The Lie Algebra of the Antisymmetric Plain Algebra

Introduction

The antisymmetric plain algebra of $\mathbb{H}$ carries the cross product $\tilde q_1\times\tilde q_2=\tfrac12(\tilde q_1\tilde q_2-\tilde q_2\tilde q_1)=\vec q_1\times\vec q_2$ of Introduction to the Antisymmetric Plain Algebra, which is alternating and satisfies the Jacobi identity (The Specificities of the Antisymmetric Plain Algebra, below this article in this category). By the definition of the Lie structure it is a real Lie algebra of dimension four, written

$$ \mathfrak{g}=(\mathbb{H},\times). $$

This article reads its structure: the centre, the derived algebra, the structure constants, the ideals and the two identifications of the derived algebra with the classical compact Lie algebras same name.

This is not the algebra $\mathbb{H}$. The Lie bracket is the cross product of the vector parts and ignores the scalars; the associative multiplication of $\mathbb{H}$ plays no role. Throughout, the natural sign ${}^{\natural}$ is the quaternion conjugation, the bar is the identity because $\mathbb{R}$ has no nontrivial involution, and the star coincides with the natural sign and gives nothing new.

The operation and its table are Introduction to the Antisymmetric Plain Algebra; the elements of the structure are The Killing Form of the Antisymmetric Plain Algebra and The Adjoint Operators and the Derivations of the Antisymmetric Plain Algebra, both below this article in this category, and the general theory of Lie algebras is Structure of Lie Algebras.

The Lie Algebra and Its Centre

Theorem. The space $\mathfrak{g}=\mathbb{H}$ with the bracket $\tilde q_1\times\tilde q_2$ is a real Lie algebra of dimension four. Its bracket vanishes on the scalar direction,

$$ e_t\times\tilde q=0\qquad\text{for every }\tilde q, $$

and on the pure elements it is the vector product of $\mathbb{R}^3$. Its centre is the scalar line,

$$ \boxed{\;\mathfrak{z}(\mathfrak{g})=\{\tilde q_1:\tilde q_1\times\tilde q_2=0\ \text{for every }\tilde q_2\}=\mathbb{R}e_t\;}, $$

of real dimension one.

Proof. The bracket is bilinear and alternating and satisfies the Jacobi identity, which is the Lie structure. The vanishing on the scalar direction is the table; the fixed elements of the cross product are the multiples of the scalar direction, since $\vec q_1\times\vec q_2=0$ for every $\vec q_2$ forces $\vec q_1=0$.

The Derived Algebra and the Identifications

Theorem (the derived algebra). The derived algebra is the vector subspace,

$$ \boxed{\;[\mathfrak{g},\mathfrak{g}]=\mathrm{Vect}(\mathbb{H})=\{\vec q\}\;}, $$

of real dimension three, and it is an ideal. The quotient $\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$ is the one-dimensional abelian algebra $\mathbb{R}e_t$, so

$$ \mathfrak{g}=\mathbb{R}e_t\oplus\mathrm{Vect}(\mathbb{H}) $$

is the direct sum of its centre and its derived algebra, two ideals.

Proof. The bracket of two elements is pure, so $[\mathfrak{g},\mathfrak{g}]\subseteq\mathrm{Vect}(\mathbb{H})$; and the pure brackets generate the vector subspace, since $\vec e_x\times \vec e_y=\vec e_z$ with its cyclic images give a basis. The pure subspace is an ideal because $[\tilde q_1,\vec q_2]$ is pure for every $\tilde q_1$. The sum is direct because the centre meets the derived algebra at $0$, and both are ideals, so the bracket is the sum of the two brackets.

