The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space
Introduction
This article reads the twelve products of Symmetric and Antisymmetric Decomposition of the Four General Products as the algebraic structures they define, and gives each structure its three-letter name, the abbreviations being read with the table of the twelve. It belongs to the Algebra group of the category Biquaternion Algebraic Structures and carries the names and the laws of the twelve.
Every one of the twelve products equips the complex space with an algebraic structure: its operation is the multiplication of a structure of its own kind, and the structure is read off the laws of that operation, which the table of Symmetric and Antisymmetric Decomposition of the Four General Products collects. The general product of a family is the multiplication of the algebra or the sesqualgebra of that family — the two bilinear families are the two algebras over $\mathbb{C}$ and the two sesquilinear families the two sesqualgebras over $(\mathbb{C},\bar{\cdot})$ — and the symmetric part of each product is the multiplication of a commutative structure and its antisymmetric part the multiplication of an alternating one, the symmetry being up to the coefficientwise conjugation in the two sesquilinear families.
The name has three letters and is read as a code.
- The first letter says which of the three operations of a family is meant: $\mathrm G$ the general product, the multiplication itself, $\mathrm S$ its symmetric part, $\mathrm A$ its antisymmetric part.
- The second letter says which of the two products of the family is meant: $\mathrm P$ the plain one and $\mathrm Q$ the quaternionic one, in the sense of the first slot, read as it stands for $\mathrm P$ and through the natural conjugation ${}^{\natural}$ for $\mathrm Q$.
- The third letter says the family: $\mathrm A$ the algebra over $\mathbb{C}$, whose second slot is read as it stands, and $\mathrm S$ the sesqualgebra over $\mathbb{C}$, whose second slot carries the Hermitian conjugation ${}^{*}$.
The twelve names are these:
| the family | general | symmetric | antisymmetric |
|---|---|---|---|
| the plain algebra | $\mathrm{GPA}$ | $\mathrm{SPA}$ | $\mathrm{APA}$ |
| the plain sesqualgebra | $\mathrm{GPS}$ | $\mathrm{SPS}$ | $\mathrm{APS}$ |
| the quaternionic algebra | $\mathrm{GQA}$ | $\mathrm{SQA}$ | $\mathrm{AQA}$ |
| the quaternionic sesqualgebra | $\mathrm{GQS}$ | $\mathrm{SQS}$ | $\mathrm{AQS}$ |
Each row is one family — the two algebras over $\mathbb{C}$ and the two sesqualgebras — and the three names of a row are the three structures of the family: the general structure of the family and its symmetrisation and its antisymmetrisation.
The four codes that begin with $\mathrm G$ name the four structures the corpus already names apart: $\mathrm{GPA}$ is the multiplication of The Biquaternion Algebra and Its Vector Spaces, $\mathrm{GPS}$ of Introduction to the General Plain Sesqualgebra, $\mathrm{GQA}$ of Introduction to the General Quaternionic Algebra, and $\mathrm{GQS}$ of Introduction to the General Quaternionic Sesqualgebra. The other eight codes are the symmetrisations and the antisymmetrisations of those four, and the body of Symmetric and Antisymmetric Decomposition of the Four General Products is where they are read.
What the last letter does and does not claim. Only the six structures whose code ends in $\mathrm A$ are algebras over $\mathbb{C}$: the three operations of the plain row and the three of the quaternionic row are $\mathbb{C}$-bilinear, so each is an algebra multiplication in the broad sense. The six whose code ends in $\mathrm S$ are the six operations of the two sesqualgebras over $(\mathbb{C},\bar{\cdot})$: the two general ones, $\mathrm{GPS}$ and $\mathrm{GQS}$, are the multiplications of the two sesqualgebras over $\mathbb{C}$, and their four parts $\mathrm{SPS}$, $\mathrm{SQS}$, $\mathrm{APS}$ and $\mathrm{AQS}$ are sesquilinear again, because the exchange that splits them carries the conjugation of the value and keeps the class. The bare interchange of the two arguments, which returns a product of the opposite parity, has its own two halves — the symmetrised sesquilinear product and the sesquilinear commutator — and those are the only $\mathbb{R}$-bilinear operations of the family. The trailing $\mathrm S$ names the class of the operation, which is the class of the parent product, and it is read that way throughout. The twelve names are read against the class-preserving exchange; §The Exchange Behind the Split of Symmetric and Antisymmetric Decomposition of the Four General Products contrasts it with the bare interchange.
