The Number Systems as Clifford Algebras

Introduction

The classical number systems over the real numbers — the complex numbers, the split complex numbers, the dual numbers, the quaternions, the split biquaternions and the biquaternions — are all Clifford algebras of quadratic forms, and the correspondence is not a collection of isolated coincidences but a single dictionary. Each system is the Clifford algebra of a one-, two- or three-dimensional form; the form is read off from the multiplication of the imaginary units; and the properties of the system that one usually establishes by hand — associativity, the existence of a conjugation, the multiplicativity of the norm, the presence of zero divisors — become corollaries of the classification of the preceding articles.

This article states that dictionary. The basic algebra of each number system is not repeated here: the complex numbers, the split complex numbers in Split Complex Algebra, the dual numbers in Dual Numbers Algebra, the quaternions, the split biquaternions, and the biquaternions. What is added here is the identification of each with a Clifford algebra, the reading of its norm from the Clifford form, and the reason the list stops where it does.

The conventions are those of the category. The Clifford algebra $\mathrm{Cl}_{p,q}$ has $p$ generators of square $+1$ and $q$ of square $-1$, so that $\mathrm{Cl}_{0,1}\cong\mathbb{C}$, $\mathrm{Cl}_{1,0}\cong\mathbb{D}$ and $\mathrm{Cl}_{0,2}\cong\mathbb{H}$; the even subalgebra is written $\mathrm{Cl}^0$; and the low-dimensional identifications are those of The Low-Dimensional Classification. The number systems are written $\mathbb{R},\mathbb{C},\mathbb{D},\mathbb{D}',\mathbb{H},\mathbb{H}_{\mathbb{D}},\mathbb{B}$, with $\mathbb{D}$ the split complex numbers (unit $j$, $j^2=+1$), $\mathbb{D}'$ the dual numbers (unit $\varepsilon$, $\varepsilon^2=0$), $\mathbb{H}$ the quaternions (units $e_1,e_2,e_3$, $e_k^2=-1$), $\mathbb{H}_{\mathbb{D}}=\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$ the split biquaternions, and $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ the biquaternions.

The Dictionary

Definition. A number system in the sense of this article is a finite-dimensional unital associative real algebra $A$ with a distinguished subspace $V$ of generators, an involution ${}^{\natural}$ fixing the scalars, and a quadratic form $N$ on $A$ such that $N(a)=a a^{\natural}$ is multiplicative.

Every such algebra is a Clifford algebra, and the correspondence is given by the following dictionary.

number system	Clifford algebra	generating form
$\mathbb{R}$	$\mathrm{Cl}_{0,0}$	empty
$\mathbb{C}$	$\mathrm{Cl}_{0,1}$	$q=-\mathrm{id}$ on $\mathbb{R}$
$\mathbb{D}$	$\mathrm{Cl}_{1,0}$	$q=+\mathrm{id}$ on $\mathbb{R}$
$\mathbb{D}'$	$\mathrm{Cl}(\mathbb{R},0)$	degenerate, $q=0$
$\mathbb{H}$	$\mathrm{Cl}_{0,2}$	$q=-(\text{quaternion norm})$ on the imaginary part
$\mathbb{H}_{\mathbb{D}}=\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$	$\mathrm{Cl}_{0,3}\cong\mathbb{H}\times\mathbb{H}$	$q=-\mathrm{id}$ on $\mathbb{R}^3$
$M_2(\mathbb{R})$	$\mathrm{Cl}_{2,0}=\mathrm{Cl}_{1,1}$	$\operatorname{diag}(+1,+1)$ or $\operatorname{diag}(+1,-1)$
$\mathbb{B}$	$\mathrm{Cl}_{3,0}=\mathbb{C}\mathrm{l}_2$	$q=+\mathrm{id}$ on $\mathbb{R}^3$

The entries are the low-dimensional classification, read from the number-system side. The one-dimensional cases have the two signs of the form; the two-dimensional cases have the three forms of signature $(2,0)$, $(1,1)$ and $(0,2)$, which are $M_2(\mathbb{R})$, $M_2(\mathbb{R})$ and $\mathbb{H}$; and the three-dimensional positive definite case is the complex $2\times2$ matrix algebra, which is the biquaternion algebra.

