The Multiplication Operators of the Symmetric Plain Algebra
Introduction
The symmetric plain algebra of $\mathbb{H}$ carries the Jordan product $\tilde q_1\bullet\tilde q_2=\tfrac12(\tilde q_1\tilde q_2+\tilde q_2\tilde q_1)$ of Introduction to the Symmetric Plain Algebra.
This is not the algebra $\mathbb{H}$. The two algebras share the space, the addition, the real scalars and the unit, but $\bullet$ is a different multiplication and the projection onto the scalar line is not multiplicative. 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 Multiplication Operators
The Definition and the Two-Sided Coincidence
Definition. The left multiplication operator of the block is $L^{\bullet}_{\tilde p}\tilde q=\tilde p\bullet\tilde q$; the right one is $R^{\bullet}_{\tilde p}\tilde q=\tilde q\bullet\tilde p$.
Proposition (the half-sum). The left multiplication of the block is the half-sum of the plain left and right multiplications,
$$ \boxed{\;L^{\bullet}_{\tilde p}=\tfrac12\bigl(L_{\tilde p}+R_{\tilde p}\bigr),\qquad L_{\tilde p}+R_{\tilde p}=2\,L^{\bullet}_{\tilde p}\;}. $$
Proof. $\tfrac12(L_{\tilde p}+R_{\tilde p})\tilde q=\tfrac12(\tilde p\tilde q+\tilde q\tilde p)=\tilde p\bullet\tilde q$. The stated sum is the operator of the doubled product and is not $L^{\bullet}_{\tilde p}$; the factor $\tfrac12$ is essential.
Proposition (the two-sided coincidence). The product is commutative, so the left and the right operators coincide:
$$ \boxed{\;L^{\bullet}_{\tilde p}=R^{\bullet}_{\tilde p}\;}. $$
Proof. $R^{\bullet}_{\tilde p}\tilde q=\tilde q\bullet\tilde p=\tilde p\bullet\tilde q=L^{\bullet}_{\tilde p}\tilde q$ by commutativity. The block therefore carries one operator per element, not two, and there is no asymmetry of the two sides.
Composition
Theorem. The composition of two multiplication operators is
$$ L^{\bullet}_{\tilde p}L^{\bullet}_{\tilde q}=\tfrac14\bigl(L_{\tilde p\tilde q}+R_{\tilde q\tilde p}+M_{\tilde p,\tilde q}+M_{\tilde q,\tilde p}\bigr), $$
and its symmetrised form is
$$ L^{\bullet}_{\tilde p}L^{\bullet}_{\tilde q}+L^{\bullet}_{\tilde q}L^{\bullet}_{\tilde p} =L^{\bullet}_{\tilde p\bullet\tilde q}+\tfrac12\bigl(M_{\tilde p,\tilde q}+M_{\tilde q,\tilde p}\bigr). $$
Proof. Writing $L^{\bullet}_{\tilde p}=\tfrac12(L_{\tilde p}+R_{\tilde p})$ and expanding the product, $L_{\tilde p}L_{\tilde q}=L_{\tilde p\tilde q}$, $R_{\tilde p}R_{\tilde q}=R_{\tilde q\tilde p}$, $L_{\tilde p}R_{\tilde q}=M_{\tilde p,\tilde q}$ and $R_{\tilde p}L_{\tilde q}=M_{\tilde q,\tilde p}$, gives the first identity; adding it to the one with $\tilde p,\tilde q$ interchanged and using $L^{\bullet}_{\tilde p\bullet\tilde q}=\tfrac14(L_{\tilde p\tilde q}+R_{\tilde p\tilde q}+L_{\tilde q\tilde p}+R_{\tilde q\tilde p})$ gives the second.
Remark (the composition is not the product of the parameters). The block is not associative, so $L^{\bullet}_{\tilde p}L^{\bullet}_{\tilde q}\neq L^{\bullet}_{\tilde p\bullet\tilde q}$; the two-sided operators $M_{\tilde p,\tilde q}+M_{\tilde q,\tilde p}$ measure the failure, and they are what the plain operators contribute and the block keeps.
The Kernel
Theorem (injectivity). The map $\tilde p\mapsto L^{\bullet}_{\tilde p}$ is injective: $L^{\bullet}_{\tilde p}=0$ if and only if $\tilde p=0$.
