List of Algebras by Dimension
Introduction
This article gathers the real algebras of dimension $1$, $2$, $4$ and $8$, with the classification in each dimension. The dimensions are the ones the doubling chain produces, and each carries a classification theorem: dimension $1$ has one division algebra, dimension $2$ has $\mathbb{C}$ as its only division algebra, Frobenius' theorem makes $\mathbb{H}$ the only four-dimensional real division algebra, and Hurwitz' theorem makes $\mathbb{O}$ the only eight-dimensional normed division algebra.
Every entry points to the article that introduces the algebra and states the classification it belongs to. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being an algebra of the given dimension in which a product fails to be a division algebra, or an expected algebra that does not exist in the dimension, with the failure named and the article that records it.
Dimension 1
A one-dimensional real algebra is $\mathbb{R}$ with the product $xy = cxy$ for a scalar $c$; it is a field exactly for $c > 0$, up to isomorphism, and it is a division algebra exactly when $c \neq 0$. There is no other one-dimensional real division algebra, and $\mathbb{R}$ is the field the whole chain starts from.
| Algebra | The property it has, and the classification | Introduced in |
|---|---|---|
| $\mathbb{R}$ | dimension $1$; the unique one-dimensional real division algebra; a field, ordered and complete | The Real Numbers |
| $\mathbb{R}$ as a real algebra | the commutative associative unital algebra of dimension $1$ | Algebras |
| $\mathbb{R}$ as a Clifford algebra | $\mathrm{Cl}_{0,0} \cong \mathbb{R}$; the start of the Clifford description | The Number Systems as Clifford Algebras |
| Non-example: the one-dimensional algebra with $xy = 0$ | fails the unit and the division property: it is a zero algebra | Examples of Algebras |
| Non-example: $F[x]$ as a one-dimensional algebra | fails: the polynomial ring is infinite-dimensional over $F$ | Polynomial Rings and Rational Functions |
Dimension 2
The two-dimensional real algebras are $\mathbb{C}$, the split-complex numbers $\mathbb{D} \cong \mathbb{R} \times \mathbb{R}$, and the dual numbers $\mathbb{D}' = \mathbb{R}[\varepsilon]/(\varepsilon^2)$, so $\mathbb{R}[t]/(t^2 + 1)$, $\mathbb{R}[t]/(t^2 - 1)$ and $\mathbb{R}[t]/(t^2)$ exhaust the commutative associative unital cases. Only $\mathbb{C}$ is a field; $\mathbb{D}$ is reduced with zero divisors and $\mathbb{D}'$ is local with a nilpotent.
| Algebra | The property it has, and the classification | Introduced in |
|---|---|---|
| $\mathbb{C}$ | dimension $2$; the unique two-dimensional real division algebra; algebraically closed | The Complex Numbers |
| $\mathbb{D} = \mathbb{R}[t]/(t^2-1)$ | dimension $2$; split-complex numbers, $\cong \mathbb{R}\times\mathbb{R}$; reduced with zero divisors, not a field | Split-Complex Algebra |
| $\mathbb{D}' = \mathbb{R}[\varepsilon]/(\varepsilon^2)$ | dimension $2$; dual numbers, local and non-reduced, nilpotent $\varepsilon$ | Dual Numbers Algebra |
| The three as quotients of $\mathbb{R}[t]$ | $\mathbb{R}[t]/(t^2+1)$, $\mathbb{R}[t]/(t^2-1)$ and $\mathbb{R}[t]/(t^2)$ | Quotients of the Tensor Algebra |
| Complex conjugation | the nontrivial $\mathbb{R}$-automorphism of $\mathbb{C}$, fixing $\mathbb{R}$ pointwise | The Complex Numbers |
| $\mathbb{C}$ as a Clifford algebra | $\mathbb{C} \cong \mathrm{Cl}_{0,1}$, generated by one element with square $-1$ | The Number Systems as Clifford Algebras |
| $\mathbb{C}$ and $\mathbb{D}$ as the two real forms | the definite and the split form of the same composition law | Normed Division Algebras and the Hurwitz Theorem |
| Non-example: the split-complex numbers as a field | fails: $(1-t)(1+t) = 0$ with both factors nonzero | Split-Complex Algebra |
| Non-example: the dual numbers as reduced | fails: $\varepsilon^2 = 0$ with $\varepsilon \neq 0$ | Dual Numbers Algebra |
Dimension 4
The four-dimensional real algebras of the corpus are the quaternions $\mathbb{H}$ and the matrix algebra $M_2(\mathbb{R})$, which is the split quaternion algebra of the classical literature, $\mathrm{Cl}_{1,1}$; the corpus's split quaternions $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$ are eight-dimensional and are listed in the section below; the biquaternions $\mathbb{B} \cong M_2(\mathbb{C})$ are four-dimensional over $\mathbb{C}$, hence eight-dimensional over $\mathbb{R}$. Frobenius' theorem states that $\mathbb{H}$ is the only four-dimensional real division algebra, and that every real division algebra is $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$.