Remark (one collision). In the physics part of the corpus the letters $\mathrm{APS}$ also denote the algebra of physical space, the paravector space $\mathrm{span}_{\mathbb{R}}\{e_t,\gamma_k\}$ of Paravectors and the Geometry of Spacetime. In this article and its siblings the same letters denote $\mathrm{APS}$, the antisymmetric plain sesqualgebra, and the two readings occur in different parts of the corpus.
The article of each structure is its own: Introduction to the Symmetric Plain Algebra for $\mathrm{SPA}$, Introduction to the Symmetric Plain Sesqualgebra for $\mathrm{SPS}$, Introduction to the Symmetric Quaternionic Algebra for $\mathrm{SQA}$ and Introduction to the Symmetric Quaternionic Sesqualgebra for $\mathrm{SQS}$; and Introduction to the Antisymmetric Plain Algebra for $\mathrm{APA}$, Introduction to the Antisymmetric Plain Sesqualgebra for $\mathrm{APS}$, Introduction to the Antisymmetric Quaternionic Algebra for $\mathrm{AQA}$ and Introduction to the Antisymmetric Quaternionic Sesqualgebra for $\mathrm{AQS}$. Each of those articles names the product of its structure in the same sentence.
The Twelve Structures, One by One
Each structure is read with its product and with its decomposition: the operation itself is written in Symmetric and Antisymmetric Decomposition of the Four General Products, and its value is written here on the coordinates as the scalar part and the vector part, with, for the four general products, the split into the two parts. The notation is that of that article.
The General Plain Algebra (GPA)
Its product is the general plain bilinear product. Its decomposition is the scalar part $Q_{1,t}Q_{2,t}-(\vec Q_1,\vec Q_2)$ and the vector part $Q_{1,t}\vec Q_2+Q_{2,t}\vec Q_1+\vec Q_1\times\vec Q_2$, and it splits into its two parts:
$$ \boxed{\;\mathrm{GPA}=\mathrm{SPA}+\mathrm{APA}\;}. $$
The multiplication is associative, $e_t$ is a unit on both sides, and the values fill the whole space.
The General Plain Sesqualgebra (GPS)
Its product is the general plain sesquilinear product. Its decomposition is the scalar part $Q_{1,t}\overline{Q_{2,t}}+(\vec Q_1,\overline{\vec Q_2})$ and the vector part $-Q_{1,t}\overline{\vec Q_2}+\overline{Q_{2,t}}\vec Q_1-\vec Q_1\times\overline{\vec Q_2}$, and it splits into its two parts, $$ \boxed{\;\mathrm{GPS}=\mathrm{SPS}+\mathrm{APS}\;} $$ The multiplication is sesquilinear over $(\mathbb{C},\bar{\cdot})$, it is not associative, $e_t$ is a unit on the right only, and the values fill the whole space.
The General Quaternionic Algebra (GQA)
Its product is the general quaternionic bilinear product. Its decomposition is the scalar part $Q_{1,t}Q_{2,t}+(\vec Q_1,\vec Q_2)$ and the vector part $Q_{1,t}\vec Q_2-Q_{2,t}\vec Q_1-\vec Q_1\times\vec Q_2$, and it splits into its two parts, $\mathrm{GQA}=\mathrm{SQA}+\mathrm{AQA}$. The multiplication is not associative, $e_t$ is a unit on the left only, and the values fill the whole space.
