Biquaternion Lie Algebra

Introduction

The biquaternion algebra $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ carries a natural bracket, the commutator, and with it the structure of a Lie algebra. This article reads that structure: the algebra as the Lie algebra of the group of units, its centre, the trace-free subalgebra, the derived subalgebra and its identity with the vector subspace, the bracket of two vectors as a cross product, the action of the bracket on the six distinguished subspaces, and the real form of the trace-free part with its two three-dimensional real summands. The adjoint maps close the account.

The metric identification of the trace-free part with the Lorentz algebra, and of the group it exponentiates to with the spin group of Lorentzian signature, is Biquaternion Lie Group and Exponential Structure and Biquaternion Rotations and Lorentz Transformations; it is metric and is not made here. The topology of the group is The Biquaternion Unit Group as a Topological Group, and the automorphisms are Biquaternion Automorphisms and Derivations.

Conventions. The algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, with basis $e_0=1,e_1,e_2,e_3$ and central scalar imaginary $i$; a general element is $\tilde{Q}=\sum_{\mu=0}^{3}Q_\mu e_\mu$ with $Q_\mu\in\mathbb{C}$, written $\tilde{Q}=Q_0e_0+\mathbf{Q}$ with $\mathbf{Q}=\sum_{k=1}^{3}Q_ke_k$. The six distinguished subspaces are those of Biquaternion Algebra and Relations Between Subspaces.

The Algebra as a Lie Algebra

The trace functional is $\mathrm{Tr}(\tilde{Q})=2Q_0$. The set $\mathbb{B}$ carries the commutator bracket

$$ [\tilde{P},\tilde{Q}]=\tilde{P}\tilde{Q}-\tilde{Q}\tilde{P}, $$

under which it is a complex Lie algebra, written $\mathrm{G}$, of dimension $4$ over $\mathbb{C}$ and $8$ over $\mathbb{R}$. Its centre is the scalar line

$$ \mathrm{Z}(\mathrm{G})=\mathbb{C}e_0, $$

of complex dimension $1$ and real dimension $2$; every scalar multiple of $e_0$ commutes with all of $\mathbb{B}$. Removing the centre gives the decomposition

$$ \mathrm{G}=\mathrm{B}_0\oplus\mathbb{C}e_0, $$

with complex dimensions $4=3+1$ and real dimensions $8=6+2$, where

$$ \mathrm{B}_0=\{\tilde{Q}\in\mathbb{B} : Q_0=0\}=\mathrm{span}_\mathbb{C}\{e_1,e_2,e_3\} $$

is the trace-free subalgebra, the complex pure-vector part.

Physical reading. $\mathrm{G}$ is the Lie algebra of the group of units, which is $\mathrm{GL}(2,\mathbb{C})$ over $\mathbb{C}$; the centre $\mathbb{C}e_0$ is the generator of the central phase, the one direction that commutes with everything and therefore the one direction that cannot be gauged away by the algebra's own action. In physics terms the centre is the global phase generator, and the trace-free part is where the non-abelian content of the framework lives. The centre is also the scalar line of the physical dictionary: a general element carries $ct'+ict$ on its scalar coefficient, the real part the informational time and the imaginary part the material time, so the complex line $\mathbb{C}e_0$ is where both sectors put their time.

The Trace-Free Subalgebra

The Lie algebra of the norm-one group is the trace-free subalgebra

$$ \mathrm{B}_0=\{\tilde{Q}\in\mathbb{B} : Q_0=0\}=\mathrm{span}_\mathbb{C}\{e_1,e_2,e_3\}, $$

of complex dimension $3$ and real dimension $6$, closed under the bracket $[e_j,e_k]=2\sum_l\epsilon_{jkl}e_l$. Over the reals it splits into two three-dimensional real subspaces,

$$ \mathrm{B}_0=\mathrm{span}_\mathbb{R}\{e_1,e_2,e_3\}\;\oplus\;\mathrm{span}_\mathbb{R}\{ie_1,ie_2,ie_3\}, $$

with real dimensions $6=3+3$. The first summand is

$$ \mathrm{K}=\mathrm{span}_\mathbb{R}\{e_1,e_2,e_3\}, $$

on which the bracket is (twice) the cross product; the second is $\mathrm{span}_\mathbb{R}\{ie_1,ie_2,ie_3\}$. The two are non-isomorphic real Lie algebras; which of them is compact is a topological statement, made in The Biquaternion Unit Group as a Topological Group.

In the fixed-point subspaces of the basic algebra article the first summand is generated by $e_k\in\mathbb{H}_{\mathbb{B}}\cap\mathbb{M}_-$ and the second by $ie_k\in\mathbb{M}_+$. These are exactly the two real three-dimensional pieces into which $\mathrm{B}_0$ splits, and the central direction $\mathbb{C}e_0$ completes the picture, $\mathrm{G}=\mathrm{B}_0\oplus\mathbb{C}e_0$.

