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      Jordan Superalgebras

       Consuelo MARTINEZ1 and Efim ZELMANOV2

       1Department of Mathematics, University of Oviedo, Spain

       2Department of Mathematics, University of California San Diego, USA

1.1. Introduction

      Superalgebras appeared in a physical context in order to study, in a unified way, supersymmetry of elementary particles. Jordan algebras grew out of quantum mechanics and gained prominence due to their connections to Lie theory. In this chapter, we survey Jordan superalgebras focusing on their connections to other subjects. In this section we introduce some basic definitions and in section 1.2 we give the Tits–Kantor–Koecher construction that shows the way in which Lie and Jordan structures are connected. In section 1.3, we show examples of some basic superalgebras (the so-called classical superalgebras). Section 1.4 is about the notion of brackets and explains how to construct superalgebras using different types of brackets. Section 1.5 explains Cheng–Kac superalgebras, an important class of superalgebras that appeared for the first time in the context of superconformal algebras. The classification of Jordan superalgebras is explained in section 1.6, and it includes the cases of an algebraically closed field of zero characteristics, the case of prime characteristic, both for Jordan superalgebras with semisimple even part and with non-semisimple even part, and the case of non-unital Jordan superalgebras. Finally, in section 1.7, we give some general ideas about Jordan superconformal algebras. Throughout the chapter, all algebras are considered over a field F, charF ≠ 2.

      DEFINITION 1.1.– A (linear) Jordan algebra is a vector space J with a linear binary operation (x, y) ↦ xy satisfying the following identities:

      (J1) xy = yx (commutativity);

      (J2) (x²y)x = x²(yx) ∀x, y ∈ J (Jordan identity).

      Instead of (J2) we can consider the corresponding linearized identity:

      (J’2) (xy)(zu) + (xz)(yu) + (xu)(yz) = ((xy)z)u + ((xu)z)y + ((yu)z)x ∀x, y, z, u ∈ J.

      REMARK 1.1.– A Lie algebra L is a vector space with a linear binary operation (x, y) ↦ [x, y] satisfying the following identities:

      (L1) [x, y] = –[y, x] (anticommutativity);

      (L2) [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 for arbitrary elements x, y, z ∈ J (Jacobi identity).

      EXAMPLE 1.1.– If A is an associative algebra, then (A⁽⁺⁾, ∙), where a ∙ b = ab + ba is a Jordan algebra, and (A^(–), [, ]), where [a, b] = ab – ba is a Lie algebra. Both A⁽⁺⁾ and A^(–) have the same underlying vector space as A.

      DEFINITION 1.2.– A superalgebra A is an algebra with a ℤ/2ℤ-grading. So

      Elements of

are called homogeneous elements. The parity of a homogeneous element a, denoted |a|, is defined by |a| = 0 if

and |a| = 1 if

.

      Elements in

are called even and elements in

are called odd.

      Note that

is a subalgebra of A, but

is not, instead it can be seen as a bimodule over Скачать книгу