Abdenacer Makhlouf - Algebra and Applications 1

THEOREM 1.5 (Zelmanov (2000)).– Let J be a finite dimensional Jordan superalgebra. Then J is semisimple if and only if

where J (1),…, J (t)are simple Jordan superalgebras and for every i = 1,…, s , the superalgebras Jij are simple non-unital Jordan superalgebras over the field extension Ki of F .

Jacobson (1968) developed the theory of bimodules over semisimple finite dimensional Jordan algebras.

In this section, we discuss representations (bimodules) of finite dimensional Jordan superalgebras.

DEFINITION 1.15.– The rank of a Jordan superalgebra J is the maximal number of pairwise orthogonal idempotents in the even part .

Unless otherwise stated we will assume char F = 0.

DEFINITION 1.16.– Let V be a ℤ/2ℤ- graded vector space with bilinear mappings V × J → V , J × V → V . We call V a Jordan bimodule if the split null extension V + J is a Jordan superalgebra.

Recall that in the split null extension the multiplication extends the multiplication on J , products V ∙ J and J ∙ V are defined via the bilinear mappings above and V ∙ V = (0).

Let be a Jordan bimodule over J . Consider the vector space , where is a copy of with different parity. Define the action of J on V opvia

Then V opis also a Jordan bimodule over J . We call it the opposite module of V .

Let V be the free Jordan J -bimodule on one free generator.

DEFINITION 1.17.– The associative subsuperalgebra U ( J ) of End F V generated by all linear transformations RV ( a ) : V → V, v ↦ va, a ∈ J, is called the universal multiplicative enveloping superalgebra of J .

Every Jordan bimodule over J is a right module over U ( J ).

DEFINITION 1.18.– A bimodule V over J is called a one-sided bimodule if { J , V , J } = (0).

Let V ( 1/2 ) be the free one-sided Jordan J -bimodule on one free generator.

DEFINITION 1.19.– The associative subsuperalgebra S ( J ) of End F V (1/2) generated by all linear transformations RV (1/2)( a ) : V (1/2) → V (1/2), v ↦ va, a ∈ J, is called the universal associative enveloping algebra of J .

Every one-sided Jordan J-bimodule is a right module over S ( J ).

Finally, let J be a unital Jordan superalgebra with the identity element e. Let V (1) denote the free unital J-bimodule on one free generator. The associative subsuperalgebra U 1( J ) of End FV (1) generated by { RV (1/2)( a )} a ∈ J is called the universal unital enveloping algebra of J .

For an arbitrary Jordan bimodule V , the Peirce decomposition

is a decomposition of V into a direct sum of unital and one-sided bimodules. Hence U ( J ) ≅ U 1( J ) ⊕ S ( J ).

In this section, we consider Jordan bimodules over finite dimensional simple Jordan superalgebras of rank ≥ 3, that is, superalgebras Josp( n , 2 m ), n + m ≥ 3; Q ( n ) (+), n ≥ 3; JP( n ), n ≥ 3.

In this case, the universal multiplicative enveloping superalgebra U ( J ) is finite dimensional and semisimple (Martin and Piard 1992). Hence every Jordan bimodule is completely reducible, as in the case of Jordan algebras.

The superalgebras Josp( n , 2 m ) and JP( n ) are of the type

where A is a simple finite dimensional associative superalgebra and ∗ is an involution.

EXAMPLE 1.25.– An arbitrary right module over A is a one-sided module over H ( A , ∗).

EXAMPLE 1.26.– The subspace K ( A , ∗) = { k ∈ A | k ∗= – k } with the action k ∙ a = ka + ak ; k ∈ K ( A , ∗), a ∈ H ( A , ∗) is a unital H ( A , ∗)-bimodule.

THEOREM 1.6 (Martin and Piard (1992)).– An arbitrary irreducible Jordan bimodule over Josp( n , 2 m ), n + m ≥ 3, or JP( n ), n ≥ 3, is a bimodule of examples 1.25and 1.26or the regular bimodule.

The superalgebras , Q ( n ) (+)are of the type A (+), where A is a simple finite dimensional associative superalgebra.

EXAMPLE 1.27.– Every right module over A gives rise to a one-sided Jordan bimodule over A (+).

Suppose now that the superalgebra A is equipped with an involution ∗.

EXAMPLE 1.28.– The subspaces H ( A , ∗) and K ( A , ∗) become Jordan A (+)- bimodules with the actions:

1 1) h ○ a = ha + a∗h;

2 2) h ○ a = ha∗ + ah;

3 3) k ○ a = ka + a∗k;

4 4) k ○ a = ka∗ + ak;

where h ∈ H ( A , ∗), k ∈ K ( A , ∗), a ∈ A .

The associative superalgebras Mm+n ( F ), where both m , n are odd, and Q ( n ), are not equipped with an involution, but they are equipped with a pseudoinvolution.

DEFINITION 1.20.– A graded linear map ∗ : A → A is called a pseudoinvolution if ( ab )∗ = (–1)| a|∙|b| b ∗ a ∗, ( a ∗)∗ = (–1) |a| a for arbitrary elements .