The General Quaternionic Sesqualgebra (GQS)
Its product is the general quaternionic sesquilinear product. Its decomposition is the scalar part $Q_{1,t}\overline{Q_{2,t}}-(\vec Q_1,\overline{\vec Q_2})$ and the vector part $-Q_{1,t}\overline{\vec Q_2}-\overline{Q_{2,t}}\vec Q_1+\vec Q_1\times\overline{\vec Q_2}$, and it splits into its two parts, $\mathrm{GQS}=\mathrm{SQS}+\mathrm{AQS}$. The multiplication is sesquilinear over $(\mathbb{C},\bar{\cdot})$, it is not associative, it has a unit on neither side, and the values fill the whole space.
The Symmetric Plain Algebra (SPA)
Its product is the symmetric plain bilinear product, the symmetric part of the general plain bilinear product. Its decomposition is the scalar part $Q_{1,t}Q_{2,t}-(\vec Q_1,\vec Q_2)$ and the vector part $Q_{1,t}\vec Q_2+Q_{2,t}\vec Q_1$. The multiplication is commutative and satisfies the Jordan identity, $e_t$ is a unit on both sides, and the values fill the whole space.
The Symmetric Plain Sesqualgebra (SPS)
Its product is the symmetric plain sesquilinear product, the symmetric part of the general plain sesquilinear product. Its decomposition is the scalar part $Q_{1,t}\overline{Q_{2,t}}+(\vec Q_1,\overline{\vec Q_2})$ alone, its vector part being zero. The multiplication is conjugate-commutative, its values lie in the centre, and it has no unit.
The Symmetric Quaternionic Algebra (SQA)
Its product is the symmetric quaternionic bilinear product, the symmetric part of the general quaternionic bilinear product. Its decomposition is the scalar part $Q_{1,t}Q_{2,t}+(\vec Q_1,\vec Q_2)$ alone, its vector part being zero. The multiplication is commutative, it is not a Jordan product, its values lie in the centre, and it has no unit.
The Symmetric Quaternionic Sesqualgebra (SQS)
Its product is the symmetric quaternionic sesquilinear product, the symmetric part of the general quaternionic sesquilinear product. Its decomposition is the scalar part $Q_{1,t}\overline{Q_{2,t}}-(\vec Q_1,\overline{\vec Q_2})$ and the vector part $-Q_{1,t}\overline{\vec Q_2}-\overline{Q_{2,t}}\vec Q_1$. The multiplication is conjugate-commutative, it is not a Jordan product, it has no unit, and its values lie in no subspace of the remarkable subspaces.
The Antisymmetric Plain Algebra (APA)
Its product is the antisymmetric plain bilinear product, the antisymmetric part of the general plain bilinear product. Its decomposition is the scalar part $0$ and the vector part $\vec Q_1\times\vec Q_2$ alone. The multiplication is alternating and satisfies the Jacobi identity, it has no unit, and its values lie in the vector subspace.
The Antisymmetric Plain Sesqualgebra (APS)
Its product is the antisymmetric plain sesquilinear product, the antisymmetric part of the general plain sesquilinear product. Its decomposition is the scalar part $0$ and the vector part $-Q_{1,t}\overline{\vec Q_2}+\overline{Q_{2,t}}\vec Q_1-\vec Q_1\times\overline{\vec Q_2}$. The multiplication is conjugate-alternating, it fails the Jacobi identity, it has no unit, and its values lie in the vector subspace.
The Antisymmetric Quaternionic Algebra (AQA)
Its product is the antisymmetric quaternionic bilinear product, the antisymmetric part of the general quaternionic bilinear product. Its decomposition is the scalar part $0$ and the vector part $Q_{1,t}\vec Q_2-Q_{2,t}\vec Q_1-\vec Q_1\times\vec Q_2$. The multiplication is alternating, it fails the Jacobi identity, it has no unit, and its values lie in the vector subspace.