The Isomorphism with $\mathfrak{su}(2)$

Theorem. The derived algebra is isomorphic to the real Lie algebra $\mathfrak{su}(2)$ of the traceless skew-Hermitian $2\times2$ complex matrices, by

$$ \boxed{\;\vec e_x\mapsto-\tfrac{i}{2}\sigma_1,\qquad \vec e_y\mapsto-\tfrac{i}{2}\sigma_2,\qquad \vec e_z\mapsto-\tfrac{i}{2}\sigma_3\;}. $$

Proof. The Pauli matrices satisfy $\sigma_j\sigma_k=\delta_{jk}\mathrm{I}_2+i\sum_l\varepsilon_{jkl}\sigma_l$, so $[-\tfrac{i}{2}\sigma_j,-\tfrac{i}{2}\sigma_k]=-\tfrac14[\sigma_j,\sigma_k]=\sum_l\varepsilon_{jkl}(-\tfrac{i}{2}\sigma_l)$; the images reproduce the structure constants of the vector product, and they are traceless and skew-Hermitian. The map is a Lie isomorphism onto $\mathfrak{su}(2)$.

The Isomorphism with $\mathfrak{so}(3)$

Theorem. The derived algebra is isomorphic to the real Lie algebra $\mathfrak{so}(3)$ of the skew-symmetric $3\times3$ real matrices, by $\vec q_1\mapsto C_{\vec q_1}$, the cross-product matrix $C_{\vec q_1}\vec q_2=\vec q_1\times\vec q_2$. Under this map the adjoint representation of $\mathfrak{g}$ on the derived algebra is

$$ \operatorname{ad}_{\tilde q_1}\longmapsto C_{\vec q_1},\qquad \operatorname{ad}_{\tilde q_1}=\operatorname{diag}(0,C_{\vec q_1})\ \text{on}\ \mathbb{H}=\mathbb{R}e_t\oplus\mathrm{Vect}(\mathbb{H}) . $$

Proof. The cross-product matrix is skew-symmetric because $\vec q_1\times\vec q_2\perp\vec q_1$, and $C_{\vec q_1}\vec q_2=\vec q_1\times\vec q_2$ is the defining matrix of the cross product; the identification $C_{\vec q_1}\leftrightarrow\vec q_1$ carries the bracket to the commutator, $[C_{\vec q_1},C_{\vec q_2}]=C_{\vec q_1\times\vec q_2}$, since the cross product satisfies the Jacobi identity. The adjoint vanishes on the centre and acts on the pure part by the cross product.

Remark (the three presentations). The three algebras $\mathbb{R}^3$ with the cross product, $\mathfrak{su}(2)$ and $\mathfrak{so}(3)$ are one and the same real Lie algebra of dimension three, the compact form of $\mathfrak{sl}(2,\mathbb{C})$; the isomorphism between the last two is $\vec q\mapsto C_{\vec q}$ through the vector product. This is the block's contribution to the dictionary of the chapter.

The Structure Constants

In the basis $(e_t,\vec e_x,\vec e_y,\vec e_z)$ the bracket is

$$ \boxed{\;[e_t,\vec e_\nu]=0,\qquad [\vec e_\mu,\vec e_\nu]=\sum_{\lambda=1}^{3}\varepsilon_{\mu\nu\lambda}\vec e_\lambda\quad(\mu,\nu,\lambda\in\{1,2,3\})\;}, $$

with $\varepsilon$ the Levi-Civita symbol, and the table is the one of the cross product of $\mathbb{R}^3$ extended by zero on the scalar direction.

The Ideals, the Simplicity and the Centre

Theorem. The ideals of $\mathfrak{g}$ are

$$ \boxed{\;0,\qquad \mathbb{R}e_t,\qquad \mathrm{Vect}(\mathbb{H}),\qquad \mathfrak{g}\;}, $$

and there are no others. The algebra is not simple and not semisimple; its radical is the centre $\mathbb{R}e_t$; and the derived algebra $\mathrm{Vect}(\mathbb{H})$ is simple.

