A Four-Dimensional Algebra Construction Adapted to Fibre Bundle Topology and Its Structural Correspondence to Fundamental Interactions

Author: Zhang Suhang, Luoyang, Henan

Abstract

This paper proposes a four-dimensional algebraic labelling tool for the “base manifold–fibre” layered topology. We construct a real four-dimensional associative algebra \mathcal A=\operatorname{span}_{\mathbb R}\{e_0,e_1,e_2,e_3\}, where e_0 is the multiplicative identity. The three nilpotent basis elements e_1,e_2,e_3 satisfy e_1^2=e_2^2=e_3^2=0, with only one set of non-commutative couplings retained: e_1e_2=e_3,\ e_2e_1=-e_3. While formally adopting the “one-real, three-imaginary” layout, this algebra inherits no multiplication rules from Hamilton’s quaternions. A complete multiplication table and associativity verification are provided. We only discuss its correspondence to the layered structure of fibre bundles in the sense of labelling and stratification. This paper does not formulate physical field theories, derive field equations, or claim group isomorphism or the unification of fundamental interactions.

Keywords: fibre bundle; layered topology; four-dimensional algebra; associative algebra; structural correspondence; exploratory modelling

1. Introduction

The “base manifold–fibre” layered structure of fibre bundles furnishes a topological language for describing the hierarchy of spacetime geometry and gauge degrees of freedom. Among classical algebraic tools, quaternions feature a distinctive “one-real, three-imaginary” form. Nevertheless, their built-in multiplication rules (imaginary units squared to -1, cyclic non-commutativity) are predefined structures and are not suitable as the layered labels required here.

This work does not modify quaternions, nor does it claim that the new algebra outperforms existing ones. Instead, we define a four-dimensional associative algebra from scratch, supply its full multiplication table, verify associativity, and then examine its utility as a labelling tool for fibre bundle topology.

2. Labelling Requirements for the Layered Structure of Fibre Bundles

We adopt only the following layered description of fibre bundles:

1. Base manifold B: the spacetime substrate, marking the geometric tier where gravity resides;

2. Fibre F: an intrinsic space attached at each point of B, carrying gauge degrees of freedom. Gauge fields are described by connections on principal bundles;

3. The fibre is partitioned into sectors according to gauge groups: the internal gauge group of the Standard Model is \mathrm{SU}(3)_C\times \mathrm{SU}(2)_L\times \mathrm{U}(1)_Y, and gravity is not incorporated into this internal gauge group.

The algebraic labelling tool we seek shall satisfy these criteria:

- one “real” direction corresponding to the base-manifold tier;

- three “imaginary” directions serving as labels for the three gauge sectors;

- a clear formal distinction between the real and imaginary directions;

- limited algebraic couplings permitted among the imaginary directions, which shall not be interpreted as fibre-bundle connections;

- multiplication must be associative, with rules amenable to line-by-line verification.

3. Full Construction of the Four-Dimensional Algebra

3.1 Vector Space and Basis

Define a real four-dimensional vector space:

\mathcal A=\operatorname{span}_{\mathbb R}\{e_0,e_1,e_2,e_3\}.

Any element of the algebra takes the form:

q=a e_0+b e_1+c e_2+d e_3,\qquad a,b,c,d\in\mathbb R.

The term a e_0 is referred to as the real part, and b e_1+c e_2+d e_3 as the imaginary part. The labels “real/imaginary” are purely formal and carry no inherited meaning from Hamilton’s quaternions.

3.2 Complete Multiplication Table

e_0 is stipulated as the identity element:

e_0e_i=e_ie_0=e_i,\qquad i=0,1,2,3.

Only one non-commutative coupling is preserved among the imaginary basis elements:

e_1e_2=e_3,\qquad e_2e_1=-e_3,

and all remaining products of imaginary basis elements vanish:

e_1^2=e_2^2=e_3^2=0,\quad e_1e_3=e_3e_1=e_2e_3=e_3e_2=0.

Multiplication is extended bilinearly over the entire algebra \mathcal A.

Multiplication table for basis elements:

Left   Right         

3.3 Associativity Check

We verify the identity (e_ie_j)e_k=e_i(e_je_k) for all basis elements.

Cases involving e_0 follow immediately from the definition of multiplicative identity. We only examine combinations without e_0 below:

1. If e_je_k=0, the right-hand side e_i(e_je_k)=0. On the left-hand side (e_ie_j)e_k, e_ie_j can only be \pm e_3 when non-zero, and e_3e_k=0, so the left-hand side is also zero.

