Note that
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,
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
DEFINITION 1.4.– The algebra
Let us consider V the variety of associative, commutative, anticommutative, Jordan or Lie algebras, respectively. Then we get:
EXAMPLE 1.3.– A superalgebra
EXAMPLE 1.4.– A superalgebra
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
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
DEFINITION 1.6.– An
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
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
