Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space
Introduction
This article collects, in one place, the definitions of the classes of elements and of the derived operations with which the twelve algebraic structures of The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space are studied. It is a definitions article: each notion is stated for a general product, and its statement is then read on the products of the space. It belongs to the Algebra group of the category Biquaternion Algebraic Structures, follows the article that names the twelve structures and precedes the articles that study them. Each of the twelve thus cites one owner for its vocabulary rather than restating it.
The point of collecting them is that every definition is relative to a product, and this is easy to lose when each structure is read on its own. The words zero divisor, idempotent, nilpotent, unit each name an equation, and the equation names a product; an element that satisfies the equation of one product need not satisfy that of another. The definitions are therefore stated for a general product $\star$, and every statement that singles out a product names it.
- The elements, the basis and the four conjugations are The Biquaternion Algebra and Its Vector Spaces.
- The four general products, their co-ordinate forms and the tables of the twelve are The Four General Products and Their Relations over the Biquaternion Algebra and Symmetric and Antisymmetric Decomposition of the Four General Products.
- The twelve structures and their three-letter names are The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space.
- The general algebra vocabulary — the unit, the inverse, the Peirce decomposition — is Unital Algebras, and the corpus-wide readings of the words are Conventions in Mathematics.
This article defines. It states the relativity of the definitions as the principle it is, and it proves the one structural fact the definitions turn on, the transport of §Transport Along an Isomorphism; every specialization is cited to the article that owns it.
1. The Product, its Slots and its Class
A product on $\mathbb{B}$ is a map $$ \star \;:\; \mathbb{B}\times\mathbb{B}\longrightarrow\mathbb{B}, $$ and the twelve products of the space are its instances. A product is read by its two slots: the first slot is read either as it stands (plain, letter $\mathrm P$) or through the natural conjugation (quaternionic, letter $\mathrm Q$), and the second slot is read either as it stands (the algebra, letter $\mathrm A$) or through the star (the sesqualgebra, letter $\mathrm S$). The four general products are the four combinations, and each of their symmetric and antisymmetric parts is one of the twelve.
Definition (class of a product). A product $\star$ is read in each slot through an involution: it is $\sigma$-semilinear in the first slot and $\varsigma$-semilinear in the second, $$ \star(\lambda x,y)=\sigma(\lambda)\star(x,y),\qquad \star(x,\lambda y)=\varsigma(\lambda)\star(x,y), $$ where each of $\sigma,\varsigma$ is the identity or the complex conjugation. The product is $\mathbb{C}$-bilinear when both are the identity, and sesquilinear when the second slot carries the conjugation; the first slot carries one too or not according to the family. Of the four general products only the plain bilinear carries no conjugation at all; the quaternionic bilinear carries it in the first slot, the plain sesquilinear in the second, and the quaternionic sesquilinear in both.
Definition (the laws of a product). A product $\star$ is associative when $(x\star y)\star z=x\star(y\star z)$ for all $x,y,z$; commutative when $x\star y=y\star x$; alternating when $x\star x=0$ for every $x$, which for a bilinear product is equivalent to anticommutativity, $x\star y=-y\star x$. It satisfies the Jacobi identity when $x\star(y\star z)+y\star(z\star x)+z\star(x\star y)=0$, and the Jordan identity when $(x\star x)\star(y\star x)=\bigl((x\star x)\star y\bigr)\star x$. The laws decide which classical structure the product defines, and the reading of the twelve one by one is The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space.
The Relativity of the Definitions
Every class below is defined by an equation whose symbol is the product, so the product is part of the definition and not a background convention.
Principle (relativity). Let $\star$ be a product and let $\mathcal E_\star(x)$ be any equation in the element $x$ whose only operation is $\star$ and whose constants are fixed elements or scalars. The class of solutions of $\mathcal E_\star$ depends on $\star$: replacing the product replaces the equation and can replace the class.
The definitions of this article are all of that form — $\tilde Q\star\tilde Q_2=0$ for the zero divisor, $\tilde\Pi\star\tilde\Pi=\tilde\Pi$ for the idempotent, $\tilde\Upsilon\star\tilde\Upsilon=0$ for the nilpotent, and so on — and none of them is a property of the element alone. The word is therefore always read with of: a zero divisor of $\star$, an idempotent of $\star$. Two products may select the same class and two may not, and when they do not the difference is a computation, of which §9 gives the table for the four general products.