Remark. The row of $\mathbb{H}_{\mathbb{D}}$ in the dictionary records an isomorphism of algebras, and the isomorphism deserves a word, because the presentation of $\mathbb{H}_{\mathbb{D}}$ as a tensor product does not exhibit it as the Clifford algebra of the form of its tensor-product generators. The generators $j\otimes1$ and $1\otimes e_k$ have squares $+1$ and $-1$, so the form they span is $\operatorname{diag}(+1,-1,-1,-1)$; the Clifford algebra of that $(1,3)$-form is the graded tensor product $\mathrm{Cl}_{1,0}\,\hat{\otimes}\,\mathrm{Cl}_{0,3}\cong\mathrm{Cl}_{1,3}\cong M_2(\mathbb{H})$, which is not $\mathbb{H}_{\mathbb{D}}$, since the algebra of a tensor product of forms is the graded tensor product while the ordinary tensor product $\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}\cong\mathbb{H}\times\mathbb{H}$ is a different algebra. The isomorphism of the dictionary is with $\mathrm{Cl}_{0,3}\cong\mathbb{H}\times\mathbb{H}$, realised on a different generating subspace of $\mathbb{H}_{\mathbb{D}}$; the presentation by $j$ and the $e_k$ describes the algebra, not the Clifford algebra with those generators.

Remark. The symbol $\mathbb{H}_{\mathbb{D}}$ denotes the tensor product $\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$, of real dimension eight, as fixed by the shared conventions of the category. The four-dimensional algebra $\mathrm{Cl}_{1,1}\cong M_2(\mathbb{R})$ is also called the split quaternions in the classical literature, since it is the split form of $\mathbb{H}$ in the sense of composition algebras; the two algebras are not isomorphic and the corpus reserves the name $\mathbb{H}_{\mathbb{D}}$, the split-biquaternions, for the eight-dimensional one. The dictionary lists both, the four-dimensional algebra under its matrix description and the eight-dimensional one under its tensor-product description.

Remark. The form of the Clifford algebra is not the norm of the number system; it is the norm with the opposite sign on the generators. For $\mathbb{C}$ the norm is $N(x+ye)=x^2+y^2$, positive definite, while the Clifford form is $q(e)=-1$; for $\mathbb{H}$ the norm is the sum of squares on the coefficients, while $q(e_k)=-1$. The reason is that the Clifford relation is $e^2=q(e)$, whereas the number system has $e^2=-1$ for an imaginary unit, so the two differ by a sign and one writes $q=-N$ on the span of the generators. The dictionary above records the Clifford form, and the norm is recovered as the Clifford form with the sign reversed on the generators and extended multiplicatively.

The One-Dimensional Systems

The complex numbers. With one generator $e$ of square $-1$, the algebra is $\mathrm{Cl}_{0,1}=\mathbb{R}[e]/(e^2+1)=\mathbb{C}$. The Clifford form is $q(e)=-1$; the norm is $N(x+ye)=x^2+y^2$, multiplicative, and non-vanishing on nonzero elements, so $\mathbb{C}$ is a division algebra. The conjugation ${}^{\natural}$ is the Clifford conjugation $e^{\natural}=-e$, the norm satisfies $N(z)=z\bar z$, and the units are the nonzero elements.

The split complex numbers. With one generator $j$ of square $+1$, the algebra is $\mathrm{Cl}_{1,0}=\mathbb{R}[j]/(j^2-1)=\mathbb{D}=\mathbb{R}\times\mathbb{R}$, with the idempotents $\pi_\pm=\tfrac12(1\pm j)$. The Clifford form is $q(j)=+1$. The norm $N(x+yj)=x^2-y^2$ is indefinite and vanishes on the isotropic lines $x=\pm y$, so $\mathbb{D}$ is not a division algebra; the zero divisors are the nonzero multiples of the idempotents. The appearance of the zero divisors is thus exactly the statement that the form of the Clifford algebra is indefinite, or equivalently that the algebra is a nontrivial direct product.

