible representations for such groups. Since no...

The notion of structural completeness has received considerable attention for many years. A translation to algebra gives: a quasivariety is structurally complete if it is generated by its free algebras. It appears that many deductive systems (quasivarieties), like S5 or MVn fails structural completeness for a rather immaterial reason. Therefore the adjusted notion was introduced: almost structu...

It is proved that the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A (1) 1 of level k coincides with a certain W -algebra. In particular, a set of generators for the parafermion vertex operator algebra is determined.

Let Uq(ŝl2) ≥0 denote the Borel subalgebra of the quantum affine algebra Uq(ŝl2). We show that the following hold for any choice of scalars ε0, ε1 from the set {1,−1}. (i) Let V be a finite-dimensional irreducible Uq(ŝl2) -module of type (ε0, ε1). Then the action of Uq(ŝl2) ≥0 on V extends uniquely to an action of Uq(ŝl2) on V . The resulting Uq(ŝl2)-module structure on V is irreducible and of ...

Let L = H(2r; n) be a graded Lie algebra of Hamiltonian type in the Cartan type series over an algebraically closed field of characteristic p > 2. In the generalized restricted Lie algebra setup, any irreducible representation of L corresponds uniquely to a (generalized) p-character χ. When the height of χ is no more than min{pni − pni−1 | i = 1, 2, . . . , 2r} − 2, the corresponding irreducibl...