Algebra and Applications 1. Abdenacer Makhlouf

Algebra and Applications 1

EXAMPLE 1.2.– If V is a vector space of countable dimension, then G = G(V) denotes the Grassmann (or exterior) algebra over V, that is, the quotient of the tensor algebra over the ideal generated by the symmetric tensors v ⊗ w + w ⊗ v, v, w ∈ V. This algebra G(V) is ℤ/2ℤ-graded. Indeed,

, where the “even part” is the linear span of all tensors of even length and the “odd part”

is the linear span of all tensors of odd length.

G(V) is an example of a superalgebra.

DEFINITION 1.3.– Consider a variety of algebras V defined by homogeneous identities (see Jacobson (1968) or Zhevlakov et al. (1982)). We say that a superalgebra

is a V-superalgebra if the even part of A ⊗F G(V) lies in the variety, that is

DEFINITION 1.4.– The algebra

is called the Grassmann envelope of the superalgebra A and will be denoted as G(A).

Let us consider V the variety of associative, commutative, anticommutative, Jordan or Lie algebras, respectively. Then we get:

EXAMPLE 1.3.– A superalgebra

is an associative superalgebra if and only if it is a ℤ/2ℤ-graded associative algebra.

EXAMPLE 1.4.– A superalgebra

is a commutative superalgebra if it satisfies:

for any x, y homogeneous elements of A.

EXAMPLE 1.5.– A superalgebra A is an anticommutative superalgebra if

for every x, y homogeneous elements of A.

EXAMPLE 1.6.– A Jordan superalgebra is a superalgebra that is commutative and satisfies the graded identity:

for every homogeneous elements

EXAMPLE 1.7.– An anticommutative superalgebra A is a Lie superalgebra if it satisfies:

for every

DEFINITION 1.5.– If

is a Jordan superalgebra and , then their triple product is defined by:

Note that every algebra is a superalgebra with the trivial grading, that is,

1.2. Tits–Kantor–Koecher construction

Tits (1962, 1966) made an important observation that relates Lie and Jordan structures. Let L be a Lie superalgebra whose even part

contains an

-triple {e, f, h}, that is,

DEFINITION 1.6.– An

-triple e, f, h is said to be “good” if ad(h) : L → L is diagonalizable and the eigenvalues are only –2, 0, 2.

In such a case, L = L_{– 2} + L₀ + L₂ decomposes as a direct sum of eigenspaces. We can define a new product in L₂ by:

With this new product, J = (L₂, ○) becomes a Jordan superalgebra.

Moreover, (Tits 1962, 1966; Kantor 1972) and (Koecher 1967) showed that every Jordan superalgebra can be obtained in this way. The corresponding Lie superalgebra is not unique, but any two such Lie superalgebras are centrally isogenous, that is, they have the same central cover. Let us recall the construction of L = TKK(J), the universal Lie superalgebra in this class (see Martin and Piard (1992)).

CONSTRUCTION.– Consider J a unital Jordan superalgebra, and {ei}_i∈I a basis of J.

Let

Define a Lie superalgebra K by generators

and relations

This Lie superalgebra has a short grading K = K_–1 + K₀+ K₁ where

K is the universal Tits–Kantor–Koecher Lie superalgebra of the unital Jordan superalgebra J:

1.3. Basic examples (classical superalgebras)

Let

be an associative superalgebra. The new operation in the underlying vector space A given by:

defines a structure of a Jordan superalgebra on A that is denoted A⁽⁺⁾.

DEFINITION 1.7.– Those Jordan superalgebras that can be obtained as subalgebras of a superalgebra A⁽⁺⁾, with A an associative superalgebra, are called special. Superalgebras that are not special are called exceptional.

REMARK 1.2.– If we consider in the original associative superalgebra the new product given by:

we get a Lie superalgebra that is denoted as A^(–).

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