The Lie Algebra of the Antisymmetric Plain Algebra

Introduction

The antisymmetric part of the plain product of the biquaternions, introduced in Introduction to the Antisymmetric Plain Algebra,

$$ \tilde Q_1\wedge\tilde Q_2=\tfrac12\bigl(\tilde Q_1\tilde Q_2-\tilde Q_2\tilde Q_1\bigr)=\vec Q_1\times\vec Q_2, $$

is $\mathbb{C}$-bilinear, alternating and satisfies the Jacobi identity, so it makes the underlying complex space $\mathbb{B}$ a Lie algebra, written

$$ \mathfrak{g}=\bigl(\mathbb{B},\wedge\bigr), $$

of complex dimension four and real dimension eight. This article develops the structure of that Lie algebra: its centre, which is the centre $\mathbb{C}_{\mathbb{B}}$ of the biquaternion algebra, and the quotient by it; its derived algebra $\mathrm{Vect}(\mathbb{B})$, stable under the bracket, of complex dimension three and real dimension six; the identifications of that derived algebra with the cross-product algebra of $\mathbb{C}^3$ and with $\mathfrak{sl}(2,\mathbb{C})$; the readings of the quaternion subspace and the anti-Hermitian subspace on the real forms, carrying $\mathbb{R}e_t\oplus\mathfrak{su}(2)$ and $\mathfrak{u}(2)$; the derived series, the readings of solvability, nilpotency and simplicity; the ideals and the quotients; and the enveloping algebra.

The bracket and its Jacobi identity are established in Introduction to the Antisymmetric Plain Algebra and are used here, not reproved. The general theory of Lie algebras, of their ideals, of their solvable and nilpotent series, of simplicity and of the enveloping algebra is Lie Algebras and Structure of Lie Algebras; the associativity of the plain product, from which the Jacobi identity descends, is Associative Algebras and The Commutator Operator; the remarkable subspaces and their names are Introduction to the Remarkable Subspaces, and their readings under the four general products are Remarkable Subspaces and the Four General Products, Forms and Norms; the skew-Hermitian elements and the group of linear maps preserving their form are The Unitary Lie Algebra; and the splitting of the plain product into the two halves is The Symmetric and Antisymmetric Parts of an Algebra Product. The article reads the Lie algebra as a module over $\mathbb{C}$ with an alternating bracket, and no group, no exponential and no space enters.

The Centre and the Quotient

The Centre of the Bracket

Definition. The centre of $\mathfrak{g}$ is

$$ \mathfrak{z}(\mathfrak{g})=\{\tilde Q_1\in\mathbb{B}:\tilde Q_1\wedge\tilde Q_2=0\text{ for all }\tilde Q_2\in\mathbb{B}\}. $$

Theorem (the centre is the centre of the algebra). For $\tilde Q_1\in\mathbb{B}$ the following are equivalent: $\tilde Q_1$ lies in $\mathfrak{z}(\mathfrak{g})$; its vector part is zero, $\vec Q_1=0$; and $\tilde Q_1$ lies in $\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_t$, the centre of the biquaternion algebra. Hence

$$ \boxed{\; \mathfrak{z}(\mathfrak{g})=\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_t \;}, $$

of complex dimension one and real dimension two.

Proof. The bracket depends only on the vector parts, $\tilde Q_1\wedge\tilde Q_2=\vec Q_1\times\vec Q_2$, so it vanishes for every $\tilde Q_2$ exactly when $\vec Q_1\times\vec Q_2=0$ for every vector $\vec Q_2$. If $\vec Q_1\neq0$, a vector $\vec Q_2$ that is not parallel to $\vec Q_1$ gives a nonzero cross product, so the condition forces $\vec Q_1=0$; conversely $\vec Q_1=0$ gives $\tilde Q_1\in\mathbb{C}e_t$. The centre of the biquaternion algebra is $\mathbb{C}e_t$, the complex multiples of the unit, because the quaternion units anticommute off the scalar direction; the second equivalence is that standard fact.

Remark. The centre of the bracket and the centre of the algebra are therefore the same subspace, and it is the first structural coincidence of the block: the elements the bracket cannot see are exactly the central elements of the multiplication, the complex multiples of $e_t$. The centre is abelian under the bracket, and it is annihilated by the adjoint of every element.