Proof. If $L^{\bullet}_{\tilde p}=0$ then $\tilde p\bullet\tilde q=0$ for every $\tilde q$, so $\tilde p$ is annihilated with everything; but a nonzero element annihilates only the plane of Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space when it is pure and nothing when it is non-pure, so $\tilde p=0$.
Corollary. The kernel of $L^{\bullet}_{\tilde p}$ is $\{0\}$ for a non-pure $\tilde p$ and the two-dimensional plane of Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space for a nonzero pure $\tilde p$; the operator is singular exactly at the zero divisors, with $\det L^{\bullet}_{\tilde p}=p_0^2\langle\tilde p\rangle_{\natural}$.
The Jordan Relation
The Operator Form of the Jordan Identity
Theorem. The Jordan identity of the block is equivalent to the operator commutation
$$ \bigl[\,L^{\bullet}_{\tilde p},\,L^{\bullet}_{\tilde p^{\,\bullet2}}\,\bigr]=0 \qquad\text{for every }\tilde p, $$
and, more generally, $\bigl[L^{\bullet}_{\tilde p},L^{\bullet}_{\tilde q}\bigr]$ is a derivation of the block for all $\tilde p,\tilde q$.
Proof. The equivalence of the Jordan identity with $[L_{\tilde q_4},L_{\tilde q_4^{\bullet2}}]=0$ is the standard operator form quoted in Jordan Algebras; the derivation property of $[L^{\bullet}_{\tilde p},L^{\bullet}_{\tilde q}]$ is the same statement applied to the linearised identity, and both are the algebra of the symmetrisation.
The Inner Derivations
Definition. The inner derivations of the block are the operators $D_{\tilde p,\tilde q}=[L^{\bullet}_{\tilde p},L^{\bullet}_{\tilde q}]$.
Theorem. The inner derivations span a three-dimensional real Lie algebra, isomorphic to $\mathfrak{so}(3)$; it acts on the vector subspace by its usual rotations and annihilates the scalar line.
Proof. The derivation algebra of a degree-two algebra is the skew part of the structure Lie algebra, and for the block it is the Lie algebra acting on the three-dimensional vector subspace by the rotations that preserve the Euclidean form, $\mathfrak{so}(3)$; the scalar line is central and every derivation annihilates $e_t$. A direct count from the four generators gives three independent brackets, $[L^{\bullet}_{\vec e_x},L^{\bullet}_{\vec e_y}]$, $[L^{\bullet}_{\vec e_y},L^{\bullet}_{\vec e_z}]$, $[L^{\bullet}_{\vec e_z},L^{\bullet}_{\vec e_x}]$.
The Enveloping Structure
The Enveloping Algebra Is Full
Theorem. The multiplication operators generate the full endomorphism algebra:
$$ \boxed{\;\mathrm{alg}\bigl\{\,L^{\bullet}_{\tilde p}\,\bigr\}=\mathrm{End}_{\mathbb{R}}(\mathbb{H})\cong M_4(\mathbb{R})\;}, $$
of real dimension $16$; the span of the products of length at most two is already all of it.
Proof. The identity is $L^{\bullet}_{e_t}$; the four generators together with their products of length one span a $13$-dimensional space, and adjoining the products of length two completes it to $16$, by the computation of the block's operator span. Since $\mathrm{alg}$ contains the identity and is the whole of $M_4(\mathbb{R})$, it is irreducible.
Corollary (the commutant). The commutant of the family is the scalar operators, $\{L^{\bullet}_{\tilde p}\}'=\mathbb{R}\cdot\mathrm{id}$; the block is a multiplication algebra whose multiplications generate every linear map of the space, as the plain left and right multiplications do in The Enveloping Operator Algebra of the Biquaternion Algebra and the Bi-module Structure.
Remark (the contrast with the plain algebra). In the plain algebra the left and the right multiplications generate the two-sided multiplications but not all of $\mathrm{End}(\mathbb{H})$; the block, being commutative, has a single family and its products fill the whole matrix algebra. This is the specific-theory form of the coincidence $L^{\bullet}=R^{\bullet}$.
The Derivations
The Inner Derivations and Their Dimension
Theorem. Every derivation of the block is inner:
$$ \boxed{\;\operatorname{Der}(\mathbb{H}^{\bullet})=\mathrm{span}\bigl\{[L^{\bullet}_{\tilde p},L^{\bullet}_{\tilde q}]\bigr\}\cong\mathfrak{so}(3), \qquad \dim\operatorname{Der}=3\;}, $$
and no derivation is a multiplication by an element.