Transport Along an Isomorphism
The relativity has one structural counterpart, which is why two products may share a class without sharing the equation.
Proposition (transport). Let $\star$ and $\star'$ be two products on $\mathbb{B}$ and let $\theta$ be a bijection of $\mathbb{B}$ with $$ \boxed{\;\theta(x\star y)=\theta(x)\star'\theta(y) \qquad\text{for all } x,y\;}. $$ Then $\theta$ carries every class of §2 to §5 from $\star$ onto the corresponding class of $\star'$: the $\star$-units onto the $\star'$-units, the $\star$-invertible elements onto the $\star'$-invertible elements with $\theta(x^{-1})=\theta(x)^{-1}$, the $\star$-idempotents onto the $\star'$-idempotents, the $\star$-nilpotents onto the $\star'$-nilpotents, and the $\star$-zero divisors onto the $\star'$-zero divisors.
Proof. Apply $\theta$ to the defining equation and use multiplicativity. If $x\star x=x$ then $\theta(x)\star'\theta(x)=\theta(x\star x)=\theta(x)$; if $x\star x=0$ then $\theta(x)\star'\theta(x)=\theta(0)=0$; if $x\star y=0$ with $x,y\neq0$ then $\theta(x)\star'\theta(y)=0$ with $\theta(x),\theta(y)\neq0$; and if $e$ is a unit of $\star$ then $\theta(e)\star'\theta(x)=\theta(e\star x)=\theta(x)$ and $\theta(x)\star'\theta(e)=\theta(x)$, so $\theta(e)$ is a two-sided unit of $\star'$, from which $x\star y=y\star x=e$ gives $\theta(x)\star'\theta(y)=\theta(e)$.
Two products related by such a $\theta$ have the same element theory up to $\theta$, and the definitions of this article cannot separate them. Two products not so related may differ on every class, which is what §9 computes for the four general products. The relations among the four general products under the conjugations, and the question of which pairs are isomorphic in this sense, are The Four General Products and Their Relations over the Biquaternion Algebra.
2. The Unit and the Inverse
Definition (unit of a product). Let $\star$ be a product on $\mathbb{B}$. An element $e$ is a left unit of $\star$ when $e\star x=x$ for every $x$; a right unit of $\star$ when $x\star e=x$ for every $x$; and a two-sided unit of $\star$, or an identity of $\star$, when it is both. A product that has a two-sided unit is unital. A two-sided unit, when it exists, is unique.
Definition (inverse). Let $\star$ be a product with a two-sided unit $e$. An element $x$ is left-invertible of $\star$ when there is $y$ with $y\star x=e$; right-invertible when there is $y$ with $x\star y=e$; and invertible of $\star$ when there is $y$ with $x\star y=y\star x=e$. Such a $y$ is an inverse of $x$ of $\star$, written $x^{-1}$. In an associative unital product the inverse is unique and the invertible elements form the group of units.
The inverse is defined by an equation in $\star$, so it is relative to the product in the same way as the classes of §3 to §5: the inverse element of $x$ for one product is not the inverse for another, and the symbol $x^{-1}$ is read only with the product in force.
The four general products. Only one of the four has a two-sided unit.
| product | left unit | right unit | two-sided unit |
|---|---|---|---|
| $\tilde Q\tilde Q_2$ | $e_t$ | $e_t$ | $e_t$ |
| $\tilde Q\tilde Q_2^{*}$ | none | $e_t$ | none |
| $\tilde Q^{\natural}\tilde Q_2$ | $e_t$ | none | none |
| $\tilde Q^{\natural}\tilde Q_2^{*}$ | none | none | none |
The table is verified on the identity equations and it is the unit table of The Four General Products and Their Relations over the Biquaternion Algebra. Three of the four products are therefore not unital, and for them the group of units is not available; what survives is the one-sided inverse, and its equations are in Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, §The One-Sided Inverse Equations.
The general reading, for an associative unital algebra, is Unital Algebras; the invertibility criterion of the plain product, $\tilde Q$ invertible exactly when the central square $C(\tilde Q)=\sum_\mu Q_\mu^{2}$ is nonzero, is Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra.