The Quotient

Proposition. The bracket induced on the quotient $\mathbb{B}/\mathbb{C}_{\mathbb{B}}$ is the cross product, and the projection $\mathbb{B}\to\mathbb{B}/\mathbb{C}_{\mathbb{B}}$ is a morphism of Lie algebras with kernel the centre.

Proof. The bracket of two cosets is well defined because the bracket vanishes on the centre in each slot; the representatives differ from one another by central elements and the bracket of a central element with anything is zero. The quotient is identified with the vector space of the vector parts, on which the induced bracket is the cross product.

Remark (the bracket is the direct sum of a trivial centre and the vector part). The centre and the vector subspace are complementary, $\mathbb{B}=\mathbb{C}_{\mathbb{B}}\oplus\mathrm{Vect}(\mathbb{B})$, and the bracket never leaves the vector subspace. Hence $\mathfrak{g}$ is the direct sum of Lie algebras

$$ \mathfrak{g}=\mathbb{C}_{\mathbb{B}}\oplus\mathrm{Vect}(\mathbb{B}), $$

the first an abelian Lie algebra of complex dimension one and the second the cross-product algebra of complex dimension three. The bracket is a cross product on the vector part, and the scalar direction is a central summand.

The Derived Algebra and the Identifications

The Vector Subspace is the Derived Algebra

Theorem. The image of the bracket is the vector subspace, and it is the derived algebra of $\mathfrak{g}$:

$$ [\mathfrak{g},\mathfrak{g}]=\mathrm{Vect}(\mathbb{B})=\mathbb{C}\vec e_x\oplus\mathbb{C}\vec e_y\oplus\mathbb{C}\vec e_z, $$

of complex dimension three and real dimension six; the bracket restricted to it is again the cross product, so $[\mathfrak{g},\mathfrak{g}]$ is stable under the bracket and is a Lie subalgebra of $\mathfrak{g}$.

Proof. Every value of the bracket has scalar part zero, by the scalar-part computation of Introduction to the Antisymmetric Plain Algebra, so the image lies in the vector subspace; conversely $\vec e_x\wedge \vec e_y=\vec e_z$, $\vec e_y\wedge \vec e_z=\vec e_x$ and $\vec e_z\wedge \vec e_x=\vec e_y$ generate the vector subspace, so the image and the derived algebra are equal to it. The bracket of two vector elements is a cross product and so a vector element, which is the stability. Computed exactly on the units and on general elements.

Corollary. The derived algebra is perfect for the vector part, $[\mathrm{Vect}(\mathbb{B}),\mathrm{Vect}(\mathbb{B})]=\mathrm{Vect}(\mathbb{B})$, and the derived algebra of the whole algebra is reached in one step, $[\mathfrak{g},\mathfrak{g}]=\mathrm{Vect}(\mathbb{B})$.

The Identification with the Cross Product and with $\mathfrak{sl}(2,\mathbb{C})$

Theorem. The derived algebra is isomorphic, as a complex Lie algebra, to the cross-product algebra of $\mathbb{C}^3$ and hence to $\mathfrak{sl}(2,\mathbb{C})$:

$$ \boxed{\; \bigl(\mathrm{Vect}(\mathbb{B}),\wedge\bigr)\cong\bigl(\mathbb{C}^3,\times\bigr)\cong\mathfrak{sl}(2,\mathbb{C}) \;}, $$

of complex dimension three and real dimension six.

Proof. The assignment $Q_{1x}\vec e_x+Q_{1y}\vec e_y+Q_{1z}\vec e_z\mapsto(Q_{1x},Q_{1y},Q_{1z})$ carries the bracket to the cross product, since $\vec e_x\wedge \vec e_y=\vec e_z$ and its cyclic analogues are the cross-product table. The cross-product algebra of $\mathbb{C}^3$ is the complex orthogonal algebra $\mathfrak{so}(3,\mathbb{C})$, and the classical isomorphism $\mathfrak{so}(3,\mathbb{C})\cong\mathfrak{sl}(2,\mathbb{C})$ of the structure theory of Structure of Lie Algebras applies; the isomorphism is exhibited in the two-dimensional matrix model in Introduction to the Symmetric Plain Algebra, where $\tilde Q_1\wedge\tilde Q_2$ becomes $\tfrac12[\,\mathsf{M}_2(\tilde Q_1),\mathsf{M}_2(\tilde Q_2)\,]$ on the trace-free matrices.