One four-dimensional real algebra belongs to this list and to no comparison table: the tensor product $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C} \cong \mathbb{C}\oplus\mathbb{C}$, commutative, associative, with zero divisors. It is the low-dimensional analogue of the split biquaternions, and the correspondence is exact — the algebra is to $\mathbb{C}$ what $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$ is to $\mathbb{H}$: the same decomposition by two idempotents into two copies of a division algebra, the same componentwise ring-valued norm, the same zero divisors, with $\mathbb{C}$ in place of $\mathbb{H}$. The engineering literature calls it the reduced biquaternion algebra; the same algebra appears elsewhere as the double-complex numbers, the tessarines or bicomplex numbers, and Davenport's four-dimensional commutative hypercomplex algebra. It is not a column of The Eight Algebras Compared, which fixes its columns on the ladder of that article, and it is recorded with the reductions in Harmonic Analysis over Hypercomplex Systems.
| Algebra | The property it has, and the classification | Introduced in |
|---|---|---|
| $\mathbb{H}$ | dimension $4$; the unique four-dimensional real division algebra; associative but not commutative; centre $\mathbb{R}$ | Quaternion Algebra |
| Split quaternion algebra of the classical literature, $\mathrm{Cl}_{1,1} \cong M_2(\mathbb{R})$ | dimension $4$; the split form of the quaternion composition law; associative with zero divisors, not a division algebra | The Number Systems as Clifford Algebras |
| Warning: the split biquaternions $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$ | not a four-dimensional algebra: it is eight-dimensional, and appears in the table of dimension $8$ | Split-Biquaternion Algebra |
| $M_2(\mathbb{R})$ | dimension $4$; simple, centre $\mathbb{R}$; zero divisors, not a division algebra | Examples of Algebras |
| Frobenius' theorem | the classification $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ of the real division algebras | Division Algebras |
| Quaternion units $i,j,k$ | $i^2=j^2=k^2=ijk=-1$; the quaternion relations with $ij=k$, $ji=-k$ | Quaternion Algebra |
| Lipschitz quaternion lattice | the torsion-free $\mathbb{Z}$-module of rank $4$ inside $\mathbb{H}$ | Lattices and the Quaternion Lattice |
| $\mathbb{H}$ as a Clifford algebra | $\mathbb{H} \cong \mathrm{Cl}_{0,2}$, generated by two elements with square $-1$ | The Number Systems as Clifford Algebras |
| $\mathbb{H}$ and $\mathrm{Cl}_{1,1} \cong M_2(\mathbb{R})$ as the two real forms | the definite and the split form of the same quaternion composition law | Normed Division Algebras and the Hurwitz Theorem |
| $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C} \cong \mathbb{C}\oplus\mathbb{C}$ | dimension $4$; commutative and associative with zero divisors; the reduced biquaternion algebra, equivalently the double-complex, tessarine or commutative hypercomplex algebra; not a column of The Eight Algebras Compared | Harmonic Analysis over Hypercomplex Systems |
| Non-example: a four-dimensional commutative real division algebra | does not exist: Frobenius' theorem leaves only $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ | Division Algebras |
| Non-example: the four-dimensional split quaternion algebra as a division algebra | fails: it has zero divisors and is $M_2(\mathbb{R})$, not $\mathbb{H}$ | The Number Systems as Clifford Algebras |
| Non-example: $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}$ as a Clifford algebra | fails: its two generators commute and both square to $-1$, whereas $\mathrm{Cl}_{0,2} \cong \mathbb{H}$ requires them to anticommute | The Number Systems as Clifford Algebras |
Dimension 8
The eight-dimensional real algebras of the corpus are the octonions $\mathbb{O}$, the split-octonions, the split biquaternions $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H} \cong \mathbb{H}\oplus\mathbb{H}$, the biquaternions $\mathbb{B}$ and the matrix algebra $M_2(\mathbb{C})$ over $\mathbb{R}$; the sedenions are not a division algebra. Hurwitz' theorem states that the only normed division algebras over $\mathbb{R}$ are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$, so $\mathbb{O}$ is the eight-dimensional case and the chain stops there.