The Antisymmetric Quaternionic Sesqualgebra (AQS)
Its product is the antisymmetric quaternionic sesquilinear product, the antisymmetric part of the general quaternionic sesquilinear product. Its decomposition is the scalar part $0$ and the vector part $\vec Q_1\times\overline{\vec Q_2}$. The multiplication is conjugate-alternating, it fails the Jacobi identity, it has no unit, and its values lie in the vector subspace.
The Two That Are Lie and Jordan Products
Of the twelve, exactly two structures carry the product of one of the two classical nonassociative algebras, and both belong to the plain family. A symmetrisation is a Jordan product only when it satisfies the Jordan identity, an antisymmetrisation a Lie product only when it satisfies the Jacobi identity, and of the twelve exactly one of each meets its identity. The two are these:
- $\mathrm{APA}$ is alternating and satisfies the Jacobi identity, so its product is a Lie product; it is the Lie algebra of The Unitary Lie Algebra, with the vector subspace as a Lie subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{C})$, the quaternion subspace as $\mathbb{R}e_t\oplus\mathfrak{su}(2)$ and the anti-Hermitian subspace as $\mathfrak{u}(2)$, whose derived algebra is $\mathfrak{su}(2)$ (Remarkable Subspaces and the Four General Products, Forms and Norms).
- $\mathrm{SPA}$ is commutative and satisfies the Jordan identity, so its product is a Jordan product; it is the special Jordan algebra of degree two, and its Hermitian subspace is the Hermitian Jordan algebra $J(\mathbb{B})$.
The other six parts fail the identity of their kind, and each failure is witnessed on a single triple or a single element:
| name | identity | witness | failure |
|---|---|---|---|
| $\mathrm{SPS}$ | Jordan | $x=y=\vec e_x$ | the sides are $0$ and $e_t$ |
| $\mathrm{SQA}$ | Jordan | $x=y=\vec e_x$ | the sides are $0$ and $e_t$ |
| $\mathrm{SQS}$ | Jordan | $x=y=\vec e_x$ | the sides are $-e_t$ and $e_t$ |
| $\mathrm{APS}$ | Jacobi | $(e_t,\vec e_x,\vec e_y)$ | $\vec e_z$ |
| $\mathrm{AQA}$ | Jacobi | $(e_t,\vec e_x,\vec e_y)$ | $-\vec e_z$ |
| $\mathrm{AQS}$ | Jacobi | $(\vec e_x,\vec e_x,i\vec e_y)$ | $-2i\vec e_y$ |
The last column of the three Jacobi rows is the cyclic sum declared in the Conventions of Symmetric and Antisymmetric Decomposition of the Four General Products.
The three Jacobi failures have three different reasons. For $\mathrm{AQA}$ the culprit is the non-associativity of the product it comes from; for $\mathrm{APS}$ and $\mathrm{AQS}$ it is the obstruction of Lie Algebras of Sesqualgebras, where the antisymmetrisation of a sesquilinear product is a Lie bracket only after the collapse of the two involutions, which a genuine sesqualgebra forbids. The three Jordan failures are witnessed above, the first two on the same element $\vec e_x$ because the two products agree on the real basis. In the three Jordan rows the last column gives the two sides of the identity, $(x^{2}\bullet x)\bullet x$ and $x^{2}\bullet(x\bullet x)$: they differ by $e_t$ for $\mathrm{SPS}$ and $\mathrm{SQA}$, whose symmetric square on $\vec e_x$ is $e_t$, and by $2e_t$ for $\mathrm{SQS}$, whose symmetric square on $\vec e_x$ is $-e_t$.
The four general products are neither symmetrisations nor antisymmetrisations, and neither identity is put to them: $\mathrm{GPA}$ is the one of the twelve whose product is associative, and the products of the other three are not associative.