2. If e_je_k\neq0, only the pairs (j,k)=(1,2) and (2,1) are possible. Take (j,k)=(1,2): the right-hand side e_i e_3=0. The left-hand side (e_i e_1)e_2 evaluates to zero for i=1,2,3, since e_i e_1 is either 0 or -e_3, and multiplying by e_2 yields zero. The case (j,k)=(2,1) is analogous.

3. If e_ie_j=0, the left-hand side is zero. On the right-hand side e_i(e_je_k), a non-zero result would require e_je_k=\pm e_3, yet e_i e_3=0, so the right-hand side is also zero.

Therefore associativity holds for all basis elements. \mathcal A is a unital four-dimensional associative algebra over \mathbb R. Its imaginary ideal

I=\operatorname{span}\{e_1,e_2,e_3\}

satisfies I^2\subseteq I and I^3=0; it is a nilpotent ideal.

3.4 Comparison with Hamilton’s Quaternions

Item Hamilton’s quaternions   Algebra   in this paper 

Squares of imaginary basis     

Multiplication of imaginary basis Cyclic non-commutativity:   Only  ; all other imaginary products vanish 

Division property Division algebra Contains zero divisors, not a division algebra 

Relation to quaternions — Only adopts the “one-real, three-imaginary” form; no multiplication rules inherited 

This algebra merely borrows the “one-real, three-imaginary” notation and does not inherit any multiplicative structure from Hamilton’s quaternions; it is not a quaternion algebra.

4. Structural Correspondence as Layered Labels

4.1 Correspondence Scheme

Correspondence is established solely in the labelling and stratification sense:

- Basis e_0: label for the base-manifold tier;

- Basis e_1,e_2,e_3: labels for the three gauge sectors inside the fibre;

- Coupling relations e_1e_2=e_3,\ e_2e_1=-e_3: mark the existence of limited algebraic couplings between imaginary sectors. This coupling shall not be interpreted as a fibre-bundle connection, nor does it describe gauge-field dynamics.

This algebraic framework is primarily designed for marking, classification and structured organisation of layered information on local spacetime, enabling unified encapsulation of the hierarchical relations among fundamental interactions. When precise quantitative physical calculations are required, individual components may be extracted from algebraic elements and fed into the formalism of differential geometry and gauge field theory for further computation and numerical evaluation.

4.2 Explicit Disclaimers

- We do not claim that \mathcal A is isomorphic to the gauge group \mathrm{U}(1)\times \mathrm{SU}(2)\times \mathrm{SU}(3). The dimension of the corresponding Lie algebra satisfies

\dim\mathfrak u(1)+\dim\mathfrak{su}(2)+\dim\mathfrak{su}(3)=1+3+8=12,

which is far larger than the three-dimensional imaginary subspace, so it cannot accommodate the full gauge Lie algebra.

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- We do not claim that e_1,e_2,e_3 are equivalent to electromagnetic, weak and strong interactions respectively; they are merely sector labels.

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- We do not equate algebraic multiplication with principal-bundle connections. A connection is a 1-form defined on a manifold; the two belong to distinct mathematical objects.

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- We do not assert that this algebra is unique or optimal as a layered labelling tool. Its utility remains open to subsequent testing.

5. Scope and Boundary of the Model

1. This is an exploratory mathematical construction: a verifiable four-dimensional associative algebra is presented with associativity proof, and its applicability as a labelling tool for layered structures is discussed.

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2. No action functional is constructed, no field equations derived, no physical observables predicted, and no experimental tests are proposed.

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3. This paper makes no claim of unifying the four fundamental interactions and takes no stance on open questions concerning the quantisation of gravity.

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4. It does not contradict or supersede general relativity and the Standard Model of particle physics.

6. Conclusion

We construct a unital four-dimensional associative algebra \mathcal A over the real field from scratch, supply its complete multiplication table and verify associativity. Formally retaining the “one-real, three-imaginary” layout while adopting nilpotent imaginary basis elements with only one non-commutative coupling relation, the algebra is fully decoupled from Hamilton’s quaternions. We only discuss its correspondence to the “base manifold–fibre” structure of fibre bundles in the labelling sense, without asserting Lie-group isomorphism or physical unification. Future work may examine its applicability in structured classification and formal modelling.

References

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[3] K.Z. Huang. Fibre Bundles and Gauge Field Theory[M]. Higher Education Press, 2018.
[4] S. Weinberg. The Quantum Theory of Fields, Volume I[M]. Cambridge University Press, 2005.