Remark (the two descriptions of the same Lie algebra). The identification is the reason the block is called the Lie block: the derived algebra of the antisymmetric plain algebra is $\mathfrak{sl}(2,\mathbb{C})$, the smallest simple complex Lie algebra, and the whole algebra is $\mathfrak{sl}(2,\mathbb{C})$ with a one-dimensional central summand. The identification up to the factor two between the halved bracket and the matrix commutator is the one of Introduction to the Antisymmetric Plain Algebra: the matrix commutator is twice the block's bracket, and the two presentations differ by the rescaling $\tilde Q_1\mapsto2\tilde Q_1$ of the Lie algebra.

The Real Forms

Theorem (the quaternion subspace). On the real span $\mathbb{H}_{\mathbb{B}}$ of $e_t,\vec e_x,\vec e_y,\vec e_z$, the quaternion subspace, the bracket is real and takes values in the real vector triple $\mathbf{B}_1=\mathbb{R}\vec e_x\oplus\mathbb{R}\vec e_y\oplus\mathbb{R}\vec e_z$. Hence $\mathbb{H}_{\mathbb{B}}$ is a real Lie subalgebra of real dimension four, and

$$ \mathbb{H}_{\mathbb{B}}\cong\mathbb{R}e_t\oplus\mathfrak{su}(2), $$

with $\mathbb{R}e_t$ central and $\mathbf{B}_1$ identified with $\mathfrak{su}(2)$.

Proof. The bracket of two real quaternions is the cross product of two real vectors and is therefore a real vector, so $\mathbb{H}_{\mathbb{B}}$ is stable and $[\mathbb{H}_{\mathbb{B}},\mathbb{H}_{\mathbb{B}}]=\mathbf{B}_1$. The real vector triple with the cross product is the real form $\mathfrak{so}(3,\mathbb{R})\cong\mathfrak{su}(2)$ of the derived algebra, and the scalar direction is central; the sum is direct. Computed on the real basis.

Theorem (the anti-Hermitian subspace). On the anti-Hermitian subspace

$$ \mathbb{M}_-=\mathbb{R}(ie_t)\oplus\mathbf{B}_1=\{Q_te_t+\vec Q:Q_t\in i\mathbb{R},\ \vec Q\in\mathbb{R}^3\}, $$

of real dimension four, the bracket is real, takes values in $\mathbf{B}_1$ and makes $\mathbb{M}_-$ a real Lie subalgebra,

$$ \mathbb{M}_-\cong\mathfrak{u}(2), $$

whose derived algebra is $[\mathbb{M}_-,\mathbb{M}_-]=\mathbf{B}_1\cong\mathfrak{su}(2)$.

Proof. An element of $\mathbb{M}_-$ has scalar part purely imaginary and vector part real. The bracket sees only the vector parts, so it is the cross product of two real vectors, a real vector in $\mathbf{B}_1$; hence $\mathbb{M}_-$ is stable and its derived algebra is $\mathbf{B}_1$. The real Lie algebra with a one-dimensional central direction and derived algebra $\mathfrak{su}(2)$ is $\mathfrak{u}(2)$, the Lie algebra of the skew-Hermitian matrices, by its standard description in The Unitary Lie Algebra, §The Skew-Hermitian Part as a Lie Algebra; the identification is read there on the skew-Hermitian elements, of which $\mathbb{M}_-$ is the biquaternion case. Computed on the real basis.

Remark. The two theorems are two real forms of the same complex situation: the complex derived algebra $\mathfrak{sl}(2,\mathbb{C})$ restricts to $\mathfrak{su}(2)$ on the real vector triple, and the two real subspaces that contain that triple are the quaternion subspace and the anti-Hermitian subspace, carrying the central extension $\mathbb{R}e_t$ and the central extension $\mathbb{R}(ie_t)$ respectively. The quaternion subspace is $\mathbb{R}e_t\oplus\mathfrak{su}(2)$ and the anti-Hermitian subspace is $\mathfrak{u}(2)=\mathbb{R}(ie_t)\oplus\mathfrak{su}(2)$; the two differ only in which central direction the scalar part occupies. The group of the linear maps preserving the form of the skew elements is not read here: it is The Unitary Lie Algebra.