3. The Zero Divisor and the Annihilator
Definition (zero divisor of a product). Let $\star$ be a product on $\mathbb{B}$ and let $\tilde Q\in\mathbb{B}$. The element $\tilde Q$ is a left zero divisor of $\star$ when $\tilde Q\neq0$ and there is a nonzero $\tilde Q_2$ with $\tilde Q\star\tilde Q_2=0$; a right zero divisor of $\star$ when $\tilde Q\neq0$ and there is a nonzero $\tilde Q_2$ with $\tilde Q_2\star\tilde Q=0$; and a zero divisor of $\star$ when it is one or the other. It is a two-sided zero divisor of $\star$ when it is both.
Both elements are required to be nonzero, and that is essential: the origin is not a zero divisor of any product, even though $0\star\tilde Q_2=0$ for every $\tilde Q_2$.
Definition (annihilator). Let $\star$ be a product and let $\tilde Q$ be a zero divisor of $\star$. The left annihilator and the right annihilator of $\tilde Q$ in $\star$ are the sets $$ A_\star(\tilde Q)=\{\tilde Q_1:\tilde Q\star\tilde Q_1=0\},\qquad B_\star(\tilde Q)=\{\tilde Q_1:\tilde Q_1\star\tilde Q=0\}. $$ They are the kernels of the two one-sided multiplications of §7, they are nonzero exactly when $\tilde Q$ is a zero divisor of the corresponding side, and an element is a two-sided zero divisor exactly when both are nonzero.
Definition (regular element). An element of a unital product that is neither a left nor a right zero divisor is regular.
Written on the four general products, the definition of a left zero divisor is one of $\tilde Q\tilde Q_2=0$, $\tilde Q^{\natural}\tilde Q_2=0$, $\tilde Q\tilde Q_2^{*}=0$, $\tilde Q^{\natural}\tilde Q_2^{*}=0$, and the definition of a right zero divisor is the same with the factors exchanged. The eight equations are the four definitions, and the equation is the whole of the definition. For the biquaternion algebra the four definitions select one and the same set, $$ \{\tilde Q\neq0:\ C(\tilde Q)=0\}, $$ a fact about the algebra and not a property of the definitions, proved in Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, §The Four General Products Coincide. The word is coincide, not independent: the four products are four definitions, and their agreement is a theorem. The criterion $C(\tilde Q)=0$ for the plain product, the two families and their structure are Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors; the annihilators of one zero divisor, which do depend on the product, are in The Four General Products and Their Relations over the Biquaternion Algebra, §The Annihilators, and in Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space.
4. The Idempotent, the Hermitian Idempotent and the Peirce Decomposition
Definition (idempotent of a product). Let $\star$ be a product on $\mathbb{B}$. An element $\tilde\Pi$ is an idempotent of $\star$ when $$ \tilde\Pi\star\tilde\Pi=\tilde\Pi . $$ The origin and, when the product is unital, the unit are idempotents of every product: they are the trivial idempotents, and the others are nontrivial. The set of the idempotents of $\star$ is its idempotent set.
Definition (orthogonal, complete, primitive). Idempotents $\tilde\Pi,\tilde\Pi'$ of an associative product are orthogonal when $\tilde\Pi\star\tilde\Pi'=\tilde\Pi'\star\tilde\Pi=0$; a family is pairwise orthogonal when $\tilde\Pi_i\star\tilde\Pi_j=0$ for $i\neq j$, and complete when in addition $\sum_i\tilde\Pi_i$ is the unit. A nontrivial idempotent is primitive when it is not the sum of two nonzero orthogonal idempotents, and for a semisimple algebra $\tilde\Pi$ is primitive exactly when $\tilde\Pi\star\mathbb{B}\star\tilde\Pi$ is a division ring.
Definition (Hermitian idempotent). When the product is the multiplication of a $*$-algebra, an idempotent $\tilde\Pi$ with $\tilde\Pi^{*}=\tilde\Pi$ is a Hermitian idempotent; with the plain product and the star, the Hermitian idempotents are the idempotents orthogonal with respect to the form, and the associated decomposition of §The Peirce Decomposition is an orthogonal direct sum.
Definition (Peirce decomposition). Let the product be associative with unit, and let $\tilde\Pi$ be an idempotent. The Peirce decomposition of the algebra relative to $\tilde\Pi$ is the splitting of the regular module into the four pieces indexed by the pair $(\tilde\Pi,1-\tilde\Pi)$; its two algebraic norm pieces are the corner algebras $\tilde\Pi\star\mathbb{B}\star\tilde\Pi$ and $(1-\tilde\Pi)\star\mathbb{B}\star(1-\tilde\Pi)$.
