Biquaternion Orders and Finite Groups of Units

Introduction

The biquaternion algebra carries integral structures, and the groups of units of those structures are the finite groups attached to the algebra. The real slice $\mathbb{H}_{\mathbb{B}}\cong\mathbb{H}$ contains the classical quaternion orders — the Lipschitz order and the Hurwitz order — whose groups of units are the quaternion group of order eight and the binary tetrahedral group of order twenty-four. The unit sphere of the real slice has for finite subgroups the cyclic groups, the binary dihedral groups and the three binary polyhedral groups of orders $24$, $48$ and $120$, and those groups draw the figures of the theory: the regular $24$-cell with the Hurwitz units as its vertices, and the McKay correspondence. The complex order, the integral biquaternions, behaves differently: its group of units is infinite, generated along a nilpotent direction, so the finite unit groups are the real ones.

This article is the integral and finite-group entry of the Topology group. The lattice-theoretic treatment of the quaternion orders — rank, index, covolume, duality, base change — is Lattices and the Quaternion Lattice; the order theory of the quaternion algebra over $\mathbb{Q}$, with maximality and the arithmetic of the norm, is Division Algebras; the Clifford lift of the finite reflection groups and the McKay correspondence are Reflection Groups and Clifford Algebras with Signed Inner Conjugation and Root Systems and Classification. Those results are cited, not re-derived, and the present article owns their biquaternion statement: the orders inside $\mathbb{B}$, their finite unit groups as abstract groups, the figures those groups determine in the real slice, and the infinite unit group of the complex order. The quotient $Sp(1)/\{\pm e_0\}\cong SO(3)$, the rotor and the rotation $\rho_v(\tilde R)=-v\tilde Rv^{-1}$ are Biquaternion Rotations and Lorentz Transformations, cited.

Conventions. The biquaternion algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, with basis $e_0=1,e_1,e_2,e_3$, central scalar imaginary $i$, products $e_1e_2=e_3$, $e_2e_3=e_1$, $e_3e_1=e_2$ and $e_k^2=-e_0$, so that $e_1e_2e_3=-e_0$. The quaternion subspace $\mathbb{H}_{\mathbb{B}}=\mathbb{R}\{e_0,e_1,e_2,e_3\}$ is the real slice (Introduction to the Six Subspaces). The norm is $N(\tilde{Q})=\sum_\mu Q_\mu^2$, and on the real slice it is the positive definite form $\sum_\mu q_\mu^2$.

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The Quaternion Orders

Definition. A subset $\Lambda\subseteq\mathbb{H}$ is an order if it is a subring and a free abelian group of rank four whose $\mathbb{Q}$-span is $\mathbb{H}$; equivalently, it is a lattice that is also closed under multiplication and contains $e_0$.

Definition. The Lipschitz order is $$ \mathcal{L}=\mathbb{Z}e_0\oplus\mathbb{Z}e_1\oplus\mathbb{Z}e_2\oplus\mathbb{Z}e_3, $$ and the Hurwitz order is the larger lattice obtained by adjoining the half-integral element $\omega=\tfrac12(e_0+e_1+e_2+e_3)$, $$ \mathcal{L}'=\mathcal{L}\oplus\mathbb{Z}\omega . $$

Theorem. The Hurwitz order contains the Lipschitz order with index two, and it is a maximal order of $\mathbb{H}$; the Lipschitz order is an order but is not maximal.

Proof. The change of basis from $(e_0,e_1,e_2,e_3)$ to $(\omega,e_1,e_2,e_3)$ has determinant $\tfrac12$, so the index is $|\det|^{-1}=2$; equivalently $\mathcal{L}'/\mathcal{L}\cong\mathbb{Z}/2$, of prime order. Maximality of $\mathcal{L}'$ is the classical statement that the Hurwitz order is a maximal order in the rational quaternion algebra, and the half-integral element of $\mathcal{L}'\setminus\mathcal{L}$ witnesses that $\mathcal{L}$ is not maximal (Division Algebras); the module-level computations of index and covolume are in Lattices and the Quaternion Lattice.

Remark. The orders are orders in the real quaternion algebra, inside the quaternion subspace of $\mathbb{B}$. The biquaternion algebra contains them and their complexification, and it is the complexification that changes the nature of the unit group, below.