The Derived Subalgebra, the Cross Product and the Subspaces

The commutator of two biquaternions has vanishing scalar part, since the scalar part commutes with everything, $[\tilde{P},\tilde{Q}]=[\mathbf{P},\mathbf{Q}]$, and the bracket of the basis vectors is $[e_j,e_k]=2\sum_l\epsilon_{jkl}e_l$. Hence the derived subalgebra is the complex span of the vector units,

$$ [\mathrm{G},\mathrm{G}]=\mathrm{B}_0=\mathrm{span}_{\mathbb{C}}\{e_1,e_2,e_3\}=\mathrm{Vect}(\mathbb{B}), $$

of complex dimension $3$ and real dimension $6$: the derived subalgebra is exactly the vector subspace, and the quotient $\mathrm{G}/[\mathrm{G},\mathrm{G}]$ is the centre $\mathbb{C}e_0$.

For two pure vectors the quaternion product splits into its scalar and vector parts, $\mathbf{P}\mathbf{Q}=-(\sum_kP_kQ_k)e_0+\mathbf{P}\times\mathbf{Q}$, so the bracket is twice the cross product,

$$ [\mathbf{P},\mathbf{Q}]=2\,\mathbf{P}\times\mathbf{Q}, $$

with the cross product taken in $\mathbb{C}^3$ under $\mathrm{Vect}(\mathbb{B})\cong\mathbb{C}^3$. The map $\mathbf{P}\mapsto\operatorname{ad}_{\mathbf{P}}|_{\mathrm{B}_0}$ is then the cross-product operator, and for a real vector part it is the antisymmetric map represented by a real antisymmetric $3\times3$ matrix.

The action of the bracket on each of the six distinguished subspaces — the centre $\mathbb{C}_{\mathbb{B}}$, the vector subspace $\mathrm{Vect}(\mathbb{B})$, the quaternion and anti-quaternion subspaces $\mathbb{H}_{\mathbb{B}}$ and $i\mathbb{H}_{\mathbb{B}}$, and the Hermitian and anti-Hermitian sectors $\mathbb{M}_+$ and $\mathbb{M}_-$ — is the commutator row of the tables of Relations Between Subspaces. In brief, the centre is central, $\mathrm{Vect}(\mathbb{B})$ is closed and the bracket on it is the cross product above, $\mathbb{H}_{\mathbb{B}}$ is closed with the bracket of the imaginary quaternions, the bracket carries $\mathbb{M}_+$ to $\mathbb{M}_-$ and $\mathbb{M}_-$ to itself, and the bracket of $\mathbb{M}_+$ with $\mathbb{M}_-$ carries the second back to the first.

Physical reading. $[\tilde{P},\tilde{Q}]=2\,\mathbf{P}\times\mathbf{Q}$ is the commutator of the angular momentum operators: the algebra of rotations is the cross-product algebra, and the factor two is the factor that the half-angle of the rotor carries. That the derived subalgebra is the whole trace-free part says the algebra is not solvable and has no abelian ideal beyond the phase, so the framework's symmetry is semisimple up to the central $U(1)$ — the algebraic reason its gauge structure has no free abelian factor beyond the phase.

The Real Structure and the Adjoint Maps

Over $\mathbb{R}$ the algebra is the Lie algebra of the real Lie group $\mathbb{B}^\times$, and the trace-free part $\mathrm{B}_0$, of real dimension $6$, is its real form, with $[\mathrm{G},\mathrm{G}]=\mathrm{B}_0$ as over $\mathbb{C}$. The adjoint maps are

$$ \operatorname{ad}_{\tilde{Q}}:X\mapsto[\tilde{Q},X], $$

and $\tilde{Q}\mapsto\operatorname{ad}_{\tilde{Q}}$ is a Lie algebra homomorphism with kernel the centre $\mathbb{C}e_0$. The derivations, the inner automorphisms they exponentiate to and the identification of the derivation algebra are Biquaternion Automorphisms and Derivations.

Summary

The biquaternion algebra, with the commutator bracket, is the complex Lie algebra $\mathrm{G}$ of complex dimension $4$ and real dimension $8$, with centre $\mathbb{C}e_0$ and trace-free part $\mathrm{B}_0=\mathrm{span}_\mathbb{C}\{e_1,e_2,e_3\}$ of complex dimension $3$ and real dimension $6$. The derived subalgebra is the vector subspace itself, $[\mathrm{G},\mathrm{G}]=\mathrm{Vect}(\mathbb{B})$, so the algebra is not solvable; the bracket of two pure vectors is twice their cross product. Over $\mathbb{R}$ the trace-free part is the sum of the two three-dimensional real summands $\mathrm{K}=\mathrm{span}_{\mathbb{R}}\{e_1,e_2,e_3\}$ and $\mathrm{span}_{\mathbb{R}}\{ie_1,ie_2,ie_3\}$, and on the first the bracket is twice the cross product.

Summary of Notation

Symbol	Meaning
$\mathrm{G}$	$\mathbb{B}$ as a Lie algebra with the commutator; complex dim $4$, real dim $8$
$\mathrm{Z}(\mathrm{G})=\mathbb{C}e_0$	Centre; the phase generator
$\mathrm{B}_0=\mathrm{span}_\mathbb{C}\{e_1,e_2,e_3\}$	Trace-free subalgebra; complex dim $3$, real dim $6$
$[\mathbf{P},\mathbf{Q}]=2\,\mathbf{P}\times\mathbf{Q}$	Bracket of two pure vectors, twice the cross product
$[\mathrm{G},\mathrm{G}]=\mathrm{Vect}(\mathbb{B})$	Derived subalgebra is the vector subspace
$\operatorname{ad}_{\tilde{Q}}=[\tilde{Q},\cdot]$	Adjoint maps