Abdenacer Makhlouf - Algebra and Applications 1

EXAMPLE 1.32.– Let α ∈ ℂ. Then S ( n, α ) = { D ∈ W (1 : n ) | div ( tαD ) = 0} < W (1 : n ). Here if

then

If α ∈ ℤ, then [ S ( n , α ), S ( n , α )] is a proper ideal in S ( n , α ) and the superalgebra [ S ( n , α ), S ( n , α )] is simple. This family of superalgebras appeared in physics literature (Ademollo et al . 1976; Schwimmer and Seiberg 1987) under the name “ SU2 -superconformal algebras”.

EXAMPLE 1.33.– Consider the associative commutative superalgebra

and the contact bracket [ , ] of example 1.19. The superalgebra K ( n ) = (Λ(1 : n ), [ , ]) is simple unless n = 4. For n = 4, the commutator ideal [ K (4), K (4)] has codimension 1 and [ K (4), K (4)] is a simple superalgebra. This series is known in physics literature as “ SOn -superconformal algebras” (Ademollo et al . 1976; Schoutens 1987).

EXAMPLE 1.34.– In section 1.5, for an arbitrary associative commutative algebra Z with a derivation d : Z → Z , we constructed the Jordan superalgebra JCK( Z , d ). The Lie superalgebra CK( Z , d ) is the Tits–Kantor–Koecher construction of JCK ( Z , d ). Taking Z = ℂ[ t , t –1] and or , we get the Cheng–Kac superalgebras CK(6) discovered by Cheng and Kac (1997) and independently by Grozman et al . (2001).

Kac and van de Leur (1989) conjectured that examples 1.31– 1.34exhaust all superconformal algebras.

The superalgebras K ( n ) and CK(6) appear as Tits–Kantor–Koecher constructions of Jordan superalgebras.

Let [ , ] be the Jordan bracket on Λ(1 : n ) = ℂ [ t , t –1, ξ 1,…, ξn ] of example 1.20, Jn = K (Λ(1 : n ), [ , ]) is the Kantor double of this bracket. Then

In Kac et al . (2001), we classified “superconformal” Jordan superalgebras.

THEOREM 1.11 (Kac et al . (2001)).– Let be a ℤ-graded Jordan superalgebra that is graded simple and the dimensions of all homogeneous components dim Ji , i ∈ ℤ are uniformly bounded. Then either

1 1) J has finitely many negative (respectively, positive) non-zero homogeneous components;

2 2) or J is isomorphic to one of the superalgebras Jn, JCK(6), n ≥ 1 or a twisted version of it.

This theorem agrees with the Kac–van de Leur conjecture on classification of superconformal algebras.

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