Ideals, Quotients and the Two Series

The Ideals

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

$$ \boxed{\; 0,\qquad \mathbb{C}_{\mathbb{B}},\qquad \mathrm{Vect}(\mathbb{B}),\qquad \mathfrak{g} \;}. $$

Proof. The centre and the derived algebra are ideals, because the bracket vanishes on the centre and the derived algebra is mapped into itself. For the converse, take a nonzero ideal $\mathfrak{a}$ and an element $\tilde Q_1\in\mathfrak{a}$. If $\vec Q_1=0$ then $\mathfrak{a}$ contains a nonzero central element, hence the whole centre. If $\vec Q_1\neq0$, then $[\mathfrak{g},\tilde Q_1]$ is the span of the cross products $\vec X\times\vec Q_1$ over all vectors $\vec X$, which is the plane orthogonal to $\vec Q_1$; and $[\mathfrak{g},[\mathfrak{g},\tilde Q_1]]$ is all of $\mathrm{Vect}(\mathbb{B})$, since the cross product of two vectors in that plane generates the third direction. Hence an ideal that contains an element with nonzero vector part contains $\mathrm{Vect}(\mathbb{B})$, and then either it stops at $\mathrm{Vect}(\mathbb{B})$ or, if it also contains a nonzero central element, it is $\mathfrak{g}$. Computed on the ideal generated by $e_t$, by $\vec e_x$ and by $e_t+\vec e_x$, of complex dimensions one, three and four.

Corollary (the quotients). The quotients of $\mathfrak{g}$ by its ideals are

$$ \mathfrak{g}/\mathbb{C}_{\mathbb{B}}\cong\mathrm{Vect}(\mathbb{B})\cong\mathfrak{sl}(2,\mathbb{C}),\qquad \mathfrak{g}/\mathrm{Vect}(\mathbb{B})\cong\mathbb{C}_{\mathbb{B}},\qquad \mathrm{Vect}(\mathbb{B})/0\cong\mathfrak{sl}(2,\mathbb{C}), $$

the first semisimple and the second abelian of complex dimension one.

The Derived Series, Solvability and Nilpotency

Theorem. The derived series of $\mathfrak{g}$ is

$$ \mathfrak{g}^{(0)}=\mathfrak{g},\qquad \mathfrak{g}^{(1)}=[\mathfrak{g},\mathfrak{g}]=\mathrm{Vect}(\mathbb{B}),\qquad \mathfrak{g}^{(2)}=[\mathfrak{g}^{(1)},\mathfrak{g}^{(1)}]=\mathrm{Vect}(\mathbb{B}), $$

and it is constant from the first step; the lower central series is likewise constant from the second step. Hence $\mathfrak{g}$ is neither solvable nor nilpotent, and it is not semisimple, its radical being the centre $\mathbb{C}_{\mathbb{B}}$.

Proof. The derived algebra is $\mathrm{Vect}(\mathbb{B})$ and it is perfect, so the derived series stabilises there and never reaches zero. The lower central series begins $\mathfrak{g}\supseteq\mathrm{Vect}(\mathbb{B})$ and $[\mathfrak{g},\mathrm{Vect}(\mathbb{B})]=\mathrm{Vect}(\mathbb{B})$ again, so it also stabilises away from zero. The radical is the largest solvable ideal; the only nonzero proper ideals are the centre, abelian and so solvable, and $\mathrm{Vect}(\mathbb{B})$, not solvable; hence the radical is the centre. Computed on the basis.

Remark. The algebra is reductive and not semisimple. It is the direct sum of an abelian Lie algebra and a simple one, so its radical is the centre and its quotient by the radical is simple; the Cartan reading of the same fact through the Killing form is The Killing Form of the Antisymmetric Plain Algebra, where the form is shown to be degenerate with the centre as its radical.

Simplicity

Theorem. $\mathfrak{g}$ is not simple, because $\mathbb{C}_{\mathbb{B}}$ is a nonzero proper ideal. The derived algebra $\mathrm{Vect}(\mathbb{B})$ is simple: it has no nonzero proper ideal.

