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Algebra and Applications 1. Abdenacer Makhlouf
Читать онлайн.Название Algebra and Applications 1
Год выпуска 0
isbn 9781119818151
Автор произведения Abdenacer Makhlouf
Жанр Математика
Издательство John Wiley & Sons Limited
graded by degrees of t is a superconformal algebra.
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
Kac and van de Leur (1989) conjectured that examples 1.31–1.34 exhaust 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, ξ1,…, 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
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.
1.9. References
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