Modules over the Biquaternion Algebra

Introduction

This article classifies the modules over the biquaternion algebra and describes the one module that the framework uses, the defining module on which the algebra acts as $M_2(\mathbb{C})$. It uses the idempotents and the Peirce decomposition of Biquaternion Ideals and Peirce Decomposition, the matrix model of The 2×2 Matrix Element Representation of Biquaternions, and the invertibility criterion of Biquaternion Norm and Invertibility. The group-theoretic representations of the algebra and of its group of units are The Lorentz Group in Biquaternionic Form — Structure and Representations, and the spinor reading of the module is Biquaternion Spin Geometry.

Scope. This is the module-theoretic article of the Algebra slot. It classifies modules and computes their endomorphisms; it does not treat the representations of the unit group, the norm, or the Clifford structure. The physical reading is attached at the point of each algebraic statement, and it is a reading.

Conventions. The algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}\cong M_2(\mathbb{C})$, with basis $e_0=1,e_1,e_2,e_3$, central scalar imaginary $i$, and $e_k^2=-e_0$. The idempotents are $\tilde\Pi_1=\tfrac12(e_0+ie_3)$ and $\tilde\Pi_2=e_0-\tilde\Pi_1$, with $\tilde R=\tfrac12(e_1+ie_2)$ and $\tilde T=\tfrac12(e_1-ie_2)$ the transition elements. Modules are left modules unless stated, and $\dim_\mathbb{C}$ is the complex dimension.

The Defining Module

Two orthogonal idempotents generate the two columns of the algebra. With $\tilde\Pi_2=e_0-\tilde\Pi_1$,

$$ \tilde\Pi_1^2=\tilde\Pi_1,\qquad \tilde\Pi_2^2=\tilde\Pi_2,\qquad \tilde\Pi_1\tilde\Pi_2=\tilde\Pi_2\tilde\Pi_1=0,\qquad \tilde\Pi_1+\tilde\Pi_2=e_0, $$

and the four Peirce corners of $\mathbb{B}$ with respect to the pair are

$$ \tilde\Pi_1\mathbb{B}\tilde\Pi_1=\mathbb{C}\tilde\Pi_1,\qquad \tilde\Pi_1\mathbb{B}\tilde\Pi_2=\mathbb{C}\tilde R,\qquad \tilde\Pi_2\mathbb{B}\tilde\Pi_1=\mathbb{C}\tilde T,\qquad \tilde\Pi_2\mathbb{B}\tilde\Pi_2=\mathbb{C}\tilde\Pi_2, $$

each of complex dimension one. The minimal left ideal generated by $\tilde\Pi_1$ is therefore

$$ S:=\mathbb{B}\tilde\Pi_1=\mathbb{C}\{\tilde\Pi_1,\,\tilde T\}, $$

of complex dimension two, the defining module of $\mathbb{B}$. The second column $\mathbb{B}\tilde\Pi_2=\mathbb{C}\{\tilde\Pi_2,\,\tilde R\}$ is a second minimal left ideal, isomorphic to the first.

The left action of $\mathbb{B}$ on $S$ is computed on the basis $(\tilde\Pi_1,\tilde T)$: multiplication by the idempotent fixes $\tilde\Pi_1$ and kills $\tilde T$, and the matrix units exchange the columns, $\tilde R\tilde\Pi_1=0$, $\tilde T\tilde\Pi_1=\tilde T$, $\tilde R\tilde T=\tilde\Pi_1$, $\tilde T^2=0$. On the algebra basis the action is

$$ \Phi(e_0)=I,\qquad \Phi(e_1)=\begin{pmatrix}0&-i\\-i&0\end{pmatrix},\qquad \Phi(e_2)=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad \Phi(e_3)=\begin{pmatrix}-i&0\\0&i\end{pmatrix},\qquad \Phi(i)=iI, $$

so that $\Phi(e_k)=-i\sigma_k$ for the Pauli matrices. The assignment $\Phi$ is an algebra isomorphism $\mathbb{B}\to M_2(\mathbb{C})$, and $S$ is the defining representation of $M_2(\mathbb{C})$ on $\mathbb{C}^2$ in the idempotent basis. Nothing depends on the choice of idempotent: all minimal left ideals are isomorphic and are indexed by the projective line $\mathbb{P}^1(\mathbb{C})$.

