Structure theorem for Jordan algebra bundles

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.

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 \in X} K (ξ_{x}) = ⋃_{x \in X} (Der ξ_{x} \oplus L (ξ_{x}) \oplus ξ_{x} \oplus \overline{ξ_{x}})

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 $\overline{ξ_{x}}$ 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 \in X} g (ξ_{x}) = ⋃_{x \in X} (Der ξ_{x} + L (ξ_{x}))

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 $K (ξ) = g (ξ) \oplus ξ \oplus \overline{ξ}$ , is a Lie algebra bundle, where $\overline{ξ}$ 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 $K (ξ) = g (ξ) \oplus ξ \oplus \overline{ξ}$ ⁠. 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 $(End ϕ) (x, f) = ϕ_{x} f ϕ_{x}^{- 1}$ ⁠. 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 $L (ξ) = h (ξ) \oplus ξ \oplus \overline{ξ}$ 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 $ξ = (R \times J, p, R)$ is a trivial vector bundle. Let θ : ξ ⊗ ξ → ξ be defined by, θ(t, (u, v)) = t(uv) for all $t \in R$ ⁠, 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 $\hat{θ} : U \times (V \otimes V) \to U \times V$ given by,

\hat{θ} (x, (v, w)) = α_{x}^{- 1} θ (α_{x} (v), α_{x} (w)) .

(3.1)

Then, from above Remark (3.3), $x \mapsto {\hat{θ}}_{x}$ 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 $J_{0} = (V, {\hat{θ}}_{x_{0}}), x_{0} \in U$ is semisimple, J₀ is rigid by [9, Corollary 1.3]. That is the orbit

G (x_{0}) = {g . θ_{x_{0}} | g \in Aut (V)},

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

Let $U^{'} = {x \in U | {\hat{θ}}_{x} \in G (x_{0})}$ ⁠. Then, for each x ∈ U′, there exists a g_x in G such that

{\hat{θ}}_{x} = g_{x} . {\hat{θ}}_{x_{0}} = g_{x} {\hat{θ}}_{x_{0}} (g_{x}^{- 1} \oplus g_{x}^{- 1}) .

(3.2)

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 ${\hat{θ}}_{x_{0}}$ ⁠. 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 \mapsto {\hat{θ}}_{x} \mapsto g_{x} G_{0} \mapsto g_{x} .

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

\begin{array}{l} ϕ (x, {\hat{θ}}_{x_{0}} (a, b)) & = α_{x} g_{x} {\hat{θ}}_{x_{0}} (a, b) \\ = α_{x} {\hat{θ}}_{x} (g_{x} (a), g_{x} (b)) by (3.2) \\ = α_{x} α_{x}^{- 1} θ_{x} (α_{x} g_{x} (a), α_{x} g_{x} (b)) by (3.1) \\ = θ_{x} (ϕ (x, a), ϕ (x, b)) . \end{array}

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 $\tilde{I}$ denotes the collection of all distinct ideals of J whose dimension is equal to that of V, then by the semisimplicity of J, $\tilde{I}$ is a finite set [13, Corollary.4.6, p. 98]. Let $\tilde{I} = {I_{1}, I_{2}, \dots, I_{n}}$ 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 $\tilde{J_{i}}$ be the vector subspace $\tilde{J_{i}} = {T \in Hom (V, J) | T (V) \subseteq I_{i}}$ ⁠. Then $\tilde{J_{i}}$ 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 $\tilde{J_{i}}$ is closed in Hom(V, J), $\tilde{X_{i}} = {x \in X | ϕ_{x} \in \tilde{J_{i}}}$ is closed in X.

To prove: $\tilde{X_{i}} = X_{i}$

$x \in X_{i} \Rightarrow ϕ_{x} (V) = I_{i} \Rightarrow ϕ_{x} \in \tilde{J_{i}} \Rightarrow x \in \tilde{X_{i}} \Rightarrow X_{i} \subseteq \tilde{X_{i}}$ ⁠. Now if $x \in \tilde{X_{i}}$ ⁠, $ϕ_{x} \in \tilde{J_{i}}$ ⁠. 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, $X_{i} = \tilde{X_{i}}$ ⁠.