The Groups of Units

Theorem. The group of units of the Lipschitz order is the eight-element quaternion group, $$ \mathcal{L}^{\times}=\{\pm e_0,\pm e_1,\pm e_2,\pm e_3\}\cong Q_8 , $$ and the group of units of the Hurwitz order is the twenty-four-element binary tetrahedral group, $$ (\mathcal{L}')^{\times}=\mathcal{L}^{\times}\cup\{\tfrac12(\pm e_0\pm e_1\pm e_2\pm e_3)\}\cong 2T . $$

Proof. On real quaternion coordinates the norm is $N(\tilde q)=\sum_\mu q_\mu^2\geq0$, an integer for $\tilde q$ in either order; the inverse is $\tilde q^{-1}=\tilde q^*/N(\tilde q)$ with $\tilde q^*$ the quaternion conjugate, and $\tilde q^*$ lies in the order whenever $\tilde q$ does, so $\tilde q$ is a unit exactly when $N(\tilde q)=1$. The norm-one elements with integer coordinates are the eight signed units $\pm e_\mu$, and with half-integer coordinates they are those together with the sixteen elements $\tfrac12(\pm e_0\pm e_1\pm e_2\pm e_3)$, of norm one. The resulting groups are closed under multiplication, have the stated orders, and are the quaternion group and the binary tetrahedral group respectively.

Remark (the two indices). The lattices have index $2$, but the unit groups have index $24/8=3$: the Lipschitz units are a proper subgroup of index three in the Hurwitz units, so the two notions of index do not agree.

The Elements of Order Six

The Lipschitz unit group is a $2$-group and the Hurwitz unit group is not, and one element exhibits the difference. The order of a unit $Q$ is the least $n\geq1$ with $Q^n=e_0$.

Example. Let $$ Q=\tfrac12(e_0-e_1-e_2+e_3). $$ This is a Hurwitz unit, being one of the sixteen half-integral norm-one elements of $(\mathcal{L}')^{\times}\setminus\mathcal{L}^{\times}$ listed above. Its square is the negative of its conjugate, $$ Q^2=-Q^*, $$ and therefore $$ Q^3=Q\,Q^2=-QQ^*=-N(Q)e_0=-e_0,\qquad Q^6=e_0 , $$ so $Q$ has order six and generates a cyclic subgroup $\langle Q\rangle\cong C_6$ of the unit group.

Remark. The identity $Q^2=-Q^*$ is a property of the half-integral units and not of this example alone, and the sign of the real part decides the order. For a half-integral unit $$ Q=\tfrac12(s_0e_0+s_1e_1+s_2e_2+s_3e_3),\qquad s_\mu=\pm1, $$ the real part is $\tfrac12s_0$ and the vector part has norm $\tfrac34$, so $Q^2=-Q^*$ when $s_0=+1$, which gives $Q^3=-e_0$ and order six, while $Q^2=+Q^*$ when $s_0=-1$, which gives $Q^3=e_0$ and order three. There are eight units of each kind, and the element orders of the two orders of units are as follows.

The eight elements of order six form four cyclic subgroups of order six, and their squares are the eight elements of order three, so the unit group has four subgroups of order three and four of order six, each subgroup of order three lying in exactly one of order six. None of these subgroups lies in the Lipschitz units, whose element orders are $1$, $2$ and $4$.

The Finite Subgroups of the Unit Sphere

The unit quaternions $Sp(1)=S^3$ form a group with centre $\{\pm e_0\}$, and the quotient $$ Sp(1)/\{\pm e_0\}\cong SO(3) $$ is the rotation group of Euclidean three-space (Biquaternion Rotations and Lorentz Transformations). The cover being twofold, a finite rotation group is the one that preserves a figure, so the figures classify the finite rotation groups first.

Theorem. The finite subgroups of $SO(3)$ are the rotation groups of the plane and the three Platonic figures: the cyclic groups, the rotation groups of a regular pyramid; the dihedral groups, the rotation groups of a regular prism; and the three polyhedral groups $$ T\ (12),\qquad O\ (24),\qquad I\ (60), $$ the rotation groups of the tetrahedron, the octahedron and the icosahedron.

Theorem. The finite subgroups of the unit sphere $Sp(1)=S^3\subset\mathbb{H}_{\mathbb{B}}$ are the twofold preimages of these: the cyclic groups, the binary dihedral groups, and the three binary polyhedral groups $$ 2T\ (24),\qquad 2O\ (48),\qquad 2I\ (120), $$ the binary tetrahedral, octahedral and icosahedral groups.