Theorem. $S$ is a simple left $\mathbb{B}$-module, and up to isomorphism it is the only one.

Proof. In the matrix model $S=\mathbb{C}^2$ with $\mathbb{B}\cong M_2(\mathbb{C})$ acting on column vectors: if $s\neq0$ the matrix units send $s$ to a basis of $\mathbb{C}^2$, so $\mathbb{B}s=\mathbb{C}^2$ and $S$ has no nonzero proper submodule. For uniqueness, $\mathbb{B}$ is a simple artinian ring with one isotypic component, and a full matrix algebra has exactly one simple module, its column space.

Physical reading. The defining module is the one-particle module, and the two minimal left ideals are its two chiralities. The algebra acts on $S=\mathbb{C}^2$ exactly as $M_2(\mathbb{C})$ on two-component spinors, so an element of $S$ is a Weyl spinor. The statement that the two columns are isomorphic says that the two chiralities carry the same module structure; they are exchanged by the matrix units, which is why a mass term, which couples the two, can be written inside the algebra. The module is not where the four-vectors live: the material four-position $\tilde{Q}=ict\,e_0+\mathbf{x}$ is an element of the algebra that acts on $S$, so the time coordinate $ict$ belongs to the algebra and not to the module, which carries no time of its own.

The Category of Left Modules

Theorem. Let $M$ be a left $\mathbb{B}$-module. Then $M$ is a direct sum of copies of $S$, $$ M\cong S^{\oplus k}, $$ with $k$ an integer when $M$ is finitely generated. Every $\mathbb{B}$-module is projective. The left regular module is free of rank one and decomposes as ${}_\mathbb{B}\mathbb{B}\cong S\oplus S$, the two summands being the two columns.

Proof. The algebra is $M_2(\mathbb{C})$, simple and artinian, hence semisimple; over a semisimple ring every module is a direct sum of simple modules, every module is projective, and the simple modules are the columns of the matrix algebra, of which there is one isomorphism class. The decomposition of the regular module is the Peirce decomposition read column by column.

For a finite-dimensional module the invariant $k$ is recovered from $\dim_\mathbb{C}S^{\oplus k}=2k$, so a finite-dimensional module is $\mathbb{C}^{2k}$ on which $\mathbb{B}$ acts block-diagonally as $k$ copies of the defining action.

Physical reading. The regular module decomposing as $S\oplus S$ is the statement that one biquaternion carries two spinors: the algebra acting on itself splits into the two chiralities. A general module $S^{\oplus k}$ is $k$ one-particle states with the same chirality, and the classification says that a biquaternion module is nothing but a complex vector space with a fixed labelling by the algebra — no further structure is added.

A caution on tensor products. For a non-commutative algebra the tensor product of two left $\mathbb{B}$-modules is not naturally a left $\mathbb{B}$-module: the two actions compete on the shared algebra, and only a diagonal action survives. The tensor-product ring structure therefore belongs to the group-theoretic side, treated in The Lorentz Group in Biquaternionic Form — Structure and Representations; the module classification above uses direct sums only.

Projectivity and the Parity of Freeness

Proposition. The module $S^{\oplus k}$ is free if and only if $k$ is even, equivalently if and only if $\dim_\mathbb{C}S^{\oplus k}=2k$ is divisible by $4$: $$ S^{\oplus k}\text{ is free}\iff 2\mid k\iff 4\mid\dim_\mathbb{C}S^{\oplus k}. $$

Proof. Since $\mathbb{B}\cong S\oplus S$ as left modules, $\mathbb{B}^{\oplus m}\cong S^{\oplus 2m}$; the free modules are therefore exactly the modules $S^{\oplus k}$ with $k$ even.

In particular $S$ itself is projective — every module is — but not free: a two-dimensional complex module with no basis over $\mathbb{B}$. The obstruction to freeness is a parity, and the same parity reappears as the obstruction to a real structure. This is the sharpest contrast with the module theory over the division ring $\mathbb{H}$, where every module is free.