The dual numbers. The degenerate case is obtained by taking the form $q=0$ on a one-dimensional space. The Clifford algebra $\mathrm{Cl}(\mathbb{R},0)$ is the exterior algebra on the generator, $\mathbb{R}[\varepsilon]/(\varepsilon^2)$ with $\varepsilon^2=0$, which is the algebra of dual numbers $\mathbb{D}'$. The radical of the form is the whole space, the algebra is not semisimple, and the "norm" degenerates: $N(a+b\varepsilon)=a^{2}$ vanishes on the line $\mathbb{R}\varepsilon$, so its radical is one-dimensional and the form is not non-degenerate. The dual numbers are therefore the Clifford algebra of a degenerate form, and they are the smallest model of the radical phenomena of the degenerate case: they are not a composition algebra, because the norm is degenerate.

Remark. The three one-dimensional systems correspond to the three orbits of the pair $(\dim V,q)$ with $\dim V=1$: the form can be negative definite, positive definite, or zero. The classification is thus complete in dimension one, and it is the sharpest illustration of the fact that the Clifford algebra detects the order of the form and not only its rank.

The Two-Dimensional Systems

The quaternions. With two generators $e_1,e_2$ of square $-1$ and anticommuting, the algebra is $\mathrm{Cl}_{0,2}\cong\mathbb{H}$, with the quaternion units $e_1,e_2,e_3=e_1e_2$ of the category convention; this is the computation of The Low-Dimensional Classification. The Clifford form is $q=-\mathrm{id}$ on the imaginary part, the norm is $N=x_0^2+x_1^2+x_2^2+x_3^2$, positive definite and multiplicative, and $\mathbb{H}$ is a division algebra. The Clifford conjugation ${}^{\natural}$ is the quaternion conjugation, and the group of units is $\mathbb{H}^{\times}=\mathbb{R}_{>0}\times Sp(1)$, with $Sp(1)$ the unit quaternions.

The commutative counter-case. The hypothesis of anticommutativity in the previous paragraph is not redundant, and dropping it gives an algebra that is not a Clifford algebra at all. Let two generators with square $-1$ commute: the algebra is $\mathbb{R}[i,e]/(i^2+1,\,e^2+1)$, of real dimension four, commutative, and isomorphic to $\mathbb{C}\oplus\mathbb{C}$ through the idempotents $\tfrac12(1\pm ie)$, the reduced biquaternion algebra of List of Algebras by Dimension. It is deliberately absent from the dictionary: $\mathrm{Cl}_{0,2} \cong \mathbb{H}$ and this algebra are generated by the same two elements with the same two squares and differ only in the sign of $e_1e_2$. The case is the sharpest instance of the difference between the two tensor products recorded for $\mathbb{H}_{\mathbb{D}}$ above — the Clifford algebra of a pair of forms is the graded tensor product, where the generators anticommute, while the ordinary tensor product of the same two algebras is commutative and is not a Clifford algebra at all. Equivalently, the dictionary's map sends anticommutativity to the Clifford relation, and every four-dimensional commutative algebra with the same generators lies outside its image.

The four-dimensional split quaternions. With generators $e_1$ of square $+1$ and $e_2$ of square $-1$, the algebra is $\mathrm{Cl}_{1,1}\cong M_2(\mathbb{R})$, the split quaternion algebra of the classical literature; it is the split form of $\mathbb{H}$ as a composition algebra and it is not the symbol $\mathbb{H}_{\mathbb{D}}$ of the corpus, which denotes the eight-dimensional $\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$. The Clifford form is $\operatorname{diag}(+1,-1)$, the norm is indefinite of signature $(2,2)$, and the algebra has zero divisors; the idempotents are the elements of rank one in the matrix model. The three two-dimensional forms are thus $M_2(\mathbb{R})$ for the signatures $(+1,-1)$ and $(+1,+1)$ and $\mathbb{H}$ for $(-1,-1)$, and the split quaternion algebra is the $(+1,-1)$ case.