Proof. Let $\Gamma\subset Sp(1)$ be finite. Its image under the quotient is a finite subgroup $\bar\Gamma\subset SO(3)$, and the cover being twofold, $|\Gamma|=2|\bar\Gamma|$; conversely the preimage of a finite subgroup of $SO(3)$ is finite, of twice the order. The preimage of a cyclic group of order $n$ is cyclic of order $2n$, since the preimage of a cyclic group is cyclic; the preimage of a dihedral group of order $2n$ is a binary dihedral group of order $4n$; and the preimages of the three polyhedral groups of orders twelve, twenty-four and sixty are $2T$, $2O$ and $2I$, of orders twenty-four, forty-eight and one hundred twenty.

The Hurwitz order realises $2T$ integrally, as above; the other two are figures of the unit sphere without an integral model of the same kind. The Clifford realisation of these groups as the even parts of the lifts of the finite reflection groups is Reflection Groups and Clifford Algebras with Signed Inner Conjugation, and the classification of the reflection groups is Root Systems and Classification.

The 24-Cell and the Twelve Rotations

The twenty-four Hurwitz units are points of the unit sphere $S^3\subset\mathbb{H}_{\mathbb{B}}\cong\mathbb{R}^4$, and they are the vertices of a regular $24$-cell, the regular polytope of four-dimensional Euclidean space whose twenty-four cells are octahedra. The unit group is drawn as a figure.

Theorem (the symmetry group). The symmetry group of the $24$-cell is larger than the unit group: the signed permutations of the four coordinates form the Weyl group $B_4$ of order $384$, and the full symmetry group is the Weyl group $F_4$ of order $1152$, whereas the unit group is $2T$ of order $24$.

Proof. A symmetry of the $24$-cell permutes the vertices, so the symmetry group acts on the twenty-four units; the group generated by sign changes and coordinate permutations is the hyperoctahedral group $B_4$ of order $2^4\cdot 4!=384$, and it preserves the vertex set; the $24$-cell is the $F_4$ root polytope, whose full symmetry group is the Weyl group $F_4$ of order $1152$. Both contain the unit group $2T$, and both are strictly larger.

The two Weyl groups are the symmetry groups of the lattice $\mathcal{L}'$ of the Hurwitz order and of the root system $F_4$; the reflection-group and root-system theory is Root Systems and Classification, and the lattice theory is Lattices and the Quaternion Lattice. The unit group reaches the algebra through the rotations $\rho_v$ of Biquaternion Rotations and Lorentz Transformations, §Reflections, not through these symmetries: the map $$ \rho_v(\tilde R) = -v\,\tilde R\,v^{-1} $$ is a rotation of determinant $+1$ of the real slice and depends on $v$ only up to sign, so the twenty-four Hurwitz units give twelve distinct rotations. Together they generate the conjugation action of $2T$ on the real slice, of kernel $\{\pm e_0\}$, so that $2T/\{\pm e_0\}\cong A_4$ of order twelve, and adjoining the central negation, which is $\rho_{e_0}$, gives a group of order $24$. Read on the vector subspace $\mathrm{Vect}(\mathbb{B})\cong\mathbb{R}^3$ instead of on the real slice, the same map $\rho_{e_1}$ is the reflection in the hyperplane orthogonal to $e_1$, which supplies the root system $A_1$ of Root Systems and Classification.

The McKay Correspondence

The five families of finite subgroups of $Sp(1)$ are the five families of simply laced Dynkin diagrams: the cyclic groups give the types $\tilde A$, the binary dihedral groups the types $\tilde D$, and the three binary polyhedral groups the exceptional types, $$ 2T \leftrightarrow \tilde E_6,\qquad 2O \leftrightarrow \tilde E_7,\qquad 2I \leftrightarrow \tilde E_8 . $$ This is the McKay correspondence, and its content is that the Platonic solids, their binary preimages in the unit sphere and the exceptional simple Lie algebras are one subject. The Clifford lift of the finite rotation groups that exhibits the correspondence is Reflection Groups and Clifford Algebras with Signed Inner Conjugation, and the classification of the root systems is Root Systems and Classification.

The Integral Biquaternions

Definition. The integral biquaternions are the elements whose coefficients lie in the Gaussian integers, $$ \Lambda=\mathbb{Z}[i]e_0\oplus\mathbb{Z}[i]e_1\oplus\mathbb{Z}[i]e_2\oplus\mathbb{Z}[i]e_3 =\mathcal{L}\otimes_{\mathbb{Z}}\mathbb{Z}[i]. $$ They form an order in $\mathbb{B}$ over $\mathbb{Z}[i]$, and the larger $\Lambda'=\mathcal{L}'\otimes_{\mathbb{Z}}\mathbb{Z}[i]$ contains it with index two.

