Structure theorem for Jordan algebra bundles

Abstract

Purpose

The aims of this paper is to prove that every semisimple Jordan algebra bundle is locally trivial and establish the decomposition theorem for locally trivial Jordan algebra bundles using the decomposition theorem of Lie algebra bundles.

Design/methodology/approach

Using the decomposition theorem of Lie algebra bundles, this paper proves the decomposition theorem for locally trivial Jordan algebra bundles.

Findings

Findings of this paper establish the decomposition theorem for locally trivial Jordan algebra bundles.

Originality/value

To the best of the author’s knowledge, all the results are new and interesting to the field of Mathematics and Theoretical Physics community.

Citation

Kumar, R. (2023), "Structure theorem for Jordan algebra bundles", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-08-2022-0186

1. Introduction

In modern mathematics, an important notion is that of non-associative algebra. In Ref. [1], we gave a relationship between two important classes of non-associative algebras, namely, Lie algebras (introduced in 1870 by the Norwegian mathematician Sophus Lie in his study of the groups of transformations) and Jordan algebras (introduced in 1932–1933 by the German physicist Pasqual Jordan (1902–1980) in his algebraic formulation of quantum mechanics [2–4]). These two algebras are interconnected, as was remarked for instance by Kevin McCrimmon [5, p. 622]:

We are saying that if you open up a Lie algebra and look inside, 9 times out of 10 there is a Jordan algebra (of pair) which makes it work.

Here, we recall some connections between Jordan algebra bundles and Lie algebra bundles [1, 6]. If ξ is a locally trivial Jordan algebra bundle in which each fibre ξ_x has a unit element then

K(ξ)=⋃x∈XK(ξx)=⋃x∈X(Derξx⊕L(ξx)⊕ξx⊕ξx‾)

is a Lie algebra bundle, where Derξ_x is the vector space of all derivation defined on ξ_x, L(ξ_x) is the vector space of all left translations of ξ_x and ξx‾ an isomorphic copy of ξ_x.

Lemma 1.1.

[1] Let ξ be a locally trivial Jordan algebra bundle over X, with each fibre ξ_x having a unit element e_x, x ∈ X. Then

g(ξ)=⋃x∈Xg(ξx)=⋃x∈XDerξx+L(ξx)

is a Lie algebra bundle.

Theorem 1.2.

[1] If ξ is a locally trivial Jordan algebra bundle in which each fibre has a unit element, then K(ξ)=g(ξ)⊕ξ⊕ξ‾, is a Lie algebra bundle, where ξ‾ is an isomorphic copy of the Jordan algebra bundle ξ. Further, ξ can be imbedded in K(ξ) such that the Jordan multiplication on ξ can be given in terms of the Lie multiplication on K(ξ).

Given a locally trivial Jordan algebra bundle ξ in which each fibre ξ_x has a unit element, consider the Lie algebra bundle K(ξ)=g(ξ)⊕ξ⊕ξ‾. Let h(ξ) = ∪_x∈Xh(ξ_x), where each h(ξ_x) = L(ξ_x) ⊕ [L(ξ_x), L(ξ_x)] is an ideal of g(ξ_x) = Derξ_x + L(ξ_x) [7]. Let ϕ : U × J → p⁻¹(U) be the local triviality of the Jordan algebra bundle ξ, where J is a Jordan algebra. Set g(ϕ) = Endϕ|_U×g(J), where the vector bundle morphism Endϕ : U ×EndJ → ∪_x∈U Endξ_x is given by (Endϕ)(x,f)=ϕxfϕx−1. Further, h(ξ) is a vector bundle since g(ϕ)|_U×h(J) maps U × h(J) onto ∪_x∈Uh(ξ_x). Hence h(ξ) is an ideal subbundle of g(ξ). We denote by L(ξ), the ideal subbundle L(ξ)=h(ξ)⊕ξ⊕ξ‾ of K(ξ).□

In this paper, we prove that any semisimple Jordan algebra bundle is locally trivial and we supply an example to show that the converse need not be true. Further, we prove that a semisimple Jordan algebra bundle can be written as the direct sum of simple ideal bundles.

1.1 Notations and terminology

All Jordan algebra bundles ξ = (ξ, p, X, θ) are over the arbitrary topological space X unless otherwise mentioned.

2. Preliminaries

Definition 2.1.

