The Specificities of the Antisymmetric Quaternionic Algebra

Introduction

The antisymmetric quaternionic multiplication $$ \tilde Q_1\diamond\tilde Q_2=\tfrac12\bigl(\tilde Q_1^{\natural}\tilde Q_2-\tilde Q_2^{\natural}\tilde Q_1\bigr) =Q_{1t}\vec Q_2-Q_{2t}\vec Q_1-\vec Q_1\times\vec Q_2 $$ is alternating, vector valued and without a unit, and it fails the Jacobi identity (Introduction to the Antisymmetric Quaternionic Algebra, The Jacobi Failure and the Associator Defect of the Antisymmetric Quaternionic Algebra). As an element-theoretic object the block is poor: every element is a nilpotent, the origin is the only idempotent, there is no root of $-1$ or of $+1$, and there is no unit and no inverse. As an operator-theoretic object it has one sharp feature, and the feature is the whole of the present article.

The feature is the action of the algebra unit. For the block, $$ L_{e_t}\tilde Q_1=e_t\diamond\tilde Q_1=\vec Q_1 , \qquad R_{e_t}\tilde Q_1=\tilde Q_1\diamond e_t=-\vec Q_1 , \qquad L_{e_t}^{2}=L_{e_t}, $$ so the left multiplication by $e_t$ is the projection onto the vector subspace along the centre, and the right multiplication is its negative. The block is the only one of the twelve structures in which some element multiplies as the projection onto the vector subspace; the element is necessarily $e_t$, and it is uniquely determined. Equivalently, the block is the only one in which the canonical splitting $$ \mathbb{B}=\mathrm{Vect}(\mathbb{B})\oplus\mathbb{C}_{\mathbb{B}} $$ is realised by the multiplication by a single element, the element being the unit of the ambient algebra, which is not a unit of the block. The criterion is stated as the identity $L_{e_t}=P_{\mathrm{Vect}}$, it is proved equivalent to the projector classification $L_{\tilde Q_1}^{2}=L_{\tilde Q_1}\Leftrightarrow\tilde Q_1\in\{0,e_t\}$, and it isolates the block among the twelve structures. The model is The Biquaternion Algebra and Its Vector Spaces, which states left alternativity as its criterion; the twinned articles are The Specificities of the General Quaternionic Algebra and The Specificities of the Symmetric Quaternionic Algebra.

Placement

The article is the closing entry of the block, after the introduction, the Jacobi failure, the invariant forms, the remarkable subspaces, the adjoint operators and the two matrix models. It uses the operation and its image of Introduction to the Antisymmetric Quaternionic Algebra of Biquaternions, the classification of the operators of The Adjoint Operators of the Antisymmetric Quaternionic Algebra, the invariant form of The Invariant Bilinear Forms of the Antisymmetric Quaternionic Algebra, and the vanishing of the alternating centre of The Jacobi Failure and the Associator Defect of the Antisymmetric Quaternionic Algebra; and it proves, for the block, the criterion that isolates it, the unit multiplying as the projection onto the vector subspace.

Conventions

$\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ has basis $e_t,\vec e_x,\vec e_y,\vec e_z$ with $\vec e_x\vec e_y=\vec e_z$ and central scalar imaginary $i$. The operation of the block is $\diamond$, $\tilde Q_1\diamond\tilde Q_2=\tfrac12(\tilde Q_1^{\natural}\tilde Q_2-\tilde Q_2^{\natural}\tilde Q_1)$; the parent product is $\tilde Q_1^{\natural}\tilde Q_2$ and the symmetric half is $\tilde Q_1\star\tilde Q_2=\tfrac12(\tilde Q_1^{\natural}\tilde Q_2+\tilde Q_2^{\natural}\tilde Q_1)$. An element is $\tilde Q_1=Q_{1t}e_t+\vec Q_1$ with $\vec Q_1=Q_{1x}\vec e_x+Q_{1y}\vec e_y+Q_{1z}\vec e_z$; $\mathrm{Vect}(\mathbb{B})$ is the vector subspace of the pure elements, of complex dimension three, and $\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_t$ is the centre. The left multiplication of the block is $L_{\tilde Q_4}\tilde Q_1=\tilde Q_4\diamond\tilde Q_1$, and $L_{\tilde Q_4}^{2}=L_{\tilde Q_4}L_{\tilde Q_4}$ is the composite. A projector is an idempotent operator, and an idempotent is an element equal to its own square; the two levels are never interchanged, and the block is the example that makes the distinction visible: its idempotents are the origin alone, while the operator of the element $e_t$ is a projector.