Consider $\tilde{X_{i}} \cap \tilde{X_{j}}$ and let i ≠ j

\begin{array}{l} x \in \tilde{X_{i}} \cap \tilde{X_{j}} & \Rightarrow ϕ_{x} \in \tilde{J_{i}} and ϕ_{x} \in \tilde{J_{j}} \\ \Rightarrow ϕ_{x} (V) \subseteq I_{i} \cap J_{j} = 0 \end{array}

Therefore, $\tilde{X_{i}} \cap \tilde{X_{j}} = \emptyset$

Thus, ${\tilde{X_{i}}}^{'} s$ 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, $h_{1} (ξ^{'}) = \cup_{x \in X} h_{1} (ξ_{x}^{'})$ is an ideal bundle of h(ξ) where

h_{1} (ξ_{x}^{'}) = {T \in h (ξ_{x}) | T (ξ_{x}) \subseteq ξ_{x}^{'}}

Proof. Each $h_{1} (ξ_{x}^{'})$ 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 $Ψ : U \times J^{'} \to \cup_{x \in X} ξ_{x}^{'}$ ⁠. 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 $\cup_{x \in U} ξ_{x}^{'}$ ⁠. Let h₁(I) = {T ∈ h(J)| T(J) ⊆ I}. Given T ∈ h₁(I), consider $g {(Φ)}_{x} (T) = Φ_{x} T Φ_{x}^{- 1}$ ⁠.

\begin{array}{l} g {(Φ)}_{x} (T) (ξ_{x}) & = (Φ_{x} T Φ_{x}^{- 1}) (ξ_{x}) \\ = (Φ_{x} T) (J) \subseteq Φ_{x} (I) = ξ_{x}^{'} . \end{array}

Hence, $g {(Φ)}_{x} (T) \in h_{1} (ξ_{x}^{'})$ ⁠. That is, g(Φ) maps U × h₁(I) onto $\cup_{x \in U} h_{1} (ξ_{x}^{'})$ ⁠. 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) = L_{1} \oplus L_{2} \oplus \dots \oplus L_{n},

where each L_i is a simple ideal bundle of L(ξ) [14]. Also each L_i is of the form $L_{i} = \cup_{x \in X} {(h_{i})}_{x} \oplus {(A_{i})}_{x} \oplus {(\overline{B_{i}})}_{x}$ ⁠, [7, Lemma 1, p. 789]. Each $ξ_{i} = \cup_{x \in X} {(A_{i})}_{x}$ 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:

ξ_{1} \oplus ξ_{2} \oplus \dots \oplus ξ_{n} = ξ = ξ_{1}^{'} \oplus ξ_{2}^{'} \oplus \dots \oplus ξ_{m}^{'},

where each ξ_i and $ξ_{j}^{'}$ are simple ideal bundles. Consider $\oplus_{1}^{n} h_{1} {(ξ_{i})}_{x}$ ⁠, where $h_{1} {(ξ_{i})}_{x} = {T \in h (ξ_{x}) | T (ξ_{x}) \subseteq {(ξ_{i})}_{x}}$ ⁠. We have $\oplus_{1}^{n} h_{1} {(ξ_{i})}_{x} \subseteq h (ξ_{x})$ ⁠. Let L(a) ∈ h(ξ_x), then a can be written as $a = \oplus_{1}^{n} a_{i}, a_{i} \in {(ξ_{i})}_{x}$ ⁠. Since ${(ξ_{i})}_{x}$ is an ideal in ξ_x, L(a_i) maps ξ_x into ${(ξ_{i})}_{x}$ ⁠. So

L (a) = \oplus_{i = 1}^{n} L (a_{i}) \in \oplus_{1}^{n} h_{1} {(ξ_{i})}_{x},

Let [L(a), L(b)] ∈ h(ξ_x), where $a = \oplus_{1}^{n} a_{i}$ and $b = \oplus_{1}^{n} b_{i}$ ⁠, $a_{i}, b_{i} \in {(ξ_{i})}_{x}$ ⁠. Since $[L (a_{i}), L (b_{j})] (u) \in {(ξ_{i})}_{x} \cap {(ξ_{j})}_{x} = 0$ for i ≠ j, we obtain that

[L (a), L (b)] = \oplus_{1}^{n} [L (a_{i}), L (b_{j})] \in \oplus_{1}^{n} h_{1} {(ξ_{i})}_{x} .