A Jordan algebra bundle is a vector bundle ξ = (ξ, p, X) together with a vector bundle morphism θ : ξ ⊗ ξ → ξ inducing a Jordan algebra structure on each fibre ξ_x, x ∈ X.

Definition 2.2.

By a trivial Jordan algebra bundle, we mean a trivial vector bundle (X × J, p, X), where J is a Jordan algebra.

Definition 2.3.

A morphism f : ξ → ζ of Jordan bundles ξ and ζ is a morphism of the underlying vector bundles such that for every x ∈ X, f_x : ξ_x → ζ_x is a Jordan algebra homomorphism. If f is bijective and f⁻¹ is continuous, then f is called an isomorphism.

Definition 2.4.

By a subalgebra (ideal) bundle of a Jordan algebra bundle ξ = (ξ, p, X), we mean a vector sub-bundle ξ′ = (ξ′, p, X) of ξ such that each fibre (ξ′)_x is a subalgebra (ideal) of ξ_x, ∀ x ∈ X.

Definition 2.5.

By a semisimple Jordan algebra bundle, we mean a Jordan algebra bundle in which each fibre is a semisimple Jordan algebra.

Definition 2.6.

If ξ is a Jordan algebra bundle with a nontrivial multiplication θ : ξ ⊗ ξ → ξ inducing the Jordan algebra bundle structure and if ξ has no ideal bundles except itself and the zero bundle, then we call ξ a simple Jordan algebra bundle.

Definition 2.7.

A Jordan algebra bundle ξ is said to be the direct sum of the ideal bundles ξ₁, ξ₂, …, ξ_n provided, ξ = ξ₁ ⊕ ξ₂ ⊕⋯ ⊕ ξ_n

3. Semisimple Jordan algebra bundles

Definition 3.1.

A Jordan algebra bundle ξ = (ξ, p, X) is locally trivial if, for every open set U ⊆ X, there exists a Jordan algebra J, together with a (vector bundle) trivialization φ : ξ|_U → U × J, which is fibrewise a homeomorphism of Jordan algebras.

Not every Jordan algebra bundle is locally trivial, as the following example shows.

Example 3.2.

Let J be a nonzero real Jordan algebra and R the field of real numbers. Then ξ=(R×J,p,R) is a trivial vector bundle. Let θ : ξ ⊗ ξ → ξ be defined by, θ(t, (u, v)) = t(uv) for all t∈R, u, v ∈ J. Then, θ is continuous and induces a Jordan algebra structure on each fibre ξ_x = x × J. Hence, ξ is a Jordan algebra bundle. But ξ is not locally trivial because ξ_t for t ≠ 0 is a nonzero algebra whereas ξ₀ is the zero algebra.

Remark 3.3.

From Atiyah [8, p. 4], we reproduce the following things without any changes. Suppose that V and W are vector spaces and that E = X × V and F = X × W are the corresponding product bundles. Then ϕ : E → F determines a map Φ : X → Hom(V, W) by formula Φ(x)(v) = ϕ(v). Moreover, if we give Hom(V, W) its usual topology, then Φ is continuous; conversely, any such continuous map Φ : X → Hom(V, W) determines a homomorphism ϕ : E → F.

Theorem 3.4.

Every semisimple Jordan algebra bundle is locally trivial.

Proof. Let ξ be a semisimple Jordan algebra bundle. The local triviality of ξ as a vector bundle is given by the vector bundle isomorphism α : U × V → p⁻¹(U). Then θ : ξ ⊗ ξ → ξ induces the morphism θ^:U×(V⊗V)→U×V given by,

(3.1)θ^(x,(v,w))=αx−1θ(αx(v),αx(w)).

Then, from above Remark (3.3), x↦θ^x defines a continuous mapping from U to M ⊆ Hom(V × V, V), the space of all Jordan multiplications defined on V endowed with the subspace topology. Since the Jordan algebra J0=(V,θ^x0),x0∈U is semisimple, J₀ is rigid by [9, Corollary 1.3]. That is the orbit

G(x0)={g.θx0|g∈Aut(V)},

with respect to the Lie group G = Aut(V) is open in M.

Let U′={x∈U|θ^x∈G(x0)}. Then, for each x ∈ U′, there exists a g_x in G such that

(3.2)θ^x=gx.θ^x0=gxθ^x0(gx−1⊕gx−1).