Proof. The centre is a nonzero proper ideal, so $\mathfrak{g}$ is not simple. In the identification $\mathrm{Vect}(\mathbb{B})\cong\mathfrak{sl}(2,\mathbb{C})$, the simplicity of $\mathfrak{sl}(2,\mathbb{C})$ is the first case of the classification of Structure of Lie Algebras; it is also read directly from the cross product, since a nonzero ideal of the cross-product algebra contains a plane of directions and the bracket of two directions in that plane generates the third. Computed on the ideal generated by $\vec e_x$, which is the whole vector subspace.

The Enveloping Algebra

Theorem. By the Poincaré–Birkhoff–Witt theorem, applied to the direct sum $\mathfrak{g}=\mathbb{C}_{\mathbb{B}}\oplus\mathrm{Vect}(\mathbb{B})$, the enveloping algebra of $\mathfrak{g}$ is, as a complex vector space,

$$ \mathrm{U}(\mathfrak{g})\cong\mathbb{C}[z]\otimes_{\mathbb{C}}U\bigl(\mathfrak{sl}(2,\mathbb{C})\bigr), $$

where $z$ is the central generator coming from the centre $\mathbb{C}_{\mathbb{B}}$.

Proof. The centre is an abelian Lie algebra of complex dimension one, and its enveloping algebra is the polynomial algebra $\mathbb{C}[z]$ in one indeterminate; the vector part has enveloping algebra $\mathrm{U}(\mathfrak{sl}(2,\mathbb{C}))$. The Lie algebra is a direct sum, so the enveloping algebra of the sum is the tensor product of the enveloping algebras, by the Poincaré–Birkhoff–Witt theorem of Lie Algebras.

Remark (the count). The associated graded object of $\mathrm{U}(\mathfrak{g})$ is the symmetric algebra of $\mathfrak{g}$,

$$ \operatorname{gr}\mathrm{U}(\mathfrak{g})\cong S(\mathfrak{g})\cong \mathbb{C}[z]\otimes_{\mathbb{C}}\mathbb{C}[w_1,w_2,w_3], $$

of which the degree-$n$ part has dimension $\binom{n+3}{3}$, the dimension of $S^n(\mathbb{C}^4)$, the degree-$n$ symmetric power of the four-dimensional space. The identity

$$ \sum_{k=0}^{n}\binom{k+2}{2}=\binom{n+3}{3} $$

is the numerical form of the count, with $\binom{k+2}{2}$ the degree-$k$ part of $\mathbb{C}[w_1,w_2,w_3]$, and it was checked for $n=0,\dots,6$. The centre contributes one polynomial variable and the derived algebra three scaled variables, and the four together are the degree count of the four-dimensional Lie algebra. Over the reals the same statement reads $U_{\mathbb{R}}(\mathfrak{g})\cong\mathbb{R}[z_1,z_2]\otimes_{\mathbb{R}}U_{\mathbb{R}}(\mathfrak{sl}(2,\mathbb{C}))$, with the two real coordinates of the centre.

Remark (the centre of the enveloping algebra). The centre of $\mathrm{U}(\mathfrak{g})$ is $\mathbb{C}[z]$ tensored with the centre of $\mathrm{U}(\mathfrak{sl}(2,\mathbb{C}))$, which is the polynomial algebra in the Casimir element of the derived algebra; the Casimir element itself is computed in The Killing Form of the Antisymmetric Plain Algebra, where it is read on the vector subspace and on the matrix models. No further central element of $\mathrm{U}(\mathfrak{sl}(2,\mathbb{C}))$ is independent, by the theorem on the centre of the enveloping algebra of a semisimple Lie algebra.

Worked Examples

A central bracket. For $\tilde Q_1=2e_t+3ie_t$ the vector part is zero, so $\tilde Q_1\wedge \vec e_x=0$: the two complex coordinates of the centre both stand in the kernel of the adjoint.

A bracket with a scalar part. For $\tilde Q_1=e_t+\vec e_x$ and $\tilde Q_2=e_t+\vec e_y$ the bracket is $\vec e_z$, computed in Introduction to the Antisymmetric Plain Algebra; the scalar parts of the two elements do not enter, and the value lies in the derived algebra.