The idempotent is the root of $+1$ in the scaled sense of §6, and the idempotent sets of the four general products differ: the plain product has the family $\tfrac12(e_t+\tilde\Xi i)$ over the roots $\tilde\Xi$ of $-1$, the plain sesquilinear product has the Hermitian idempotents, the natural product has the trivial idempotents alone, and the quaternionic sesquilinear product has the family $-\tfrac12e_t+\tfrac{\sqrt3}{2}\vec u$. The table is in Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, §The Four Sets; the classification for the plain product is Idempotents of the General Plain Algebra, the general theory is Unital Algebras, and the Peirce decomposition of the biquaternion algebra is Ideals and Peirce Decompositions of the Biquaternion Algebra.
5. The Nilpotent
Definition (nilpotent of a product). Let $\star$ be a product on $\mathbb{B}$. An element $\tilde\Upsilon$ is nilpotent of $\star$ when some power of it vanishes, $$ \tilde\Upsilon^{\star k}=0 \quad\text{for some } k\geq1 , $$ the power being read in the sense of §6. The least such $k$ is the index of nilpotency, and an element of index two, $\tilde\Upsilon\star\tilde\Upsilon=0$ with $\tilde\Upsilon\neq0$, is a square-zero element of $\star$. The origin is a nilpotent, the trivial one, and the article is concerned with the others.
The nilpotent is the root of $0$ in the sense of §6. In the plain product the index is two or nothing — an element with vanishing cube already has vanishing square, so the nonzero nilpotents there are exactly the square-zero elements (Nilpotents of the General Plain Algebra) — but the index is not bounded by two in every product of the twelve, and the square-zero element is the case the comparison of the four general products is made in.
The square-zero sets of the four general products differ, and the difference is sharp.
| product | square-zero elements |
|---|---|
| $\tilde Q\tilde Q_2$ | the pure isotropic cone $\{Q_0=0,\ (\vec Q,\vec Q)=0\}$ |
| $\tilde Q\tilde Q_2^{*}$ | the origin alone |
| $\tilde Q^{\natural}\tilde Q_2$ | the whole zero-divisor cone $\mathcal{N}$ |
| $\tilde Q^{\natural}\tilde Q_2^{*}$ | the solutions of $\overline{\tilde Q}\tilde Q=0$ |
The table is verified on the squares and is the table of Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, where the four sets, their chain $Z_1\subseteq Z_2$, $Z_4\subseteq Z_2$, $Z_3=\{0\}$ and their incomparable pair are proved. The implication "nilpotent implies zero divisor" is not uniform: it holds for the plain product and the natural product, and it fails for the plain sesquilinear product, whose only square-zero element is the origin. It is a statement of a product and not of the twelve.
6. Powers, the Square and the Roots
Definition (power). Let $\star$ be a product. The square of an element is $\tilde Q^{\star2}=\tilde Q\star\tilde Q$. When $\star$ is associative the higher powers are unambiguous, $\tilde Q^{\star k}=\tilde Q\star\cdots\star\tilde Q$, and they obey $\tilde Q^{\star(i+j)}=\tilde Q^{\star i}\star\tilde Q^{\star j}$; when $\star$ is not associative the notation needs a bracketing, and the left-nested reading is $\tilde Q^{\star k}=\bigl(\cdots\bigl((\tilde Q\star\tilde Q)\star\tilde Q\bigr)\cdots\star\tilde Q\bigr)$. Of the four general products only the plain one is associative, so only there is $\tilde Q^{\star k}$ bracketing-free.
Definition (root of a central value). Let $\lambda\in\mathbb{C}$. A root of $\lambda$ in $\star$ is a nonzero element with $$ \tilde Q\star\tilde Q=\lambda e_t . $$ The three central values the biquaternion algebra singles out are $-1$, $+1$ and $0$: the roots of $-1$ are the imaginary units, the roots of $+1$ are the involutions, and the roots of $0$ are the nonzero nilpotents of §5.
Definition (square root of an element). For an element $a\in\mathbb{B}$, a square root of $a$ in $\star$ is an element with $\tilde Q\star\tilde Q=a$; the roots of a central value are the special case $a=\lambda e_t$.