Further, since G and G(x₀) satisfy the hypothesis of Aren’s theorem [10] G/G₀ is homeomorphic to G(x₀), where G₀ is the stability subgroup corresponding to θ^x0. Also G → G/G₀ is a principal bundle [11, p. 33] together with a local cross section given by gG₀ → g [12, p. 126]. Hence, the map x↦g_x becomes the composition of the continuous maps

x↦θ^x↦gxG0↦gx.

Therefore, we can define the vector bundle isomorphism ϕ : U × J₀ → p⁻¹(U) by ϕ(x, v) = α_x(g_x(v)). The map ϕ preserves the Jordan multiplication

ϕ(x,θ^x0(a,b))=αxgxθ^x0(a,b)=αxθ^x(gx(a),gx(b))by(3.2)=αxαx−1θx(αxgx(a),αxgx(b))by(3.1)=θx(ϕ(x,a),ϕ(x,b)).

Hence, ϕ gives the required local triviality of the Jordan algebra bundle ξ.□

4. Decomposition theorem for Jordan algebra bundle

Lemma 4.1.

Let (X × V, q, X) be a trivial vector bundle and (X × J, p, X) a trivial semisimple Jordan bundle. Suppose ϕ : X × V → X × J is a vector bundle monomorphism such that for each x ∈ X, ϕ(x, V) is an ideal in J. Then, there exists a finite open partition ∪_iX_i = X such that ϕ_x(V) = ϕ_y(V) for x, y ∈ X_i. In particular, if X is connected, for all x, y ∈ X, ϕ_x(V) = ϕ_y(V).

Proof. The map ϕ : X × V → X × J being a vector bundle morphism, x↦ϕ_x is a continuous map from X to Hom(V, J), the vector space of all linear transformations from V to J. If Ĩ denotes the collection of all distinct ideals of J whose dimension is equal to that of V, then by the semisimplicity of J, Ĩ is a finite set [13, Corollary.4.6, p. 98]. Let Ĩ={I1,I2,…,In} and let X_i = {x ∈ X| ϕ_x(V) = I_i} for i = 1, 2, …, n. Let x ∈ X. Since ϕ_x(V) is an ideal in J, and dim ϕ_x(V) = dim V, we have ϕ_x(V) = I_i for some i. Thus, x ∈ X_i, and consequently X = ∪_iX_i.

It is enough to prove that each X_i is open in X. Let Jĩ be the vector subspace Jĩ={T∈Hom(V,J)|T(V)⊆Ii}. Then Jĩ is a closed subset of Hom(V, J) being a linear subspace of the vector space Hom(V, J). Since x↦ϕ_x is continuous and Jĩ is closed in Hom(V, J), Xĩ={x∈X|ϕx∈Jĩ} is closed in X.

To prove: Xĩ=Xi

x∈Xi⇒ϕx(V)=Ii⇒ϕx∈Jĩ⇒x∈Xĩ⇒Xi⊆Xĩ. Now if x∈Xĩ, ϕx∈Jĩ. Then, ϕ_x(V) ⊆ I_i. But since dim ϕ_x(V) = dim I_i, ϕ_x(V) = I_i. So x ∈ X_i. Thus, Xi=Xĩ.

Consider Xĩ∩Xj̃ and let i ≠ j

x∈Xĩ∩Xj̃⇒ϕx∈Jĩandϕx∈Jj̃⇒ϕx(V)⊆Ii∩Jj=0

Therefore, Xĩ∩Xj̃=∅

Thus, Xĩ′s are all disjoint collection of closed sets. Hence, X_j is the complement of ∪_i≠jX_i. So X_j is open in X. We here note that if X is connected, then for all x, y ∈ X, ϕ_x(V) = ϕ_y(V). □

Lemma 4.2.