Physical reading. Projective but not free is why a single Weyl spinor is not a wave function: it is a module that cannot be given a biquaternion basis, and a biquaternion basis is what a free module, a full state, requires. The parity is physical: one chirality is projective, a pair of chiralities is free, and it is the pair — a massive Dirac field, which needs both — that admits a basis. The framework's fermions are therefore chiral by module theory, not by a postulate.

Endomorphisms and the Standard Bimodule

Because $S$ is simple, Schur's lemma gives $\operatorname{End}_\mathbb{B}(S)=\mathbb{C}$, the scalars acting by the complex structure. A $\mathbb{B}$-linear map $S^{\oplus m}\to S^{\oplus n}$ is determined by the images of the $m$ summands, so

$$ \operatorname{Hom}_\mathbb{B}\bigl(S^{\oplus m},S^{\oplus n}\bigr)\cong M_{n\times m}(\mathbb{C}),\qquad \operatorname{End}_\mathbb{B}\bigl(S^{\oplus k}\bigr)\cong M_k(\mathbb{C}), $$

with composition the matrix product; the automorphisms of $S^{\oplus k}$ are therefore $\mathrm{GL}_k(\mathbb{C})$. The double centralizer is the mirror statement,

$$ \operatorname{End}_\mathbb{C}(S)=\mathbb{B},\qquad \{\tilde A\in\mathbb{B} : \tilde As=s\tilde A\text{ for all }s\in S\}=\mathbb{C}. $$

The module $S$ is consequently a bimodule ${}_\mathbb{B}S_\mathbb{C}$: the left action is the algebra action, the right action is multiplication by the central scalars, and the two commute because $\mathbb{C}$ is the centre.

The dual is not a new module. With $S^*=\operatorname{Hom}_\mathbb{C}(S,\mathbb{C})$ carrying the contragredient action, quaternion conjugation identifies $\mathbb{B}^{\mathrm{op}}$ with $\mathbb{B}$, and $S^*\cong S$. In the matrix model the identification is the adjugate, and with $\varepsilon=i\sigma_2$ one has, for every $\tilde{Q}\in\mathbb{B}$,

$$ \Phi\bigl(\tilde{Q}^{\natural}\bigr)=\varepsilon\,\Phi(\tilde{Q})^{\mathrm{T}}\,\varepsilon^{-1}, $$

so quaternion conjugation on $\mathbb{B}$ is transposition in the basis selected by $\varepsilon$: the row space is the dual of the column space, the same module $S$.

Physical reading. The endomorphism algebra being $\mathbb{C}$ says that the only intertwiners of the one-particle module are the complex scalars, which is the algebraic form of a superselection rule for the phase. The double centralizer says the converse: the algebra is recovered from the module exactly, which is why the defining module is a faithful carrier of the whole structure. And $S^*\cong S$ through quaternion conjugation is the algebraic content of the Dirac adjoint: the dual of a spinor is again a spinor, and the conjugation that identifies them is the one that defines the invariant form.

Morita Equivalence with the Complex Field

The pair $(\mathbb{B},\mathbb{C})$ is a Morita pair, and the equivalence is implemented by $S$: the functors $M\mapsto S\otimes_\mathbb{B}M$ and $V\mapsto\operatorname{Hom}_\mathbb{C}(S,V)$ give an equivalence of categories $$ \operatorname{Mod}(\mathbb{B})\cong\operatorname{Mod}(\mathbb{C}), $$ mapping the defining module $S$ to $\mathbb{C}$. Every module over $\mathbb{B}$ is therefore a complex vector space in disguise, and every complex dimension is doubled when it is read as a biquaternion module.

Physical reading. The Morita equivalence is the algebraic reason the framework's linear algebra is complex linear algebra: a biquaternion module is a complex vector space together with the identification $\mathbb{B}\cong M_2(\mathbb{C})$, and the identification is the whole of the extra structure. It is why the representation theory of the framework reproduces the complex representation theory of quantum mechanics, with the doubling as the spin degree of freedom, and why no new linear invariants appear: the module category is the complex one. The field $\mathbb{C}$ of the equivalence is the centre $\mathbb{C}e_0$ of the algebra, the scalar line that carries the two times $ct'+ict$ of the physical dictionary.