The reading of the roots of $-1$, of $0$ and of $+1$ in the plain product, and the classification of the roots of a general element, are Square Roots of Minus One, Zero and Plus One in the General Plain Algebra and Square Roots of a General Element in the Biquaternion Algebra. The idempotents of §4 are the roots of $+1$ halved: the map $\tilde\Xi\mapsto\tfrac12(e_t+\tilde\Xi)$ carries the roots of $+1$ onto the idempotents of the plain product, and the nilpotents of §5 are the roots of $0$.
7. The Derived Operations
Definition (the commutator and the symmetrisation). For a product $\star$, $$ [x,y]_\star=x\star y-y\star x , \qquad x\circ_\star y=x\star y+y\star x , $$ and the symmetric part and the antisymmetric part of the product are $\tfrac12(x\star y+y\star x)$ and $\tfrac12(x\star y-y\star x)$. Each of the four general products splits into its two parts, and the split is the source of the twelve: four general products, three operations to a family. The method and the class-preserving exchange are The Symmetric and Antisymmetric Parts of an Algebra Product and The Symmetric and Antisymmetric Parts of a Sesqualgebra Product.
Definition (the associator and the ternary product). For a product $\star$, $$ (x,y,z)_\star=(x\star y)\star z-x\star(y\star z) $$ is the associator, and the ternary product is the trilinear operation $x\star y\star z$ read with the same bracketing. The product is associative exactly when the associator vanishes identically, so the associator is the obstruction to associativity and, through it, to the unambiguous power of §6.
Definition (the two identities). The Jacobi identity and the Jordan identity of §1 are the statements that single out the two classical non-associative structures among the twelve; the two, their witnesses and the failure of the other six parts are The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, §The Two That Are Lie and Jordan Products, and Jordan Algebras.
Definition (left and right multiplication). For $a\in\mathbb{B}$, $$ L_a^{\star}(x)=a\star x , \qquad R_a^{\star}(x)=x\star a . $$ They are the two one-sided multiplications of §3, their kernels are the two annihilators, and their composition measures the failure of associativity and of multiplicativity. Definition (sandwich). For a product with a two-sided unit and an involution, the sandwich of $a$ is the two-sided operator $x\mapsto a\star x\star a^{*}$. The operators of the biquaternion algebra are Two-Sided Operators on the Biquaternion Algebra, One-Sided Operators on the General Plain Algebra of Biquaternions and The Left and Right Multiplications of the General Quaternionic Sesquilinear Product.
Definition (derivation). An operator $D$ on $\mathbb{B}$ is a derivation of $\star$ when $D(x\star y)=Dx\star y+x\star Dy$ for all $x,y$.
8. The Form-Theoretic Classes
The classes so far are those of the multiplication alone. The structures also carry forms and norms, and the definitions they use are collected here in the same product-relative spirit; the forms themselves are The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra and the norms The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra.
Definition (centre and commutant). The commutant of a subset $S\subseteq\mathbb{B}$ in a product $\star$ is $\{x: x\star s=s\star x\ \text{for all }s\in S\}$; the centre is the commutant of the whole space. For the plain product the centre is the scalar line $\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_t$ (The Biquaternion Algebra and Its Vector Spaces).
Definition (isotropic element, radical). Let $\langle\cdot,\cdot\rangle$ be a form on $\mathbb{B}$. An element is isotropic for the form when $\langle x,x\rangle=0$, and the radical is the set of elements orthogonal to the whole space. The isotropic elements of the four forms are The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra, and the isotropic cone of the quaternionic product is The Isotropic Structure of the General Quaternionic Algebra.
Definition (adjoint, Hermitian, normal). Let $\langle\cdot,\cdot\rangle$ be a form with an involution ${}^{*}$. The adjoint of $a$ is the element $a^{\dagger}$ with $\langle a\star x,y\rangle=\langle x,a^{\dagger}\star y\rangle$; an element is Hermitian when $a^{\dagger}=a$, anti-Hermitian when $a^{\dagger}=-a$, and normal when $a\star a^{\dagger}=a^{\dagger}\star a$. The four adjoints of the two algebras and the two sesqualgebras are worked in The Four Adjoints of the Biquaternion Algebra in Examples; the trace form is The Trace Form and the Invariance of the Symmetric Plain Algebra, and the invariant form of the antisymmetric product is The Killing Form of the Antisymmetric Plain Algebra.