The matrix algebra. The signature $(2,0)$ gives $\mathrm{Cl}_{2,0}\cong M_2(\mathbb{R})$, so the same algebra arises from two inequivalent forms. This is the two-dimensional instance of the general fact that $\mathrm{Cl}_{p,q}$ depends on the difference of the signatures modulo eight and not on the individual signs; the two forms are real forms of the same complexification and give the same algebra here because the classes $d\equiv2$ and $d\equiv0$ both lie in the real block of the eightfold table, where the algebra is a full matrix algebra over $F$ of the same size.

The Biquaternion Algebra

The three-dimensional positive definite form gives the algebra that the applications of this category use most.

Theorem. $\mathrm{Cl}_{3,0}\cong M_2(\mathbb{C})\cong\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, and an explicit isomorphism is $\gamma_k\mapsto ie_k$, where $\gamma_k$ are the Clifford generators of $\mathrm{Cl}_{3,0}$ (of square $+1$) and $e_k$ are the quaternion units.

Proof. The images $ie_k$ square to $(ie_k)^2=i^2e_k^2=(-1)(-1)=+1$ and anticommute, so the assignment extends to an algebra homomorphism; it is injective because the images generate the eight-dimensional algebra, and both sides are eight-dimensional.

Under this identification the quaternion units of the embedded copy are the bivectors of the Clifford algebra, $e_1=-\gamma_2\gamma_3$ and cyclically, and the central scalar imaginary of $\mathbb{B}$ is the volume element, $i\mapsto\gamma_1\gamma_2\gamma_3$. The Clifford form on the generators is $q(\gamma_k)=+1$, while the norm of the biquaternion algebra is the complex-valued form

$$ N(\tilde Q)=\tilde Q\tilde{Q}^{\natural}=\sum_{\mu=0}^{3}Q_\mu^2, $$

with $Q_\mu\in\mathbb{C}$. Writing $Q_\mu=a_\mu+ib_\mu$ this is $N=\sum_\mu(a_\mu^2-b_\mu^2)+2i\sum_\mu a_\mu b_\mu$, so the real part of the norm is a quadratic form of signature $(4,4)$ and the Hermitian form $\tilde Q\tilde{Q}^{*}$ has scalar part the positive definite form $\sum_\mu|Q_\mu|^2$. The biquaternion algebra is thus a system whose norm is complex-valued and indefinite in its real part, and it is not a division algebra; its zero divisors are not covered here.

Remark. The biquaternion algebra is a complex algebra and an eight-dimensional real algebra at once, and the Clifford identification fixes the complex structure: in $\mathrm{Cl}_{3,0}$ the volume element $\omega=\gamma_1\gamma_2\gamma_3$ is central with $\omega^{2}=-1$, so that $\mathbb{R}[\omega]\cong\mathbb{C}$ is the center and $\mathrm{Cl}_{3,0}$ is a complex algebra with $\omega$ in the role of the scalar imaginary; under the identification $\gamma_k\mapsto ie_k$ one has $\omega\mapsto i$, and $\mathrm{Cl}_{3,0}$, $\mathbb{B}$ and $M_2(\mathbb{C})$ are the same algebra in three presentations. The complexification is the larger algebra $\mathbb{C}\mathrm{l}_3=\mathrm{Cl}_{3,0}\otimes_{\mathbb{R}}\mathbb{C}\cong M_2(\mathbb{C})\times M_2(\mathbb{C})$, of complex dimension eight, and it is the complexification, not the biquaternion algebra, that has two simple factors. This is the algebraic origin of the appearance of the pair of conjugate spinor modules in the complexified picture, recorded in Real Spinors and Reality Conditions with Inner Conjugation.

Norms and Composition

The multiplicity of the norm is the composition property, and it is possible only in dimensions one, two, four and eight.

Definition. A composition algebra over $\mathbb{R}$ is a real algebra $A$ with a non-degenerate quadratic form $N$ satisfying

$$ N(xy)=N(x)N(y) \qquad (x,y\in A), $$

together with the associated bilinear form and the conjugation $x^{\natural}$ defined by $x x^{\natural}=N(x)1$.