The derived algebra in one step. $(e_t+\vec e_x)\wedge \vec e_x=0$ and $\vec e_x\wedge \vec e_y=\vec e_z$: the central direction of $\tilde Q_1=e_t+\vec e_x$ contributes nothing, and the derived algebra is generated by the vector parts alone.

The quaternion real form. For $\tilde q=e_t+\vec e_x$ and $\tilde q_1=e_t+\vec e_y$ in $\mathbb{H}_{\mathbb{B}}$ the bracket is $\tilde q\wedge\tilde q_1=\vec e_z$, a real vector element: the quaternion subspace is stable, and its derived algebra is the real vector triple.

The anti-Hermitian real form. For $x=ie_t+\vec e_x$ and $y=ie_t+\vec e_y$ in $\mathbb{M}_-$ the bracket is $x\wedge y=\vec e_z$, again a real vector element: the scalar directions $ie_t$ commute with everything and the derived algebra is again the real vector triple, which is the identification $\mathfrak{u}(2)$ with derived algebra $\mathfrak{su}(2)$.

An element with rank-two adjoint. For $\tilde Q_1=\vec e_x+i\vec e_y$ the adjoint is the complex matrix of rank two on the four-dimensional space, with kernel the complex plane spanned by $e_t$ and $\vec e_x+i\vec e_y$, of complex dimension two; the rank and the kernel are read in The Adjoint Operators and the Derivations of the Antisymmetric Plain Algebra.

Summary

The antisymmetric plain algebra makes $\mathbb{B}$ a complex Lie algebra $\mathfrak{g}$ of dimension four, whose centre is the centre $\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_t$ of the algebra, of dimension one, and whose derived algebra is the vector subspace $\mathrm{Vect}(\mathbb{B})$, of complex dimension three and real dimension six, stable under the bracket and identified with the cross-product algebra of $\mathbb{C}^3$ and with $\mathfrak{sl}(2,\mathbb{C})$. The algebra is the direct sum $\mathfrak{g}=\mathbb{C}_{\mathbb{B}}\oplus\mathrm{Vect}(\mathbb{B})$, its quotient by the centre is $\mathfrak{sl}(2,\mathbb{C})$, its quotient by the derived algebra is the abelian centre, and its only ideals are $0$, the centre, the derived algebra and the whole algebra. The derived series and the lower central series stabilise at the derived algebra, so the algebra is neither solvable nor nilpotent; it is reductive and not semisimple, with the centre as radical, and its derived algebra is simple. On the real forms the quaternion subspace is $\mathbb{R}e_t\oplus\mathfrak{su}(2)$ and the anti-Hermitian subspace is $\mathfrak{u}(2)$, of derived algebra $\mathfrak{su}(2)$. By Poincaré–Birkhoff–Witt the enveloping algebra is $\mathbb{C}[z]\otimes_{\mathbb{C}}\mathrm{U}(\mathfrak{sl}(2,\mathbb{C}))$ as a complex vector space, with associated graded object $\mathbb{C}[z]\otimes\mathbb{C}[w_1,w_2,w_3]$ and degree-$n$ dimension $\binom{n+3}{3}$.

Summary of Notation

Further Reading

• Introduction to the Antisymmetric Plain Algebra, for the bracket, its Jacobi identity and the sixteen brackets of the basis
• The Killing Form of the Antisymmetric Plain Algebra, for the invariant form, its radical and the Casimir element
• The Adjoint Operators and the Derivations of the Antisymmetric Plain Algebra, for the adjoints, their kernels, the derivations and the automorphisms of the bracket
• The Unitary Lie Algebra, for the skew-Hermitian elements, the unhalved commutator and the group of linear maps preserving their form
• Remarkable Subspaces and the Four General Products, Forms and Norms, for the remarkable subspaces read under the four general products
• Lie Algebras: A General Introduction, for the axioms, the ideals, the series, the simplicity and the enveloping algebra
• Structure of Lie Algebras, for the classical algebras, the isomorphism $\mathfrak{so}(3,\mathbb{C})\cong\mathfrak{sl}(2,\mathbb{C})$, the radical and the semisimplicity criteria
• Associative Algebras and The Commutator Operator, for the associativity from which the Jacobi identity descends