Real representations of finite Clifford algebras. I. Classification

Abstract

A classification of real matrix irreducible representations of finite-dimensional real Clifford algebras has been made. In contrast to the case of complex representation, three distinct types of representations can be obtained which we call normal, almost complex, and quaternionic. The dimension of the latter two cases is twice as large as that of the normal representation. A criteria for a given Clifford algebra to possess a particular type of the representations is also given with some applications.

On the classification of N-extended supersymmetric quantum mechanical systems

“…The construction along the lines (3.2)-(3.9) for representations of the type {d, d} in the case of positively definite signature (p, q) = (N, 0) was performed in [10] (see also [11]) where the dimensionalities as well as realizations of the Γ-matrices (3.6) were described. In the case of pseudo-Euclidean metric with signature (p, q) such construction extensively uses the considerations of the papers [23]. The results will be presented in the next section.…”

“…For what concerns real matrix representations of the Clifford algebras we borrow the results of [23]. Three cases have to be distinguished for real representations, specified by the type of most general solution allowed for a real matrix S commuting with all the Clifford Γ i matrices, i.e.…”

A proposal for 0 extended supersymmetry in integrable systems

“…Once again if one studies the case of N = 2 utilizing the parametrization in section 2, then (6.1) is found to contain the N = 1 theory [10] as well as the proposal for the N = 2 [12] theory.…”

“…For example, we propose that the Kadomtsev-Petviashvili (KP) system [10] works much the same way and for exactly the same multiplet. We begin by now assuming that each field in the multiplet of (2.1) depends on bosonic variables (x, y, t) but we use exactly the same set of transformation laws.…”

“…An explicit representation of these GR(d, N) algebras can be found in our works [7,9]. The first discussion of this type of real Clifford algebra was given by Okubo [10] and this algebraic structure has also been noted in a study of 3D non-linear σ-models [11].…”