9. The Dictionary: The Classes and the Four General Products
The four general products are four definitions to a class, and the table says which of the four agree. The entry coincide means that the four definitions select one and the same subset of $\mathbb{B}$; differ means that they do not, and the specializing article carries the sets.
| class of § | $\tilde Q\tilde Q_2$ | $\tilde Q\tilde Q_2^{*}$ | $\tilde Q^{\natural}\tilde Q_2$ | $\tilde Q^{\natural}\tilde Q_2^{*}$ | verdict |
|---|---|---|---|---|---|
| unit | $e_t$ two-sided | right only | left only | none | differ |
| zero divisor | $\mathcal{N}\setminus\{0\}$ | same | same | same | coincide |
| idempotent | $\tfrac12(e_t+\tilde\Xi i)$ | the Hermitian ones | the trivial pair | $-\tfrac12e_t+\tfrac{\sqrt3}{2}\vec u$ | differ |
| square-zero | the pure cone | $\{0\}$ | the whole cone | $\{\overline{\tilde Q}\tilde Q=0\}$ | differ |
| annihilator of a fixed zero divisor | one plane | another | another | another | differ |
| invertibility | $C\neq0$ | no two-sided notion | no two-sided notion | no two-sided notion | — |
The one row on which the four products agree is the zero-divisor set. It is the reason the zero divisor is the subject of a synthesis article of its own, Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, and it is the exception that proves the rule of §The Relativity of the Definitions: the definitions are four, and their agreement in this one row is a theorem about the biquaternion algebra, not a feature of the definitions. Every other row shows the four products separating, and each is computed in Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space and The Four General Products and Their Relations over the Biquaternion Algebra.
Summary
A definition of the element theory names a product. This article states, once and for a general product $\star$, the unit and the inverses of a product; the left, right and two-sided zero divisor and the two annihilators; the idempotent, its orthogonal, complete and primitive families and the Hermitian idempotent; the nilpotent, its index and the square-zero element; the powers and their bracketing, the root of a central value and the square root of an element; the commutator and the symmetrisation with the two parts of a product; the associator and the ternary product with the two classical identities; the left and right multiplications, the sandwich and the derivation; and the centre, the commutant, the isotropic element, the radical and the adjoint. Each is then read on the products of the space, and the dictionary of §9 records that the zero-divisor set is the single row on which the four general products agree, while the units, the idempotents, the square-zero elements and the annihilators separate them.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $\star$ | a general product, one of the twelve, with the class of §1 |
| $\tilde Q\tilde Q_2$ | the general plain bilinear product, the multiplication of the algebra |
| $\tilde Q^{\natural}\tilde Q_2$, $\tilde Q\tilde Q_2^{*}$, $\tilde Q^{\natural}\tilde Q_2^{*}$ | the other three general products |
| $e$ | a unit; left, right or two-sided, per §2 |
| $x^{-1}$ | the inverse of $x$, read in the product in force |
| $C(\tilde Q)=\sum_\mu Q_\mu^{2}$ | the central square, deciding the zero divisor of the plain product |
| $A_\star(\tilde Q),B_\star(\tilde Q)$ | the left and the right annihilator of $\tilde Q$ in $\star$ |
| $\tilde\Pi$ | an idempotent; $\tilde\Pi^{*}=\tilde\Pi$ makes it a Hermitian idempotent |
| $\tilde\Upsilon$ | a nilpotent of the plain product; index two, hence square-zero |
| $[x,y]_\star$, $x\circ_\star y$ | the commutator and the symmetrisation of $\star$ |
| $(x,y,z)_\star$ | the associator of $\star$ |
| $L_a^{\star},R_a^{\star}$ | the left and the right multiplication by $a$ in $\star$ |
| $\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_t$ | the centre of the plain product |
Further Reading
- Symmetric and Antisymmetric Decomposition of the Four General Products and The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the products and the structures the definitions here serve, and for the tables of the laws, the units and the images.
- Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the product-relative definition read on the four products, the coincidence of the four zero-divisor sets, and the product-dependent readings of the annihilators, the square, the idempotents and the units.
- Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, Idempotents of the General Plain Algebra and Nilpotents of the General Plain Algebra, for the three element classes of the plain product.
- Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the two readings of §5 and §9 that the four products decide.
- Unital Algebras and Conventions in Mathematics, for the general algebra vocabulary and the corpus-wide readings of the words, and The Biquaternion Algebra and Its Vector Spaces for the elements and the conjugations.