The Governing Identity

Proposition (the unit acts as the projection onto the vector subspace). For every element $\tilde Q_1$, $$ \boxed{\; L_{e_t}\tilde Q_1=\vec Q_1 , \qquad R_{e_t}\tilde Q_1=-\vec Q_1 , \qquad L_{e_t}^{2}=L_{e_t} \;}. $$ The image of $L_{e_t}$ is $\mathrm{Vect}(\mathbb{B})$, its kernel is $\mathbb{C}_{\mathbb{B}}$, and $\mathbb{B}=\mathrm{Vect}(\mathbb{B})\oplus\mathbb{C}_{\mathbb{B}}$.

Proof. The explicit form at $\tilde Q_1=e_t$, where $Q_{1t}=1$ and $\vec Q_1=\mathbf 0$, gives $e_t\diamond\tilde Q_1=\vec Q_1$, and at $\tilde Q=e_t$ it gives $\tilde Q_1\diamond e_t=-\vec Q_1$. The map $\tilde Q_1\mapsto\vec Q_1$ is the identity on the vector subspace and vanishes on the centre, so it is a projector, its square is itself, its image is $\mathrm{Vect}(\mathbb{B})$ and its kernel is $\mathbb{C}_{\mathbb{B}}$; the two are complementary because every element is the sum of its scalar and vector parts.

Remark (the element is not a unit). The element $e_t$ satisfies $e_t\diamond\tilde Q_1=\vec Q_1$, which is $\tilde Q_1$ only when $\tilde Q_1$ is pure; at a non-pure element the scalar part is discarded. The block has no unit (Introduction to the Antisymmetric Quaternionic Algebra of Biquaternions), and $e_t$ is a left identity on the vector subspace and not a unit: it acts as the identity there and annihilates the centre, and its right action is the negative of its left action. The identity $R_{e_t}=-L_{e_t}$ is the alternation of the block read at the unit.

The Projector Classification

Theorem (the criterion of the block). The left multiplication of an element is a projector exactly at the origin and at the algebra unit, $$ \boxed{\; L_{\tilde Q_1}^{2}=L_{\tilde Q_1}\iff \tilde Q_1=0\ \text{or}\ \tilde Q_1=e_t \;}. $$ The two projectors are the zero operator and the projection onto the vector subspace.

Proof. Write $L_{\tilde Q_1}\tilde Q_2=\tilde{Q}_3$ with $\tilde{Q}_3=Q_{1t}\vec Q_2-Q_{2t}\vec Q_1-\vec Q_1\times\vec Q_2$; the value is pure, so $L_{\tilde Q_1}^{2}\tilde Q_2=L_{\tilde Q_1}\tilde{Q}_3=Q_{1t}\tilde{Q}_3-\vec Q_1\times\tilde{Q}_3$, and the projector condition reads $Q_{1t}\tilde{Q}_3-\vec Q_1\times\tilde{Q}_3=\tilde{Q}_3$ for every $\tilde Q_2$. At $\tilde Q_2=e_t$, where $Q_{2t}=1$ and $\vec Q_2=\mathbf 0$, one has $\tilde{Q}_3=-\vec Q_1$, and the condition becomes $(Q_{1t}-1)\vec Q_1=\mathbf 0$; so either $\vec Q_1=\mathbf 0$ or $Q_{1t}=1$. If $\vec Q_1=\mathbf 0$, then $L_{\tilde Q_1}\tilde Q_2=Q_{1t}\vec Q_2$ and the projector condition is $Q_{1t}^{2}=Q_{1t}$, giving $Q_{1t}=0$ or $Q_{1t}=1$, that is $\tilde Q_1=0$ or $\tilde Q_1=e_t$. If $\vec Q_1\neq\mathbf 0$ and $Q_{1t}=1$, the condition reads $\vec Q_1\times\tilde{Q}_3=0$ for every $\tilde Q_2$, so $\tilde{Q}_3$ is parallel to $\vec Q_1$ at every $\tilde Q_2$; taking $\tilde Q_2$ pure with $\vec Q_2$ perpendicular to $\vec Q_1$ gives $\tilde{Q}_3=\vec Q_2-\vec Q_1\times\vec Q_2$, a vector perpendicular to $\vec Q_1$ and nonzero, against the requirement, so this case is impossible. The two admissible elements give $L_0=0$ and $L_{e_t}=P_{\mathrm{Vect}}$ by the governing identity.

