Algebra and Applications 1. Abdenacer Makhlouf

Algebra and Applications 1

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where Sr = (0) for r < 0, for r ≥ 1. Clearly, .

THEOREM 1.10 (Martin and Piard (1992)).–

1 1) For every r ≥ 1, Sr/Sr–2 is a unital irreducible Jordan bimodule over J.

2 2) Let V′= Fu ⊕ V, where |u| = 0. Extend the bilinear form 〈 , 〉 to V′ via 〈u, u〉 = 1, 〈u, V〉 = (0). Then for every even r ≥ 0, the quotient uSr /uSr–2 is a unital irreducible Jordan bimodule over J.

3 3) Every unital irreducible finite dimensional J-bimodule is isomorphic to Sr/Sr–2 or to uSr/uSr–2 for even r.

The classification of irreducible Jordan bimodules over M₁₊₁(F)⁽⁺⁾, D(t), K₃, JP(2) is too technical for an Encyclopedia survey. For a detailed description of finite dimensional irreducible Jordan bimodules, (see Martínez and Zelmanov (2003), Martin and Piard (1992), Martínez and Zelmanov (2006), Martínez and Shestakov (2020)). We will make only some general comments.

1.7.2(d) The universal associative enveloping superalgebra of J = M₁₊₁(F)⁽⁺⁾ is infinite dimensional, and finite dimensional one-sided Jordan bimodules over J are not necessarily completely reducible.

There is a family of 4-dimensional unital Jordan J-bimodules V (α, β, γ), which are parameterized by scalars α, β, γ ∈ F. If γ² – 1 – 4αβ ≠ 0, then the bimodule V (α, β, γ) is irreducible. If γ2 – 1 – 4αβ = 0, then it has a composition series with 2-dimensional irreducible factors.

Every irreducible finite dimensional unital Jordan J-bimodule is isomorphic to V (α, β, γ), γ² – 1 – 4αβ ≠ 0, or to a factor of a composition series of V (α, β, γ), γ² – 1 – 4αβ = 0 (see Martínez and Zelmanov (2009); Martínez and Shestakov (2020)).

1.7.2(e) Now let us discuss the superalgebras D(t) and K₃. Recall that

Then D(0) = F ∙ 1 + K₃, D(–1) ≅ M₁₊₁(F)⁽⁺⁾, D(1) is a Jordan superalgebra of a superform.

We will assume therefore that t ≠ –1, 1.

One-sided bimodules. The superalgebra K₃ does not have any non-zero one-sided Jordan bimodules (it has non-zero one-sided bimodules if char F > 0). All finite dimensional one-sided Jordan bimodules over D(t), t ≠ –1, 1, are completely reducible. The superalgebra D(t) does not have non-zero one-sided bimodules unless or , where m ∈ ℤ, m ≥ 1. For , there exists one (up to opposites) irreducible one-sided J-bimodule of dimension 2m + 3; for , there exists one (up to opposites) irreducible one-sided J-bimodule of dimension 2m + 1.

Unital bimodules. If J = D(t) and t cannot be represented as or , 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 , or , then there is one (up to opposites) additional irreducible bimodule.

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 a superconformal algebra if

1 i) L is graded simple;

2 ii) ;

3 iii) the dimensions dim Li, i ∈ ℤ are uniformly bounded.

EXAMPLE

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