| Algebra | The property it has, and the classification | Introduced in |
|---|---|---|
| Octonions $\mathbb{O}$ | dimension $8$; the unique normed eight-dimensional real division algebra; non-associative | Octonion Algebra |
| Octonion units $e_0,\dots,e_7$ | the multiplication table of the Fano plane; $e_0 = 1$ and $e_i^2 = -1$ | Octonion Algebra |
| Split-octonions | dimension $8$; a non-associative algebra with zero divisors and a split form | Normed Division Algebras and the Hurwitz Theorem |
| Split biquaternions $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$ | dimension $8$; centre $\mathbb{D}$; associative with zero divisors, $\cong \mathbb{H}\oplus\mathbb{H}$, and no nonzero nilpotents | Split-Biquaternion Algebra |
| $\mathbb{O}$ and the split octonions as the two real forms | the definite and the split form; only the definite one is a division algebra | Normed Division Algebras and the Hurwitz Theorem |
| Biquaternions $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ | dimension $4$ over $\mathbb{C}$, $8$ over $\mathbb{R}$; $\cong M_2(\mathbb{C})$; associative with zero divisors | Biquaternion Algebra |
| $\mathbb{B}$ as a Clifford algebra | $\mathbb{B} \cong \mathrm{Cl}_{3,0}$ and $\cong \mathrm{Cl}_{0,3}$ in the appropriate form | The Clifford Structure of the Biquaternion Algebra |
| $\mathbb{B}$ as a complexification | the complexification $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$; real dimension $8$ | Biquaternion Algebra |
| Hurwitz' theorem | the four normed division algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$, and no others | Normed Division Algebras and the Hurwitz Theorem |
| Cayley–Dickson doubling | the construction producing each dimension from the last and the property lost at each step | Normed Division Algebras and the Hurwitz Theorem |
| Non-example: the octonions as associative | fails associativity: $(e_1 e_2)e_4 \neq e_1(e_2 e_4)$ | Octonion Algebra |
| Non-example: an eight-dimensional associative real division algebra | does not exist: Hurwitz' theorem and Frobenius' theorem leave only $\mathbb{O}$ at dimension $8$ | Division Algebras |
| Non-example: $\mathbb{O}$ as a Clifford algebra | not a Clifford algebra: every Clifford algebra is associative, and $\mathbb{O}$ is not | The Number Systems as Clifford Algebras |
The Doubling and the Property Lost
The four dimensions are linked by the Cayley–Dickson doubling: each algebra of dimension $2n$ is built from one of dimension $n$, and the list records the algebra produced and the property that does not survive. The doubling continues beyond dimension $8$, but the norm stops being multiplicative at the sedenions and no larger dimension contributes a normed division algebra.
| Step | The algebra produced | The property lost at the step | Introduced in |
|---|---|---|---|
| $\mathbb{R} \to \mathbb{C}$ | dimension $2$ | order: $\mathbb{C}$ is not orderable | The Complex Numbers |
| $\mathbb{C} \to \mathbb{H}$ | dimension $4$ | commutativity: $ij \neq ji$ | Quaternion Algebra |
| $\mathbb{H} \to \mathbb{O}$ | dimension $8$ | associativity: the associator is nonzero | Octonion Algebra |
| $\mathbb{O} \to \mathbb{S}$ | dimension $16$ | the multiplicative norm, and with it the division property | Normed Division Algebras and the Hurwitz Theorem |
| The split variants | dimensions $2$, $4$, $8$ | anisotropy of the norm, producing zero divisors | Normed Division Algebras and the Hurwitz Theorem |
The Classification in Each Dimension
The four dimensions carry four classification statements, and the table gathers them together with the algebra that is the extreme case.