The classification is short and rigid: two projectors, the zero operator at the origin and the projection onto the vector subspace at the unit, and nothing between. It is the smallest projector class a structure can have while carrying a nonzero projector, and it is the exact opposite of the general plain algebra, whose projector class is indexed by the whole idempotent family.

The Splitting Realised by One Element

Corollary (the splitting is a multiplication). The canonical splitting of the space into the vector subspace and the centre is realised by the multiplication of the single element $e_t$: $$ \boxed{\; \mathbb{B}=\mathrm{Vect}(\mathbb{B})\oplus\mathbb{C}_{\mathbb{B}}, \qquad \mathrm{Vect}(\mathbb{B})=\operatorname{im}L_{e_t}, \qquad \mathbb{C}_{\mathbb{B}}=\ker L_{e_t} \;}. $$ The vector subspace is the image of the block and a left ideal of the block, and the centre is the kernel of $L_{e_t}$.

Proof. The splitting and the identifications are the governing identity. The image of the block is $\mathrm{Vect}(\mathbb{B})$, and it is a left ideal because for $\vec A$ pure and any $\tilde Q_2$ the value $\vec A\diamond\tilde Q_2=-Q_{2t}\vec A-\vec A\times\vec Q_2$ is pure; the centre is the kernel of $L_{e_t}$ by the same identity.

The corollary is the useful form of the criterion. In the block the two subspaces the theory names — the derived subalgebra, on which the operation is a Lie bracket, and the centre, on which it vanishes — are the image and the kernel of the multiplication by one element, and the element is the unit of the ambient algebra. No other structure of the twelve has this: elsewhere the splitting, when it exists, is carried by an idempotent, a family, or not at all.

The Isolation among the Twelve

The criterion of the block is the projector classification of §The Projector Classification: the only projectors are the zero operator and the projection onto the vector subspace, the latter at the unit $e_t$, and the splitting of §The Splitting Realised by One Element is carried by the multiplication of that one element.

Theorem (the isolation). Of the twelve structures of The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, the block is the only one in which the left multiplication by the algebra unit is the projection onto the vector subspace, and the only one admitting an element $\tilde E$ with $\tilde E\star\tilde Q_1=\vec Q_1$ for every $\tilde Q_1$; the element is $\tilde E=e_t$, uniquely.

Proof. The block's entry is the governing identity. For the other eleven, group them by the action of the unit. In GPA, GQA and SPA the unit acts as the identity, whose image is the whole space and not the vector subspace; in SQA it acts as the projection onto the centre; in APA and AQS it acts as zero. In the six sesquilinear structures the left multiplication by $e_t$ is conjugate-linear or zero, and a conjugate-linear map is not the $\mathbb{C}$-linear projection onto the vector subspace; a conjugate-linear $L$ with $L^{2}=L$ is linear and conjugate-linear at once, hence zero, so in the six the only projector is the zero operator. For the element $\tilde E$, the equation at $\tilde Q_1=e_t$ reads $\tilde E\star e_t=0$; in GPA, SPA and GPS this forces $\tilde E=0$, in GQA, GQS and SQS it forces $\tilde E^{\natural}=0$, and in SQA and SPS it forces $E_te_t=0$, none compatible with the value $\vec Q_1$ at a pure $\tilde Q_1$; in APA and AQS the value at $e_t$ vanishes identically and the pure values are the cross products $\vec E\times\vec Q_1$ and $\vec E\times\overline{\vec Q_1}$, which lie in the orthogonal plane of $\vec E$; and in APS the value at $e_t$, which is $\vec E$, forces $\tilde E=E_te_t$, after which the values at $\vec e_x$ and at $i\vec e_x$ force the contradictory $E_t=-1$ and $E_t=1$. Inside the block the element is unique: at $\tilde Q_1=e_t$ the equation gives $-\vec E=0$, so $\vec E=0$, and at a pure $\tilde Q_1$ it gives $E_t\tilde Q_1=\tilde Q_1$, so $E_t=1$.

The block is alone in its class, and the reason is the two mixed terms $Q_{1t}\vec Q_2-Q_{2t}\vec Q_1$ that it carries and that the plain antisymmetrisation does not: those terms make the unit act on the scalar direction and so turn its multiplication into a proper projection.

The Other Criteria Fail Here

Two criteria adjacent to the block's are the criterion of the general plain algebra and the criterion of the symmetric quaternionic algebra, and neither holds for the block.