Consequently, $h (ξ_{x}) \subseteq \oplus_{1}^{n} h_{1} {(ξ_{i})}_{x}$ ⁠. Therefore, h(ξ) = h₁(ξ₁) ⊕ h₁(ξ₂) ⊕⋯ ⊕ h₁(ξ_n). Similarly, $h (ξ) = \oplus_{1}^{n} h_{1} (ξ_{j}^{'})$ ⁠. Hence

\begin{array}{l} L (ξ) & = L_{1} (ξ_{1}) \oplus L_{1} (ξ_{2}) \oplus \dots \oplus L_{1} (ξ_{n}) \\ = L_{1} (ξ_{1}^{'}) \oplus L_{1} (ξ_{2}^{'}) \oplus \dots \oplus L_{1} (ξ_{m}^{'}) \end{array}

where L₁(ξ_i) = h₁(ξ_i) ⊕ ξ_i ⊕ ξ_i and $L_{1} (ξ_{j}^{'}) = h_{1} (ξ_{j}^{'}) \oplus ξ_{j}^{'} \oplus ξ_{j}^{'}$ 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 $L_{1} (ξ_{j}^{'})$ ⁠. Then, obviously $ξ_{j}^{'}$ coincides with ξ_i except for the order.□

The author would like to thank the referee for constructive remarks that improve the presentation of the paper and for spotting several errors in the previous version of the paper. Also, the author would like to thank REVA University for its continuous support and encouragement.

References

Kumar

On Wedderburn principal theorem for Jordan algebra bundles

Commun Algebra

2021

;

(

1431

Google Scholar

Crossref

Jordan

Uber eine Klasse nichtassoziatiever hyperkomplexen Algebren

Gott Nachr

;

1932

569

Google Scholar

Jordan

Uber Verallgemeinerungsm oglichkeiten des Formalismus der Quantenmechanik, Gott

Nachr

;

1933

209

Google Scholar

Jordan

em Uber die Multiplikation quantenmechanischen Grossen

Z Phys

1933

;

285

Google Scholar

Crossref

McCrimmon

Jordan algebras and their applications

Bull Amer Math Soc

1978

;

612

Google Scholar

Crossref

Prema

Kiranagi

Lie algebras bundles defined by Jordan algebra bundles

Bull Math Soc Sci Math Roumanie

1987

;

(

255

Google Scholar

Koecher

Imbedding of Jordan algebras into Lie algebras-I

Amer J Math

1967

;

787

816

Google Scholar

Crossref

Atiyah

K-Theory

New York, Amsterdam

W.A.Benjamin, Inc.

;

1967

Google Scholar

Finston

Rigidity and compact real forms of semisimple complex Jordan algebras

Commun Algebra

1990

;

(

3323

Google Scholar

Crossref

Arens

Topologies for homeomorphism groups

Amer J Math

1946

;

593

610

Google Scholar

Crossref

Steenrod

The topology of fibre bundles

Princeton, New Jersy

Princeton University Press

;

1974

Google Scholar

Cohn

Lie Groups

Cambridge

Cambridge University Press

;

1965

Google Scholar

Schafer

Introduction to non-associative algebras

New York

Acd. Press

;

1966

Google Scholar

Kiranagi

Semisimple Lie algebra bundles

Bull Math Soc Sci Math Roumanie

1983

;

(

253

Google Scholar

Structure theorem for Jordan algebra bundles

1. Introduction

1.1 Notations and terminology

2. Preliminaries

3. Semisimple Jordan algebra bundles

4. Decomposition theorem for Jordan algebra bundle

References

Further reading

Data & Figures

Contents

Supplements

References

New and popular articles

Email Alerts

Cited By

Languages

Structure theorem for Jordan algebra bundles

1. Introduction

1.1 Notations and terminology

2. Preliminaries

3. Semisimple Jordan algebra bundles

4. Decomposition theorem for Jordan algebra bundle

References

Further reading

Data & Figures

Contents

Supplements

References

Related

New and popular articles

Email Alerts

Recommended for you

Cited By

Languages

Sharing Unavailable