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 $\mathrm{GL}(2,\mathbb{C})$ over $\mathbb{C}$ with its centre, the trace-free part as $\mathrm{SL}(2,\mathbb{C})$, the derived subalgebra and its identity with the vector subspace, the bracket of two vectors as twice their outer product, equivalently twice the 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 bracket is the infinitesimal counterpart of the group. The group of units, the exponential and its parametrisation are in Biquaternion Lie Group and Exponential Structure; the topology of the group is in The Biquaternion Unit Group as a Topological Group; and the motions the bracket generates are in Biquaternion Rotations and Lorentz Transformations. The algebra, its basis and its six subspaces are from Biquaternion Algebra, and the behaviour of the product and the bracket on each subspace is tabulated in Biquaternion Relations Between Subspaces, cited below.

Conventions. The biquaternion algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, with basis $e_0=1,e_1,e_2,e_3$ and central scalar imaginary $i$; a general element is $\tilde{Q}=\sum_{\mu=0}^{3} Q_\mu e_\mu$ with $Q_\mu\in\mathbb{C}$. Throughout, a biquaternion is written $\tilde{Q}=Q_0e_0+\mathbf{Q}$ with $\mathbf{Q}=\sum_{k=1}^3 Q_k e_k$.

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The Commutator Bracket

The trace functional is

$$ \mathrm{Tr}(\tilde{Q}) = 2Q_0 . $$

Dimension. As a complex vector space, $\mathbb{B} \cong \mathbb{C}^4$ has complex dimension $4$; as a real vector space it has real dimension $8$. The trace-free part has complex dimension $3$ and real dimension $6$.

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 center 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 center gives the decomposition

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

with complex dimensions $4 = 3 + 1$ and real dimensions $8 = 6 + 2$. Here $\mathrm{B}_0=\{\tilde{Q}:Q_0=0\}$ is the trace-free subalgebra, the complex pure-vector part, the subject of the next section.

The Trace-Free Subalgebra

The Lie algebra of $\mathbb{B}^\times_1$ 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 Outer Product and the Cross Product

The product of two biquaternions splits into the two halves of Biquaternion Multiplication,

$$ \tilde{P}\tilde{Q} = \tilde{P}\bullet\tilde{Q} + \tilde{P}\wedge\tilde{Q}, $$

the symmetrized product $\tilde{P}\bullet\tilde{Q}=\tfrac{1}{2}(\tilde{P}\tilde{Q}+\tilde{Q}\tilde{P})$ and the outer product $\tilde{P}\wedge\tilde{Q}=\tfrac{1}{2}(\tilde{P}\tilde{Q}-\tilde{Q}\tilde{P})$. The commutator is twice the outer product,

$$ [\tilde{P},\tilde{Q}] = \tilde{P}\tilde{Q}-\tilde{Q}\tilde{P} = 2\,\tilde{P}\wedge\tilde{Q}, $$

and the outer product of any two biquaternions is a pure vector, so the bracket depends on the vector parts alone.

Theorem. For two pure vectors the outer product is the cross product, and the bracket is twice either of them,

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

with the cross product taken in $\mathbb{C}^3$ under the identification $\mathrm{Vect}(\mathbb{B})\cong\mathbb{C}^3$.

Proof. For pure vectors the product splits as $\mathbf{P}\mathbf{Q}=-(\textstyle\sum_k P_kQ_k)e_0+\mathbf{P}\times\mathbf{Q}$ and $\mathbf{Q}\mathbf{P}=-(\textstyle\sum_k P_kQ_k)e_0-\mathbf{P}\times\mathbf{Q}$; the difference is $\mathbf{P}\mathbf{Q}-\mathbf{Q}\mathbf{P}=2\,\mathbf{P}\times\mathbf{Q}$, and the outer product is half of that difference.

The map $\mathbf{P}\mapsto\mathrm{ad}_{\mathbf{P}}|_{\mathrm{B}_0}$ is then the cross-product operator, and for a real vector part the operator so defined is antisymmetric.

The Derived Subalgebra and the Subspaces

The commutator of two biquaternions has vanishing scalar part, since the scalar part commutes with everything: $$ [\tilde{P},\tilde{Q}] = [\mathbf{P},\mathbf{Q}]. $$ 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 of Biquaternion Algebra, and the quotient $\mathrm{G}/[\mathrm{G},\mathrm{G}]$ is the centre $\mathbb{C}e_0$ of dimension $1$ over $\mathbb{C}$.

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 Biquaternion Relations Between Subspaces, read there along with the product and the symmetrized product. 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; the exact signs are those tables'.

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 $$ \operatorname{ad}_{\tilde{Q}} : \tilde{P}\mapsto[\tilde{Q},\tilde{P}], $$ and the map $\tilde{Q}\mapsto\operatorname{ad}_{\tilde{Q}}$ is a Lie algebra homomorphism whose kernel is the centre $\mathbb{C}e_0$.

Summary

Under the commutator $[\tilde{P},\tilde{Q}]=\tilde{P}\tilde{Q}-\tilde{Q}\tilde{P}$ the biquaternion algebra is a complex Lie algebra of complex dimension $4$ and real dimension $8$, isomorphic to $\mathrm{GL}(2,\mathbb{C})$; its centre is the scalar line $\mathbb{C}e_0$, of complex dimension $1$. The trace functional is $\mathrm{Tr}(\tilde{Q})=2Q_0$, and the trace-free part $\mathrm{B}_0=\{\tilde{Q}:Q_0=0\}=\mathrm{span}_{\mathbb{C}}\{e_1,e_2,e_3\}$ has complex dimension $3$ and real dimension $6$, isomorphic to $\mathrm{SL}(2,\mathbb{C})$; the algebra splits as $\mathrm{G}=\mathrm{B}_0\oplus\mathbb{C}e_0$.

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 outer product, equivalently twice their cross product, $[\mathbf{P},\mathbf{Q}]=2\,\mathbf{P}\wedge\mathbf{Q}=2\,\mathbf{P}\times\mathbf{Q}$. 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

Further Reading

• Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Springer, 2nd ed. 2015).
• John Stillwell, Naive Lie Theory (Springer, 2008).
• J. P. Ward, Quaternions and Cayley Numbers: Algebra and Applications (Kluwer, Dordrecht, 1997).
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001).