Theorem (Hurwitz). A finite-dimensional composition algebra over $\mathbb{R}$ has dimension $1$, $2$, $4$ or $8$, and the four definite examples are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and the octonions $\mathbb{O}$. The corresponding forms are the sums of $1,2,4,8$ squares, and the multiplicity identities

$$ (x_1^2+\cdots+x_n^2)(y_1^2+\cdots+y_n^2)=z_1^2+\cdots+z_n^2 $$

with $z$ bilinear in $x$ and $y$ exist exactly for $n=1,2,4,8$.

Proof sketch. The identity is equivalent to the existence of a composition algebra of dimension $n$. Each composition algebra carries an involution $x\mapsto x^{\natural}$ defined by $x x^{\natural}=N(x)1$, and the algebra is obtained from the subspace fixed by the involution by the Cayley–Dickson doubling; since $\mathbb{R}$ is the only one-dimensional composition algebra, the possible dimensions are the powers of two, and the doubling is available only while the algebra remains a normed division algebra. Hurwitz's own proof writes the identity as $z=A(x)y$ with $A(x)$ an $n\times n$ real matrix depending linearly on $x$ and shows that the matrix equation $A(x)A(x)^{T}=N(x)I$ has a solution only for $n=1,2,4,8$; the octonionic case is the last, at which associativity is lost. The details are standard.

The Clifford algebras among the composition algebras. Of the four composition algebras, the first three are Clifford algebras: $\mathbb{R}=\mathrm{Cl}_{0,0}$, $\mathbb{C}=\mathrm{Cl}_{0,1}$, $\mathbb{H}=\mathrm{Cl}_{0,2}$. The octonions are not associative, and every Clifford algebra is associative, so $\mathbb{O}$ is not a Clifford algebra. The Clifford algebras of larger dimensions are matrix algebras or products of them and are therefore not division algebras; the four-fold and eight-fold phenomena are instead visible in the spinor representations of $\mathrm{Cl}_{0,3}$ and $\mathrm{Cl}_{0,8}$, not in a further multiplication on the algebra itself.

The split composition algebras. Replacing the definite form by an indefinite one gives the split versions: $\mathbb{D}$, the four-dimensional split quaternions $M_2(\mathbb{R})$ and the split octonions. They satisfy the same composition law but have isotropic vectors, so they have zero divisors and are not division algebras. In the Clifford language the first two are $\mathrm{Cl}_{1,0}$ and $\mathrm{Cl}_{1,1}$; the split octonions are not a Clifford algebra, since they are not associative, so the associative part of the dictionary stops at the split quaternions. The split forms of the composition algebras are exactly the ones whose norms are isotropic. The eight-dimensional algebra $\mathbb{H}_{\mathbb{D}}=\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}\cong\mathbb{H}\times\mathbb{H}$ is a further split system of the dictionary, obtained by tensoring the quaternions with the split complex numbers rather than by taking the split form of the composition law.

The Cayley–Dickson Doubling and Its Limit

The list of number systems that are Clifford algebras is generated by a doubling, and the doubling is the one that appears in the Clifford chain.

Theorem (Cayley–Dickson). Let $A$ be an algebra with an involution ${}^{\natural}$ and let $A\oplus A$ carry the product and conjugation

$$ (a,b)(c,d)=(ac-d^{\natural} b,\ a^{\natural} d+cb), \qquad \overline{(a,b)}=(a^{\natural},-b). $$

If $A$ is a composition algebra, that is $N(xy)=N(x)N(y)$ for $N(x)=x x^{\natural}$, then so is $A\oplus A$, and the doubling preserves the norm. Starting from $A=\mathbb{R}$ and doubling three times gives

$$ \mathbb{R}\longrightarrow\mathbb{C}\longrightarrow\mathbb{H}\longrightarrow\mathbb{O}, $$

where each step loses a property: the first doubling loses order, the second loses commutativity and the third loses associativity. The process stops at $\mathbb{O}$, by Hurwitz's theorem that a real composition algebra has dimension $1,2,4$ or $8$.