| Dimension | The algebras of the corpus | The classification | Introduced in |
|---|---|---|---|
| $1$ | $\mathbb{R}$ | the only one-dimensional real division algebra | The Real Numbers |
| $2$ | $\mathbb{C}$, $\mathbb{D}$, $\mathbb{D}'$ | $\mathbb{C}$ the only two-dimensional division algebra; $\mathbb{R}\times\mathbb{R}$ and the dual numbers the two others | Split-Complex Algebra |
| $4$ | $\mathbb{H}$, $M_2(\mathbb{R})$ | Frobenius: $\mathbb{H}$ the only four-dimensional real division algebra; $M_2(\mathbb{R})$ is the split quaternion algebra of the classical literature | Division Algebras |
| $8$ | $\mathbb{O}$, split-octonions, $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H}\oplus\mathbb{H}$, $\mathbb{B} \cong M_2(\mathbb{C})$ | Hurwitz: $\mathbb{O}$ the only eight-dimensional normed division algebra | Normed Division Algebras and the Hurwitz Theorem |
Summary
The list gathers the real algebras of dimension $1$, $2$, $4$ and $8$ with the classification in each dimension. Dimension $1$ contains $\mathbb{R}$ and its degenerate products, with $\mathbb{R}$ the only division algebra. Dimension $2$ contains $\mathbb{C}$, the split-complex numbers $\mathbb{R}\times\mathbb{R}$ and the dual numbers, with $\mathbb{C}$ the only field and the only division algebra. Dimension $4$ contains the quaternions $\mathbb{H}$, Frobenius' unique four-dimensional real division algebra, together with the matrix algebra $M_2(\mathbb{R})$, which is the split quaternion algebra of the classical literature. Dimension $8$ contains the octonions $\mathbb{O}$, Hurwitz' unique eight-dimensional normed division algebra, together with the split-octonions, the split biquaternions $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H}\oplus\mathbb{H}$ and the biquaternions $\mathbb{B} \cong M_2(\mathbb{C})$. The Cayley–Dickson doubling runs down the chain and the Hurwitz theorem brings it to a stop after dimension $8$. The non-examples — the zero one-dimensional algebra, the split-complex numbers and the dual numbers as non-fields, a commutative four-dimensional division algebra that does not exist, the four-dimensional split quaternion algebra as a non-division algebra, the octonions as non-associative, and an eight-dimensional associative division algebra that does not exist — each name the failure or the absence.
Summary of Notation
The article denotes its objects by name; the symbols appearing in the tables are those of the introducing articles.
| Symbol | Meaning |
|---|---|
| $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ | real, complex, quaternion and octonion algebras |
| $\mathbb{D}$, $\mathbb{D}'$ | split-complex numbers, dual numbers |
| $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$ | split biquaternions, of real dimension $8$ |
| $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ | biquaternions |
| $\mathbb{S}$ | the sedenions |
| $\mathbb{Z}$ | the integers, as in the rank-$4$ Lipschitz lattice |
| $M_2(\mathbb{R})$, $M_2(\mathbb{C})$ | matrix algebras |
| $i, j, k$ | quaternion units |
| $e_0,\dots,e_7$ | octonion basis |
| $\mathbb{R}[t]/(t^2+1)$ etc. | the quadratic quotients of the polynomial ring |
Further Reading
- Ferdinand Georg Frobenius, Über lineare Substitutionen und bilineare Formen (Journal für die reine und angewandte Mathematik, 1878), for the classification of the real division algebras.
- Adolf Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln (Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1898), for the theorem on the normed division algebras.
- Richard Schafer, An Introduction to Nonassociative Algebras (Dover, 1995), for the Cayley–Dickson construction and the octonions.
- Ian Porteous, Clifford Algebras and the Classical Groups (Cambridge University Press, 1995), for the Clifford-algebraic description of the real algebras of each dimension.
- Soo-Chang Pei, Ja-Han Chang and Jian-Jiun Ding, "Commutative reduced biquaternions and their Fourier transform for signal and image processing applications", IEEE Transactions on Signal Processing 52 (2004) 2012–2022, for the reduced biquaternion algebra — the commutative four-dimensional algebra $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}\cong\mathbb{C}\oplus\mathbb{C}$, equivalently the double-complex, tessarine or commutative hypercomplex algebra — and for its defining relations, its determinant norm and its place among the four-dimensional real algebras.