then there is one (up to opposites) series of irreducible finite dimensional unital J-bimodules parameterized by positive integers. All finite dimensional bimodules in this case are completely reducible.
If
Remark. A finite dimensional irreducible bimodule over K3 is a unital finite dimensional irreducible bimodule over D(0). Hence the above description of unital finite dimensional irreducible bimodules over D(0) applies to K3.
The detailed description of irreducible and indecomposable D(t)-bimodules is contained in Martínez and Zelmanov (2003), Martínez and Zelmanov (2006). Trushina (2008) extended the description above to superalgebras over fields of positive characteristics.
1.7.2(f) Jordan bimodules over JP(2) Representation theory of JP(2) is essentially different from that of JP(n), n ≥ 3.
The universal associative enveloping superalgebra S(JP(2)) is isomorphic to M2+2(F[t]), where F[t] is the polynomial algebra in one variable (see Martínez and Zelmanov (2003)). Hence, irreducible one-sided bimodules are parameterized by scalars α ∈ F and have dimension 4, whereas indecomposable bimodules are parameterized by Jordan blocks.
Let V be an irreducible finite dimensional bimodule over JP(2). Let L = K(JP(2)) be the Tits–Kantor–Koecher Lie superalgebra of JP(2), L = L–1 + L0 + L1. The superalgebra L has one-dimensional center. Fix 0 ≠ z ∈ L0, then L/Fz ≅ P(3) (see Martinez and Zelmanov (2001)). The Lie superalgebra L0 acts on the module V (see Jacobson (1968); Martin and Piard (1992)), and the element z acts as a scalar multiplication.
DEFINITION 1.21.– We say that V is a module of level α ∈ F if z acts on V as the scalar multiplication by α.
For an arbitrary scalar α ∈ F, there are exactly two (up to opposites) non-isomorphic unital irreducible finite dimensional Jordan bimodules over JP(2) of level α. For their explicit realization, see (Martínez and Zelmanov (2014)).
Kashuba and Serganova (2020) described indecomposable finite dimensional Jordan bimodules over Kan(n), n ≥ 1 and JP(2).
1.8. Jordan superconformal algebras
In this section, we will discuss connections between infinite dimensional Jordan superalgebras and the so-called superconformal algebras that originated in mathematical physics.
In view of importance of the Virasoro algebra and (especially) its central extensions in physics, (Neveu and Schwarz 1971; Ramond 1971) and others considered superextensions of the algebra Vir. These superextensions became known as superconformal algebras. Kac and van de Leur (1989) put the theory on a more formal footing and recognized that all known superconformal algebras are in fact infinite dimensional superalgebras of Cartan type considered in Kac (1977b). Following (Kac and van de Leur 1989), we call a ℤ-graded Lie superalgebra
1 i) L is graded simple;
2 ii) ;
3 iii) the dimensions dim Li, i ∈ ℤ are uniformly bounded.
EXAMPLE 1.31.– The Lie superalgebra of superderivations
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