Proposition. The Clifford chain doubles in the same dimensions but does not lose associativity; it splits instead. The even subalgebra of $\mathrm{Cl}_{0,n+1}$ is $\mathrm{Cl}_{0,n}$,

$$ \mathrm{Cl}_{0,n+1}^{0}\cong\mathrm{Cl}_{0,n}, $$

so that $\mathrm{Cl}_{0,1}^{0}=\mathbb{R}$, $\mathrm{Cl}_{0,2}^{0}=\mathbb{C}$, $\mathrm{Cl}_{0,3}^{0}=\mathbb{H}$ and $\mathrm{Cl}_{0,4}^{0}=\mathrm{Cl}_{0,3}=\mathbb{H}\oplus\mathbb{H}$. The fourth step therefore produces the split algebra $\mathbb{H}\oplus\mathbb{H}$ rather than the octonions.

Proof. The even subalgebra of the Clifford algebra of a form of dimension $n+1$ is the Clifford algebra of the restriction of the form to a hyperplane, a standard isomorphism. Reading the chain gives the listed algebras; the last is $\mathrm{Cl}_{0,3}=\mathbb{H}\oplus\mathbb{H}$, a direct product, not a division algebra.

Corollary. The octonions are not a Clifford algebra: they are not associative, while every Clifford algebra is associative. The associative doubling of the number systems therefore stops at the quaternions, and the next Clifford algebra in the chain is the product of two quaternion algebras, $\mathbb{H}\oplus\mathbb{H}=\mathrm{Cl}_{0,3}=\mathbb{H}_{\mathbb{D}}$. The split versions of the systems — $\mathbb{D}=\mathrm{Cl}_{1,0}$ and the four-dimensional split quaternions $M_2(\mathbb{R})=\mathrm{Cl}_{1,1}$ — are obtained from the definite ones by changing the sign of one generator of the form, which keeps associativity but introduces zero divisors at the first indefinite step.

Remark. The dichotomy is worth stating plainly: the composition algebras double without bound in norm but lose their algebraic properties, whereas the Clifford algebras double without bound in associativity but eventually fail to be division algebras. The two doublings agree for the first three steps — $\mathbb{C}$, $\mathbb{H}$ and their Clifford counterparts — and diverge at the fourth, where the octonions and $\mathbb{H}\oplus\mathbb{H}$ appear. This is why the dictionary of this article has exactly the entries it has and why the biquaternion algebra, not the octonions, is the largest number system of the applications.

The Units and the Norm-One Groups

The groups of units of the number systems are the finite-dimensional Pin and Spin groups of the category.

Theorem. For the positive definite systems,

$$ \mathbb{C}^{\times}\cong\mathbb{R}_{>0}\times U(1), \qquad \mathbb{H}^{\times}\cong\mathbb{R}_{>0}\times Sp(1), \qquad \mathbb{B}^{\times}\cong GL(2,\mathbb{C}), $$

and the norm-one groups are

$$ \{z\in\mathbb{C}:N(z)=1\}=U(1)=\mathrm{Spin}(2), \qquad \{q\in\mathbb{H}:N(q)=1\}=Sp(1)=\mathrm{Spin}(3). $$

Proof. The norm is multiplicative, so $N$ is a group homomorphism from the units to $\mathbb{R}^{\times}$ (or to $\mathbb{C}^{\times}$ for $\mathbb{B}$), and its kernel is the norm-one group. The identifications of the kernels with $U(1)$ and $Sp(1)$ are immediate from the explicit forms; the spin-group identifications are those of the previous articles, in which the unit complex numbers double-cover $SO(2)$ and the unit quaternions double-cover $SO(3)$. For $\mathbb{B}$ the norm takes values in $\mathbb{C}$ and is not a division-algebra norm; the unit group is the group of invertible $2\times2$ complex matrices, and its norm-one subgroup is $SL(2,\mathbb{C})$.

Remark. The norm-one groups of the definite systems reproduce the compact spin groups of low dimension. The biquaternion unit group $\mathbb{B}^{\times}\cong GL(2,\mathbb{C})$ is not a spin group; its norm-one subgroup, cut out by $N(Q)=\sum_\mu Q_\mu^2=\det\Phi(Q)=1$, is $SL(2,\mathbb{C})$, the spin group of the signature $(1,3)$ form, and this is the content. For the biquaternions the algebra norm and the reduced norm of the matrix algebra coincide, since the norm is the determinant in the identification $\mathbb{B}\cong M_2(\mathbb{C})$; what fails, compared with $\mathbb{C}$ and $\mathbb{H}$, is the positive definiteness of the norm. The norm is complex-valued, so the norm-one group is not compact and is the six-dimensional $SL(2,\mathbb{C})$ rather than a compact spin group.