Let ξ′ be an ideal bundle of a semisimple Jordan algebra bundle ξ. Then, h1(ξ′)=∪x∈Xh1(ξx′) is an ideal bundle of h(ξ) where

h1(ξx′)={T∈h(ξx)|T(ξx)⊆ξx′}

Proof. Each h1(ξx′) is an ideal in h(ξ_x). It is enough to prove that h₁(ξ′) is a vector bundle. Since the bundle ξ is semisimple and ξ′ is semisimple being its ideal bundle, ξ and ξ′ are locally trivial by Theorem (3.4). So we have Jordan algebras J and J′ and Jordan bundle isomorphisms Φ : U × J → ∪_x∈Xξ_x and Ψ:U×J′→∪x∈Xξx′. Then, Φ⁻¹Ψ : U × J′ → U × J is a vector bundle monomorphism satisfying the hypotheses of Lemma (4.1). Hence, there exists a finite open cover {U_i} for U such that (Φ⁻¹Ψ)(x, J′) = (Φ⁻¹Ψ)(x′, J′) for all x, x′ ∈ U_i. Consequently, the neighbourhood U can be shrunk so that there exists an ideal I of J and such that Φ maps U × I onto ∪x∈Uξx′. Let h₁(I) = {T ∈ h(J)| T(J) ⊆ I}. Given T ∈ h₁(I), consider g(Φ)x(T)=ΦxTΦx−1.

g(Φ)x(T)(ξx)=(ΦxTΦx−1)(ξx)=(ΦxT)(J)⊆Φx(I)=ξx′.

Hence, g(Φ)x(T)∈h1(ξx′). That is, g(Φ) maps U × h₁(I) onto ∪x∈Uh1(ξx′). That is, h₁(ξ′) is an ideal sub-bundle of h(ξ).□

Theorem 4.3.

Every semisimple Jordan algebra bundle can be uniquely written as the direct sum of simple ideal bundles.

Proof. Let ξ be a semisimple Jordan algebra bundle. Each fibre ξ_x has a unit element being a semisimple Jordan algebra [13, Theorem 4.7, p. 99]. Hence, the corresponding Lie algebra bundle L(ξ) exists [1]. The semisimplicity of ξ_x implies that of L(ξ_x) [7, p. 805] and so L(ξ) is a semisimple Lie algebra bundle. Then, L(ξ) can be written as follows:

L(ξ)=L1⊕L2⊕⋯⊕Ln,

where each L_i is a simple ideal bundle of L(ξ) [14]. Also each L_i is of the form Li=∪x∈X(hi)x⊕(Ai)x⊕(Bi‾)x, [7, Lemma 1, p. 789]. Each ξi=∪x∈X(Ai)x is an ideal bundle of ξ and each ξ_i is simple since L_i is simple. Hence, ξ = ξ₁ ⊕ ξ₂ ⊕⋯ ⊕ ξ_n.

Let us prove the uniqueness of the decomposition. Let, ξ be expressed as follows:

ξ1⊕ξ2⊕⋯⊕ξn=ξ=ξ1′⊕ξ2′⊕⋯⊕ξm′,

where each ξ_i and ξj′ are simple ideal bundles. Consider ⊕1nh1(ξi)x, where h1(ξi)x={T∈h(ξx)|T(ξx)⊆(ξi)x}. We have ⊕1nh1(ξi)x⊆h(ξx). Let L(a) ∈ h(ξ_x), then a can be written as a=⊕1nai,ai∈(ξi)x. Since (ξi)x is an ideal in ξ_x, L(a_i) maps ξ_x into (ξi)x. So

L(a)=⊕i=1nL(ai)∈⊕1nh1(ξi)x,

Let [L(a), L(b)] ∈ h(ξ_x), where a=⊕1nai and b=⊕1nbi, ai,bi∈(ξi)x. Since [L(ai),L(bj)](u)∈(ξi)x∩(ξj)x=0 for i ≠ j, we obtain that

[L(a),L(b)]=⊕1n[L(ai),L(bj)]∈⊕1nh1(ξi)x.

Consequently, h(ξx)⊆⊕1nh1(ξi)x. Therefore, h(ξ) = h₁(ξ₁) ⊕ h₁(ξ₂) ⊕⋯ ⊕ h₁(ξ_n). Similarly, h(ξ)=⊕1nh1(ξj′). Hence

L(ξ)=L1(ξ1)⊕L1(ξ2)⊕⋯⊕L1(ξn)=L1(ξ1′)⊕L1(ξ2′)⊕⋯⊕L1(ξm′)

where L₁(ξ_i) = h₁(ξ_i) ⊕ ξ_i ⊕ ξ_i and L1(ξj′)=h1(ξj′)⊕ξj′⊕ξj′ are ideal bundles of L(ξ) by Lemma (4.2). Then by [14, Theorem 2.8], we obtain that m = n and L₁(ξ_i) coincides with one of the L1(ξj′). Then, obviously ξj′ coincides with ξ_i except for the order.□