Proof. An ideal is a subspace stable under the brackets. The centre and the derived algebra are ideals, and the sum is direct; a nonzero proper ideal $\mathfrak{a}$ that is not the centre meets the derived algebra, and then its intersection with the derived algebra is an ideal of the derived algebra, which is $\mathfrak{so}(3)$, simple of dimension three, so the intersection is the whole derived algebra and $\mathfrak{a}$ contains it; if $\mathfrak{a}$ also contains the centre it is $\mathfrak{g}$, otherwise $\mathfrak{a}=\mathrm{Vect}(\mathbb{H})$. The radical is the largest solvable ideal, and the algebra is the sum of the abelian centre and the simple derived algebra, so the radical is the centre and $\mathfrak{g}$ is reductive and not semisimple.

The Derived Series, Solvability and Nilpotency

Proposition. The derived series stabilises at the derived algebra, $\mathfrak{g}^{(1)}=[\mathfrak{g},\mathfrak{g}]=\mathrm{Vect}(\mathbb{H})$, $\mathfrak{g}^{(2)}=[\mathrm{Vect}(\mathbb{H}),\mathrm{Vect}(\mathbb{H})]=\mathrm{Vect}(\mathbb{H})$, and the lower central series does the same, so $\mathfrak{g}$ is neither solvable nor nilpotent. Its derived algebra is perfect, $[\mathrm{Vect}(\mathbb{H}),\mathrm{Vect}(\mathbb{H})]=\mathrm{Vect}(\mathbb{H})$.

Proof. The derived algebra is $\mathfrak{so}(3)$, which is perfect, so both series freeze at it; a solvable or nilpotent algebra would have a vanishing terminal term.

Remark (the enveloping algebra). Because $\mathfrak{g}=\mathbb{R}e_t\oplus\mathrm{Vect}(\mathbb{H})$ is a direct sum of ideals, the universal enveloping algebra is $\mathrm{U}(\mathfrak{g})\cong\mathbb{R}[z]\otimes\mathrm{U}(\mathfrak{su}(2))$: the centre contributes the polynomial ring in one variable and the derived algebra the enveloping algebra of $\mathfrak{su}(2)$, whose centre is the Casimir of The Killing Form of the Antisymmetric Plain Algebra, below this article in this category.

Summary

The antisymmetric plain block is the real Lie algebra $\mathfrak{g}=(\mathbb{H},\times)$ of dimension four, the cross product of the vector parts extended by zero on the scalar direction. Its centre is the scalar line $\mathbb{R}e_t$ and its derived algebra the vector subspace $\mathrm{Vect}(\mathbb{H})$, so $\mathfrak{g}=\mathbb{R}e_t\oplus\mathrm{Vect}(\mathbb{H})$ is the direct sum of two ideals, reductive with radical the centre and not semisimple. The derived algebra is the three-dimensional simple algebra $\mathfrak{so}(3)$, also isomorphic to $\mathfrak{su}(2)$ by the Pauli matrices and to $\mathbb{R}^3$ with the cross product; the structure constants are the Levi-Civita symbol extended by zero. The ideals are $0$, the centre, the derived algebra and $\mathfrak{g}$; the algebra is neither solvable nor nilpotent, its derived algebra being perfect. Over $\mathbb{C}$ the same structure has $\mathfrak{sl}(2,\mathbb{C})$ in the place of the compact form $\mathfrak{su}(2)$.

Summary of Notation

Further Reading

• Introduction to the Antisymmetric Plain Algebra, for the bracket and the basis table.
• The Specificities of the Antisymmetric Plain Algebra, below this article in this category, for the Jacobi identity and the isolation of the block.
• The Killing Form of the Antisymmetric Plain Algebra, below this article in this category, for the invariant form, the radical and the Casimir element.
• The Adjoint Operators and the Derivations of the Antisymmetric Plain Algebra, below this article in this category, for the adjoint representation, its kernel and its image.
• Structure of Lie Algebras, for the general theory of ideals, the radical and the semisimple algebras.