Remark (the alternativity criterion fails). The criterion of the general plain algebra is left alternativity, $L_{\tilde Q_1}^{2}=L_{\tilde Q_1\cdot\tilde Q_1}$ for every $\tilde Q_1$, the square being read in the product at hand; read for the block it becomes $L_{\tilde Q_1}^{2}=L_{\tilde Q_1\diamond\tilde Q_1}=L_0=0$, so the criterion would demand $L_{\tilde Q_1}^{2}=0$ for every element; at $\tilde Q_1=e_t$ the operator is the projection onto the vector subspace, whose square is itself and not zero. The criterion fails at the unit of the algebra, and the block is one of the eleven structures on which it fails.

Remark (the rank-one criterion fails). The criterion of the symmetric quaternionic algebra is the rank-one identity $L^{\bullet}_{\tilde Q_1}{}^{2}=Q_{1t}L^{\bullet}_{\tilde Q_1}$, in which the scalar coordinate is the coefficient. For the block the corresponding display is false away from the two elements of the classification: at $\tilde Q_1=\vec e_x$, where $Q_{1t}=0$, the right side is the zero operator while $L_{\vec e_x}^{2}$ is nonzero. The block's criterion is therefore not a scalar multiple of its own operator; it is the projector identity read at the two elements $0$ and $e_t$.

What the Criterion Delivers

The criterion is a classification and not a decomposition theory. What it delivers for the block is the following short list, and the block carries nothing more.

• Two projectors. The zero operator and the projection onto the vector subspace, the latter at the element $e_t$.
• One splitting. $\mathbb{B}=\mathrm{Vect}(\mathbb{B})\oplus\mathbb{C}_{\mathbb{B}}$, the image and the kernel of $L_{e_t}$.
• One left ideal. The vector subspace, which is the whole image of the block as well as a left ideal.
• No Peirce calculus. The decompositions indexed by an idempotent do not exist, there being no nontrivial idempotent; the ideal theory and the module theory of the plain product have no analogue here.

The block is thus the structure in which the projector survives as a single element and not as a family, and in which the splitting of the space is a statement about one multiplication. This is the exact sense of its isolation among the twelve: the unit carries the projector, the projector is the projection onto the vector subspace, and there is no other.

Summary

The antisymmetric quaternionic multiplication is the alternating structure whose left multiplication by the algebra unit is the projection onto the vector subspace, $L_{e_t}\tilde Q_1=\vec Q_1$, $L_{e_t}^{2}=L_{e_t}$, with $R_{e_t}=-L_{e_t}$; the projector classification of the block is the pair $L_{\tilde Q_1}^{2}=L_{\tilde Q_1}\Leftrightarrow\tilde Q_1\in\{0,e_t\}$, so the projector class is the smallest a structure with a nonzero projector can have. The projection is the canonical splitting $\mathbb{B}=\mathrm{Vect}(\mathbb{B})\oplus \mathbb{C}_{\mathbb{B}}$ realised by the multiplication of one element, the vector subspace being the image of the block and a left ideal and the centre its kernel. The block is the only one of the twelve in which the algebra unit multiplies as the projection onto the vector subspace, and the only one admitting an element $\tilde E$ with $\tilde E\star\tilde Q_1=\vec Q_1$ for every $\tilde Q_1$, the element being $e_t$ uniquely; the other eleven give the identity, the projection onto the centre, the zero operator or a conjugate-linear map at the unit, and none of them gives this projection. The alternativity criterion of the plain algebra and the rank-one criterion of the symmetric quaternionic algebra both fail for the block, and the criterion stated here is its own. What the block delivers is two projectors, one splitting and one left ideal, and no Peirce calculus.

Summary of Notation

Further Reading

• The Biquaternion Algebra and Its Vector Spaces, for the criterion of left alternativity, the model of the family
• The Specificities of the General Quaternionic Algebra, for the first border case, where the projectors are indexed by the plain idempotents
• The Specificities of the Symmetric Quaternionic Algebra, for the second border case, the hyperplane of rank-one projections onto the centre
• Introduction to the Antisymmetric Quaternionic Algebra, for the operation, its image, the one-sided action of the unit and the absence of a unit
• The Adjoint Operators of the Antisymmetric Quaternionic Algebra, for the operators $L_{\tilde Q_4}$, their matrices, their ranks and the elements for which the rank drops
• The Invariant Bilinear Forms of the Antisymmetric Quaternionic Algebra, for the form whose radical is the image of the projection
• Symmetric and Antisymmetric Decomposition of the Four General Products, for the twelve operations and their two alternating rows