Ternary generalizations of graded algebras and their applications in physics

Seminar
Speaker
Prof. Richard Kerner, University Pierre et Marie Curie - Sorbonne Universit\'es Paris, France
Date
05/03/2018 - 15:35 - 14:00Add to Calendar 2018-03-05 14:00:00 2018-03-05 15:35:00 Ternary generalizations of graded algebras and their applications in physics We discuss cubic and ternary algebras which are a direct generalization of Grassmann and Clifford algebras, but with $Z_3$-grading replacing the usual $Z_2$-grading. Elementary properties and structures of such algebras are discussed, with special interest in low-dimensional ones, with two or three generators. Invariant antisymmetric quadratic and cubic forms on such algebras are introduced, and it is shown how the $SL(2,C)$ group arises naturally in the case of lowest dimension, with two generators only, as the symmetry group preserving these forms. We also show how the calculus of differential forms can be extended to include also second differentials $d^2 x^i$, and how the $Z_3$ grading naturally appears when we assume that $d^3 = 0$ instead of $d^2 = 0$. Ternary analogue of the commutator is introduced, and its relation with usual Lie algebras investigated, as well as its invariance properties. We shall also discuss certain physical applications In particular, $Z_3$-graded gauge theory is briefly presented, as well as ternary generalization of Pauli's exclusion principle and ternary Dirac equation for quarks. 2nd floor Colloquium Room, Building 216 אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
2nd floor Colloquium Room, Building 216
Abstract

We discuss cubic and ternary algebras which are a direct generalization of Grassmann
and Clifford algebras, but with $Z_3$-grading replacing the usual $Z_2$-grading.
Elementary properties and structures of such algebras are discussed, with special interest
in low-dimensional ones, with two or three generators.
Invariant antisymmetric quadratic and cubic forms on such algebras are introduced, and it
is shown how the $SL(2,C)$ group arises naturally in the case of lowest dimension, with
two generators only, as the symmetry group preserving these forms.
We also show how the calculus of differential forms can be extended to include also second
differentials $d^2 x^i$, and how the $Z_3$ grading naturally appears when we assume that
$d^3 = 0$ instead of $d^2 = 0$.
Ternary analogue of the commutator is introduced, and its relation with usual Lie algebras
investigated, as well as its invariance properties.
We shall also discuss certain physical applications In particular, $Z_3$-graded gauge theory
is briefly presented, as well as ternary generalization of Pauli's exclusion principle and
ternary Dirac equation for quarks.

תאריך עדכון אחרון : 05/03/2018