Proof. The derivation algebra is three-dimensional and coincides with the inner derivations by the general theory of degree-two Jordan algebras; an element-multiplication $L^{\bullet}_{\tilde q_5}$ is a derivation only if $\tilde q_5\bullet(\tilde q_1\bullet\tilde q_2)=( \tilde q_5\bullet\tilde q_1)\bullet\tilde q_2+\tilde q_1\bullet(\tilde q_5\bullet\tilde q_2)$ for all $\tilde q_1,\tilde q_2$, which forces $\tilde q_5=0$ by the non-associativity at $\vec e_x,\vec e_y$.
Skewness for the Trace Form
Theorem. Every derivation is skew for the trace form:
$$ \tau(D\tilde q_1,\tilde q_2)+\tau(\tilde q_1,D\tilde q_2)=0 . $$
Proof. The trace form is associative, $\tau(\tilde q_1\bullet\tilde q_2,\tilde q_3)=\tau(\tilde q_1,\tilde q_2\bullet\tilde q_3)$, and differentiating this identity along a derivation gives the skewness; the full argument is The Trace Form and the Invariance of the Symmetric Plain Algebra, where the invariance is owned.
Summary
The multiplication operators of the block are $L^{\bullet}_{\tilde p}\tilde q=\tilde p\bullet\tilde q$, the half-sum $L^{\bullet}_{\tilde p}=\tfrac12(L_{\tilde p}+R_{\tilde p})$ of the plain left and right multiplications, and because the product is commutative the left and the right operators coincide, $L^{\bullet}_{\tilde p}=R^{\bullet}_{\tilde p}$; the map $\tilde p\mapsto L^{\bullet}_{\tilde p}$ is injective, with kernel $\{0\}$ at a non-pure element and a two-dimensional plane at a pure one. The composition is $L^{\bullet}_{\tilde p}L^{\bullet}_{\tilde q}=\tfrac14(L_{\tilde p\tilde q}+R_{\tilde q\tilde p}+M_{\tilde p,\tilde q}+M_{\tilde q,\tilde p})$, the operator form of the Jordan identity is $[L^{\bullet}_{\tilde p},L^{\bullet}_{\tilde p^{\,\bullet2}}]=0$, and every bracket $[L^{\bullet}_{\tilde p},L^{\bullet}_{\tilde q}]$ is a derivation. The operators generate the full endomorphism algebra $M_4(\mathbb{R})$, of dimension $16$, with commutant the scalars; the derivations are all inner and form the three-dimensional Lie algebra $\mathfrak{so}(3)$, and every derivation is skew for the trace form.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $L^{\bullet}_{\tilde p}=R^{\bullet}_{\tilde p}$ | the one multiplication operator of the block |
| $L^{\bullet}_{\tilde p}=\tfrac12(L_{\tilde p}+R_{\tilde p})$ | the half-sum identity |
| $L^{\bullet}_{\tilde p}L^{\bullet}_{\tilde q}=\tfrac14(L_{\tilde p\tilde q}+R_{\tilde q\tilde p}+M_{\tilde p,\tilde q}+M_{\tilde q,\tilde p})$ | the composition |
| $[L^{\bullet}_{\tilde p},L^{\bullet}_{\tilde p^{\,\bullet2}}]=0$ | the operator form of the Jordan identity |
| $\operatorname{Der}\cong\mathfrak{so}(3)$ | the derivation algebra, of dimension $3$, all inner |
| $\mathrm{alg}\{L^{\bullet}_{\tilde p}\}=M_4(\mathbb{R})$ | the enveloping algebra |
| $\tau(D\tilde q_1,\tilde q_2)+\tau(\tilde q_1,D\tilde q_2)=0$ | skewness of the derivations |
Further Reading
- Introduction to the Symmetric Plain Algebra, for the product.
- Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the kernel and the determinant of the operator.
- The Trace Form and the Invariance of the Symmetric Plain Algebra, for the invariance and the skewness.
- The Enveloping Operator Algebra of the Biquaternion Algebra and the Bi-module Structure, for the plain operators against which the fullness is read.
- Jordan Algebras, for the operator form of the Jordan identity and the derivations of a Jordan algebra.
- The Two General Products of an Algebra with Involution, for the degree-two Common Theory specialised here.