The Complexified Systems and the Odd Clifford Algebras

The dictionary of this article is over the real numbers, and the list stops before the octonions because they are not associative. Two further entries belong to the same dictionary and are recorded here: the complexifications of the systems, and the odd complex Clifford algebras they produce.

Proposition (complexification doubles at odd index). For every $n\ge 0$,

$$ \mathbb{C}\mathrm{l}_{2n+1} \cong \mathbb{C}\mathrm{l}_{2n} \oplus \mathbb{C}\mathrm{l}_{2n}, $$

the direct sum of two copies of the even complex Clifford algebra; equivalently $M_{2^n}(\mathbb{C}) \oplus M_{2^n}(\mathbb{C})$.

Proof. The volume element of an odd Clifford algebra is central; over $\mathbb{C}$ the generators can be rescaled so that it is an involution, and its two idempotents are central, orthogonal and complete, so they split the algebra. Equivalently, the classification of the complex Clifford algebras gives $\mathbb{C}\mathrm{l}_{2n}\cong M_{2^n}(\mathbb{C})$ and $\mathbb{C}\mathrm{l}_{2n+1}\cong M_{2^n}(\mathbb{C})\oplus M_{2^n}(\mathbb{C})$. $\square$

Examples.

The split biquaternions. Here $\mathbb{H}_{\mathbb{D}}=\mathrm{Cl}_{0,3}$, so the complexification is $$ \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}_{\mathbb{D}} = \mathbb{C}\mathrm{l}_3 \cong \mathbb{C}\mathrm{l}_2 \oplus \mathbb{C}\mathrm{l}_2 \cong M_2(\mathbb{C})\oplus M_2(\mathbb{C}) \cong \mathbb{B}\oplus\mathbb{B}, $$ the complex Clifford algebra of the split biquaternion algebra, split into two copies of the biquaternion algebra. The splitting is the complexified idempotent decomposition $\mathbb{H}_{\mathbb{D}} = \mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_-$ of Split-Biquaternion Ideals and Peirce Decomposition, and the construction is Complex Split Biquaternions and the Clifford Algebra Cl(3).
The biquaternions. Here $\mathbb{B} = \mathbb{C}\mathrm{l}_2 \cong M_2(\mathbb{C})$ is already complex and simple, and complexification adds nothing. The contrast between this example and the last is the sign of the Clifford form and nothing else: the positive definite three-generator algebra $\mathrm{Cl}_{3,0}\cong\mathbb{C}\mathrm{l}_2$ is simple, and the negative definite three-generator algebra $\mathrm{Cl}_{0,3}$ complexifies to a sum of two simple algebras.
The octonions and their chain algebra. The octonions are not in the dictionary, being non-associative. But the chain algebra generated by their left multiplications is associative and is the complex Clifford algebra $$ \overleftarrow{\mathbb{C}\otimes\mathbb{O}} \cong \mathrm{Cl}(6) \cong \mathbb{C}\mathrm{l}_6 \cong M_8(\mathbb{C}), $$ and the chain algebra of the split bioctonions is its doubling $$ \mathbb{C}\mathrm{l}_7 \cong \mathbb{C}\mathrm{l}_6 \oplus \mathbb{C}\mathrm{l}_6, $$ the constructions of Complex Octonions and the Clifford Algebra Cl(6) and Split Bioctonions and the Clifford Algebra Cl(7). So the odd complex Clifford algebras arise from the octonions not as a non-associative number system but as an associative chain algebra.

Remark (the pattern). The first and third examples are the same statement at the two rungs $\mathbb{C}\mathrm{l}_2$ and $\mathbb{C}\mathrm{l}_6$: a split system has an odd complex chain dimension, and its chain algebra doubles, whereas the definite system's chain algebra is even-dimensional and simple. The doubling is the algebraic content of a pair of chiralities, and it is the reason the split systems, rather than the biquaternion algebra, are the natural carriers of such a pair.

