SU(2/1) superchiral self-duality: a new quantum, algebraic and geometric paradigm to describe the electroweak interactions

We propose an extension of the Yang-Mills paradigm from Lie algebras to internal chiral superalgebras. We replace the Lie algebra-valued connection one-form A, by a superalgebra-valued polyform $$ \tilde{A} $$ A ˜ mixing exterior-forms of all degrees and satisfying the chiral self-duality condition $$ \tilde{A} =^{\ast }{\tilde{A}}_{\chi } $$ A ˜ = ∗ A ˜ χ , where χ denotes the superalgebra grading operator. This superconnection contains Yang-Mills vectors valued in the even Lie subalgebra, together with scalars and self-dual tensors valued in the odd module, all coupling only to the charge parity CP-positive Fermions. The Fermion quantum loops then induce the usual Yang-Mills-scalar Lagrangian, the self-dual Avdeev-Chizhov propagator of the tensors, plus a new vector-scalar-tensor vertex and several quartic terms which match the geometric definition of the supercurvature. Applied to the SU(2/1) Lie-Kac simple superalgebra, which naturally classifies all the elementary particles, the resulting quantum field theory is anomaly-free and the interactions are governed by the super-Killing metric and by the structure constants of the superalgebra.

Automorphisms and Inner Automorphisms

LetKbe a field of characteristic not2and letA=A0+A1be central simple superalgebra overK, and let⁎be superinvolution onA. Our main purpose is to classify the group of automorphisms and inner automorphisms of(A,⁎)(i.e., commuting with⁎) by using the classical theorem of Skolem-Noether. Also we study two examples of groups of automorphisms and inner automorphisms on even central simple superalgebras with superinvolutions.

IRREDUCIBLE REPRESENTATIONS OF A CERTAIN JORDAN SUPERALGEBRA

A classification of unital finite-dimensional irreducible Jordan representations of the simple superalgebra D(t) in the case of characteristic 0 was obtained. As a corollary we obtain a classification of finite-dimensional irreducible representations of the Kaplansky superalgebra K3 in the case of characteristic 0.