Torsion

Proposition. Every module over a semisimple ring is torsion-free. In particular $\operatorname{Mod}(\mathbb{B})$ contains no torsion: if $\tilde A m=0$ with $\tilde A$ a nonzero central element, then $m=0$.

Physical reading. There are no "hidden" relations in the framework's linear structure: a state cannot be annihilated by a nonzero central scalar, so the phase acts freely and the complex structure cannot be degenerate. Non-degeneracy of the module is the algebraic counterpart of the absence of a superselection rule for the central imaginary $i$.

Real Structures

The real structures of the modules are exactly the free ones. A module $S^{\oplus k}$ admits a real structure compatible with the algebra action if and only if $k$ is even, so that $S^{\oplus k}\cong\mathbb{H}^{\oplus m}$ after extension of scalars from $\mathbb{H}$. The modules obtained from the real quaternion algebra by extension of scalars are therefore the free ones, and the obstruction is the same parity that obstructs freeness.

Physical reading. A real structure exists exactly when the module comes from the real quaternion algebra — when it is a pair of chiralities — and it fails for a single chirality. This is the module-theoretic form of the statement that a Weyl spinor is intrinsically complex and that a real (Majorana) spinor requires the doubling.

Summary

The biquaternion algebra has, up to isomorphism, exactly one simple left module, the defining module $S=\mathbb{B}\tilde\Pi_1=\mathbb{C}\{\tilde\Pi_1,\tilde T\}\cong\mathbb{C}^2$, on which $\mathbb{B}\cong M_2(\mathbb{C})$ acts through $\Phi(e_k)=-i\sigma_k$. Every left module is a direct sum $S^{\oplus k}$ and is projective; the left regular module is free of rank one and decomposes as $S\oplus S$. The module $S^{\oplus k}$ is free exactly when $k$ is even, so $S$ is projective but not free. The endomorphism algebra of $S$ is $\mathbb{C}$, the homomorphisms are matrices over $\mathbb{C}$, the double centralizer recovers $\mathbb{B}$, and the dual of $S$ is $S$ through quaternion conjugation. The pair $(\mathbb{B},\mathbb{C})$ is a Morita pair, so $\operatorname{Mod}(\mathbb{B})\cong\operatorname{Mod}(\mathbb{C})$ and the module theory is complex linear algebra with the labelling added. Physically $S$ is the one-particle module, the two minimal left ideals are the two chiralities, the projective-but-not-free dichotomy is why one chirality is not a wave function, and the doubling is the spin degree of freedom.

Summary of Notation

Symbol	Meaning
$\tilde\Pi_1,\tilde\Pi_2$	Orthogonal primitive idempotents; the two columns
$\tilde R,\tilde T$	Transition elements; the off-diagonal Peirce corners
$S=\mathbb{B}\tilde\Pi_1=\mathbb{C}\{\tilde\Pi_1,\tilde T\}$	Defining module; the unique simple left module; complex dimension $2$
$\Phi(e_k)=-i\sigma_k$	The action of the algebra on $S$; the isomorphism $\mathbb{B}\cong M_2(\mathbb{C})$
$S^{\oplus k}$	Every left module; projective for all $k$
$S^{\oplus k}$ free $\iff 2\mid k$	Parity of freeness; $\dim_\mathbb{C}=2k$
${}_\mathbb{B}\mathbb{B}\cong S\oplus S$	Regular module; the two chiralities
$\operatorname{End}_\mathbb{B}(S)=\mathbb{C}$	Schur; only the scalars intertwine
$\operatorname{End}_\mathbb{C}(S)=\mathbb{B}$	Double centralizer
${}_\mathbb{B}S_\mathbb{C}$	Standard bimodule; $S^*\cong S$
$\Phi(\tilde{Q}^{\natural})=\varepsilon\Phi(\tilde{Q})^{\mathrm{T}}\varepsilon^{-1}$	Quaternion conjugation as the adjugate; $\varepsilon=i\sigma_2$
$\operatorname{Mod}(\mathbb{B})\cong\operatorname{Mod}(\mathbb{C})$	Morita equivalence; the field is the centre $\mathbb{C}e_0$
$\tilde{Q}=ict\,e_0+\mathbf{x}$	Material four-position; an algebra element acting on $S$, not a module element