Theorem. The group of units of the integral biquaternions is infinite.

Proof. The element $n=e_1+ie_2$ is nilpotent, $n^2=(e_1)^2+i(e_1e_2+e_2e_1)+i^2(e_2)^2=(-e_0)+0+(-1)(-e_0)=0$, so for every integer $k$ the binomial expansion truncates and $$ (e_0+n)^k=e_0+kn . $$ Each of these elements has norm $N(e_0+kn)=1+k^2+(ik)^2=1$, since the coefficients of $e_0+kn$ are $Q_0=1$, $Q_1=k$, $Q_2=ik$, $Q_3=0$; a norm-one element has inverse its conjugate and so is a unit. Hence $\Lambda^{\times}$ contains the infinite family $\{e_0+kn:k\in\mathbb{Z}\}$.

Remark. The finite unit groups are therefore those of the real order, not of the complex one. The presence of nilpotent directions in the complex order is the same phenomenon as the presence of zero divisors in the algebra at large: $\mathbb{B}\cong M_2(\mathbb{C})$ is not a division algebra, and its integral order inherits unipotent units.

Summary

The quaternion orders inside the biquaternion algebra are the Lipschitz order $\mathcal{L}$ and the Hurwitz order $\mathcal{L}'$, the second containing the first with index two and maximal. Their groups of units are the quaternion group of order eight and the binary tetrahedral group of order twenty-four; the unit-group index is three, though the lattice index is two. The element orders of the two groups differ: the Lipschitz units have orders $1$, $2$ and $4$, while the Hurwitz units have orders $1$, $2$, $3$, $4$ and $6$, the half-integral units of real part $+\tfrac12$ satisfying $Q^2=-Q^*$ and generating the four cyclic subgroups of order six.

The unit sphere $Sp(1)=S^3$ has for finite subgroups the twofold preimages of the finite rotation groups of the plane and the three Platonic figures: the cyclic groups, the binary dihedral groups and the binary polyhedral groups $2T$, $2O$, $2I$ of orders twenty-four, forty-eight and one hundred twenty. The twenty-four Hurwitz units are the vertices of the regular $24$-cell in $\mathbb{H}_{\mathbb{B}}\cong\mathbb{R}^4$, whose symmetry group is the Weyl group $F_4$ of order $1152$, containing the hyperoctahedral $B_4$ of order $384$; both are strictly larger than the unit group $2T$. The units act on the real slice by the rotations $\rho_v(\tilde R)=-v\tilde Rv^{-1}$, of determinant $+1$, and the twenty-four units give twelve distinct rotations generating a group of order $24$ with $2T/\{\pm e_0\}\cong A_4$. The five families of finite subgroups are the five families of simply laced Dynkin diagrams, the McKay correspondence.

The integral biquaternions, with coefficients in the Gaussian integers, form an order in $\mathbb{B}$ whose group of units is infinite: the nilpotent element $e_1+ie_2$ generates the unipotent family $e_0+k(e_1+ie_2)$ of norm one. The finite unit groups of the theory are thus the groups of the real quaternion orders.

Summary of Notation

Further Reading

• John H. Conway and Derek A. Smith, On Quaternions and Octonions (A. K. Peters, 2003), for the finite groups of unit quaternions, the $24$-cell and the quaternion orders.
• John H. Conway and Neil J. A. Sloane, Sphere Packings, Lattices and Groups (Springer, 3rd ed. 1999), for the integral quaternion orders, their unit groups and the root systems realised in them.
• H. S. M. Coxeter, Regular Polytopes (Dover, 3rd ed. 1973), for the $24$-cell, its symmetry group and the regular polytopes of four-dimensional space.
• Harold S. M. Coxeter and William O. J. Moser, Generators and Relations for Discrete Groups (Springer, 4th ed. 1980), for the finite rotation groups and their binary preimages.
• John McKay, Graphs, singularities and finite groups (Proceedings of Symposia in Pure Mathematics 37, 1980), for the correspondence between the finite subgroups of $SU(2)$ and the simply laced root systems.
• Marie-France Vignéras, Arithmétique des algèbres de quaternions (Springer Lecture Notes in Mathematics 800, 1980), for the order theory of the quaternion algebra, maximality and the arithmetic of the norm.