Summary

The classical real number systems are the Clifford algebras of low-dimensional quadratic forms: $\mathbb{C}=\mathrm{Cl}_{0,1}$, the split complex numbers $\mathbb{D}=\mathrm{Cl}_{1,0}$, the dual numbers $\mathbb{D}'$ the degenerate one-dimensional Clifford algebra, the quaternions $\mathbb{H}=\mathrm{Cl}_{0,2}$, the four-dimensional split quaternions, which are the two two-dimensional Clifford algebras $\mathrm{Cl}_{1,1}\cong\mathrm{Cl}_{2,0}\cong M_2(\mathbb{R})$, the eight-dimensional $\mathbb{H}_{\mathbb{D}}=\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}=\mathrm{Cl}_{0,3}\cong\mathbb{H}\times\mathbb{H}$, and the biquaternions $\mathbb{B}=\mathrm{Cl}_{3,0}=\mathbb{C}\mathrm{l}_2\cong M_2(\mathbb{C})$. In each case the Clifford form is the negative of the composition norm on the generators, and the properties of the system follow from the sign of that form: a definite form gives a division algebra, an indefinite form gives an algebra with zero divisors, and a degenerate form gives a radical and the dual numbers.

The composition property of the norm — the multiplicativity $N(xy)=N(x)N(y)$ — is possible only in dimensions $1,2,4,8$ by Hurwitz's theorem, and the four definite examples are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and the octonions. The first three are Clifford algebras; the octonions are not, because they are not associative. The unit groups of the definite systems are the low-dimensional spin groups: the norm-one complex numbers give $U(1)=\mathrm{Spin}(2)$, the unit quaternions give $Sp(1)=\mathrm{Spin}(3)$, and the biquaternion unit group is $GL(2,\mathbb{C})$ with norm-one subgroup $SL(2,\mathbb{C})$ defined by the determinant rather than by the algebra norm.

Summary of Notation

Symbol	Meaning
$\mathrm{Cl}_{p,q}$	Clifford algebra, $p$ generators of square $+1$, $q$ of square $-1$
$\mathrm{Cl}^0$	Even subalgebra
$\mathbb{C}$	Complex numbers, $\mathrm{Cl}_{0,1}$
$\mathbb{D}$	Split complex numbers, unit $j$, $j^2=+1$, $\mathbb{D}=\mathrm{Cl}_{1,0}$
$\mathbb{D}'$	Dual numbers, unit $\varepsilon$, $\varepsilon^2=0$, degenerate Clifford algebra
$\mathbb{H}$	Quaternions, units $e_1,e_2,e_3$, $e_k^2=-1$, $\mathbb{H}=\mathrm{Cl}_{0,2}$
$\mathbb{H}_{\mathbb{D}}=\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$	Split biquaternions, $\cong\mathrm{Cl}_{0,3}\cong\mathbb{H}\times\mathbb{H}$, of real dimension eight
$\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	Biquaternions, $\cong\mathrm{Cl}_{3,0}\cong M_2(\mathbb{C})$
$N(\tilde Q)=\tilde Q\tilde{Q}^{\natural}=\sum_\mu Q_\mu^2$	Norm of $\mathbb{B}$, complex-valued, signature $(4,4)$ in its real part
$\tilde Q\tilde{Q}^{*}$	Hermitian form of $\mathbb{B}$
$\langle\tilde P,\tilde Q\rangle=\sum_\mu P_{\bar{\mu}}Q_\mu$	Complex inner product on $\mathbb{B}$
$\pi_\pm=\tfrac12(1\pm j)$	Idempotents of $\mathbb{D}$
$\gamma_k\mapsto ie_k$	Isomorphism $\mathrm{Cl}_{3,0}\to\mathbb{B}$
$Sp(1)$	Unit quaternions, $\cong SU(2)\cong\mathrm{Spin}(3)$
$U(1)$	Unit complex numbers, $\cong\mathrm{Spin}(2)$
$\mathbb{O}$	Octonions, a composition algebra but not associative, hence not a Clifford algebra