Summary
The twelve structures are the twelve products of Symmetric and Antisymmetric Decomposition of the Four General Products read as multiplications, and each name is three letters: the part, $\mathrm G$ general or $\mathrm S$ symmetric or $\mathrm A$ antisymmetric; the first slot, $\mathrm P$ plain or $\mathrm Q$ quaternionic; the family, $\mathrm A$ the algebra over $\mathbb{C}$ or $\mathrm S$ the sesqualgebra over $\mathbb{C}$. The four names that begin with $\mathrm G$ are the four structures the corpus names apart, the multiplications of the four general products, and the other eight are their symmetrisations and antisymmetrisations.
Of the twelve, two are the product of a classical nonassociative algebra and both belong to the plain family: $\mathrm{APA}=\vec Q_1\times\vec Q_2$ is the one Lie product and $\mathrm{SPA}=\tilde{Q}_1\bullet\tilde{Q}_2$ the one Jordan product. The other six parts fail the identity of their kind, with the witnesses recorded above. The six structures whose code ends in $\mathrm A$ are $\mathbb{C}$-bilinear and the six whose code ends in $\mathrm S$ are sesquilinear over $(\mathbb{C},\bar{\cdot})$.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $\mathrm G$, $\mathrm S$, $\mathrm A$ (first letter) | the general product, its symmetric part, its antisymmetric part |
| $\mathrm P$, $\mathrm Q$ (second letter) | the plain first slot, the quaternionic first slot |
| $\mathrm A$, $\mathrm S$ (third letter) | the family: the algebra over $\mathbb{C}$, the sesqualgebra over $\mathbb{C}$ |
| $\mathrm{GPA}$, $\mathrm{GQA}$, $\mathrm{SPA}$, $\mathrm{SQA}$, $\mathrm{APA}$, $\mathrm{AQA}$ | the six structures of the two algebras over $\mathbb{C}$ |
| $\mathrm{GPS}$, $\mathrm{GQS}$, $\mathrm{SPS}$, $\mathrm{SQS}$, $\mathrm{APS}$, $\mathrm{AQS}$ | the six structures of the two sesqualgebras over $\mathbb{C}$ |
| $\langle\tilde{Q}_1,\tilde{Q}_2\rangle_{\natural}=Q_{1,t}Q_{2,t}+(\vec Q_1,\vec Q_2)$ | the quaternion form, the scalar part of $\mathrm{SQA}$ |
| $\mathbb{C}_{\mathbb{B}}$, $\mathrm{Vect}(\mathbb{B})$, $\mathbb{M}_{+}$, $\mathbb{M}_{-}$ | the centre, the vector subspace and the Hermitian and anti-Hermitian subspaces |
| Jacobi, Jordan | the two identities that single out $\mathrm{APA}$ and $\mathrm{SPA}$ |
| $\overline{\vec Q_2}$, $(\vec Q_1,\vec Q_2)$, $\vec Q_1\times\vec Q_2$ | the coefficientwise conjugate of the vector part, the bilinear dot product and the bilinear cross product |
Further Reading
- Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the vocabulary the twelve are studied with: the classes of elements and the derived operations defined once for a general product and read on the products of the space.
- Symmetric and Antisymmetric Decomposition of the Four General Products, for the products themselves, the method of the decomposition and the table of the twelve with their laws, units and images.
- The Four General Products and Their Relations over the Biquaternion Algebra, for the four general products that the four $\mathrm G$-names read as multiplications.
- The Symmetric and Antisymmetric Parts of an Algebra Product and The Symmetric and Antisymmetric Parts of a Sesqualgebra Product, for the two definitions of the split and the class-preserving exchange.
- The Biquaternion Algebra and Its Vector Spaces, Introduction to the General Plain Sesqualgebra, Introduction to the General Quaternionic Algebra and Introduction to the General Quaternionic Sesqualgebra, for the four structures the corpus names apart; and the eight articles of the parts, Introduction to the Symmetric Plain Algebra and its siblings, for the symmetrisations and antisymmetrisations.
- Lie Algebras of Sesqualgebras, for the obstruction that fails the Jacobi identity of the two sesquilinear antisymmetrisations, and The Unitary Lie Algebra and The Hermitian Jordan Algebra for the two structures that pass.