The Trace Form and the Invariance 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, and with it a bilinear form read off the scalar part. This article studies the trace form
$$ \tau(\tilde q_1,\tilde q_2)=\mathrm{Sc}\bigl(\tilde q_1\bullet\tilde q_2\bigr), $$
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 Trace Form
The Definition
Definition. The trace form of the block is
$$ \tau(\tilde q_1,\tilde q_2)=\mathrm{Sc}\bigl(\tilde q_1\bullet\tilde q_2\bigr). $$
Proposition. The trace form is symmetric and bilinear, and coincides with the plain bilinear form of the same space:
$$ \boxed{\;\tau(\tilde q_1,\tilde q_2)=q_{1t}q_{2t}-\vec q_1\cdot\vec q_2=\mathrm{Sc}\bigl(\tilde q_1\tilde q_2\bigr)\;}. $$
Proof. $\tilde q_1\bullet\tilde q_2$ is symmetric and bilinear, so $\tau$ is; and $\mathrm{Sc}(\tilde q_1\bullet\tilde q_2)=\mathrm{Sc}\bigl(\tfrac12(\tilde q_1\tilde q_2+\tilde q_2\tilde q_1)\bigr)=\mathrm{Sc}(\tilde q_1\tilde q_2)$ because the scalar part is invariant under reversal, $\mathrm{Sc}(\tilde q_1\tilde q_2)=\mathrm{Sc}(\tilde q_2\tilde q_1)$. Expanding the plain scalar part gives $q_{1t}q_{2t}-\vec q_1\cdot\vec q_2$.
Corollary (the polarisation). The trace form is the polarisation of the quadratic trace $\tilde q\mapsto\mathrm{Sc}(\tilde q\bullet\tilde q)=\mathrm{Sc}(\tilde q^2)=q_t^2-|\vec q|^2$:
$$ \tau(\tilde q_1,\tilde q_2)=\tfrac12\Bigl(\mathrm{Sc}(\tilde q_1+\tilde q_2)^{\bullet2}-\mathrm{Sc}\,\tilde q_1^{\bullet2}-\mathrm{Sc}\,\tilde q_2^{\bullet2}\Bigr). $$
The Gram Matrix and the Signature
Proposition. In the basis $(e_t,\vec e_x,\vec e_y,\vec e_z)$ the Gram matrix is $\operatorname{diag}(1,-1,-1,-1)$, and the signature is $(1,3)$: the trace form is indefinite and Lorentzian.
Proof. $\tau(e_t,e_t)=\mathrm{Sc}(e_t)=1$ and $\tau(\vec e_\mu,\vec e_\mu)=\mathrm{Sc}(\vec e_\mu\bullet \vec e_\mu)=\mathrm{Sc}(-e_t)=-1$ for $\mu=1,2,3$; the mixed terms vanish because $\vec e_\mu\bullet \vec e_\nu$ is pure for $\mu\neq\nu$.
Remark (the trace form is not the norm). The trace form is indefinite and has the isotropic cone $q_t^2=|\vec q|^2$; the norm $\langle\tilde q\rangle_{\natural}=q_t^2+|\vec q|^2$ is positive definite and vanishes only at the origin. The two forms of the block must not be confused: the norm polarises to the quaternion form (signature $(4,0)$) and the quadratic trace polarises to the trace form (signature $(1,3)$), as Norm and Invertibility in the Symmetric Plain Algebra records. In particular the isotropic cone of the trace form is not the zero-divisor set, which is the pure set $q_t=0$ of Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space.
The Radical
Proposition. The trace form is non-degenerate and its radical is trivial:
$$ \boxed{\;\operatorname{rad}\tau=\{\tilde q_1 : \tau(\tilde q_1,\tilde q_2)=0\ \text{for all}\ \tilde q_2\}=0\;}. $$
Proof. The Gram matrix $\operatorname{diag}(1,-1,-1,-1)$ is invertible, so the form is non-degenerate.
Corollary. The trace form identifies the algebra with its dual and provides the pairings used by the operators of the block; it is not a semidefinite form, so it is not the wall form of the block. The form that is definite is the norm, of the degree-two structure.
The Invariance
The Associativity of the Form
Theorem (associativity). The trace form is associative and invariant for the product:
$$ \boxed{\;\tau(\tilde q_1\bullet\tilde q_2,\tilde q_3)=\tau(\tilde q_1,\tilde q_2\bullet\tilde q_3)=\tau(\tilde q_2,\tilde q_1\bullet\tilde q_3)\;}. $$
Proof. $\tau(\tilde q_1\bullet\tilde q_2,\tilde q_3)=\mathrm{Sc}((\tilde q_1\bullet\tilde q_2)\tilde q_3)$. Writing $\tilde q_1\bullet\tilde q_2=\tfrac12(\tilde q_1\tilde q_2+\tilde q_2\tilde q_1)$ and using the associativity of the plain product and the invariance of the scalar part under reversal, $\mathrm{Sc}(\tilde q_1\tilde q_2\tilde q_3)=\mathrm{Sc}(\tilde q_3\tilde q_1\tilde q_2)=\mathrm{Sc}(\tilde q_2\tilde q_3\tilde q_1)$, gives the equalities.
Corollary. The trace form is a non-degenerate associative form on the block, the property that a Jordan algebra of degree two carries; equivalently, every multiplication operator is self-adjoint for $\tau$:
$$ \tau(L^{\bullet}_{\tilde q_1}\tilde q_2,\tilde q_3)=\tau(\tilde q_2,L^{\bullet}_{\tilde q_1}\tilde q_3). $$
The Invariance under the Operators
Theorem. Every derivation of the block is skew for the trace form:
$$ \boxed{\;\tau(D\tilde q_1,\tilde q_2)+\tau(\tilde q_1,D\tilde q_2)=0\;}. $$
Proof. The trace form is associative, so for a derivation $D$, $0=D\,\tau(\tilde q_1\bullet\tilde q_2,\tilde q_3)=\tau(D(\tilde q_1\bullet\tilde q_2),\tilde q_3)+\tau(\tilde q_1\bullet\tilde q_2,D\tilde q_3)$; expanding the derivation rule and using the associativity of $\tau$ twice gives $\tau(D\tilde q_1,\tilde q_2)+\tau(\tilde q_1,D\tilde q_2)=0$. The derivations are The Multiplication Operators of the Symmetric Plain Algebra, below this article in this category, and the statement is the general invariance of the trace form of a Jordan algebra quoted in Jordan Algebras.
Remark (the invariance is the content of the menu). The menu comment of the block asks for "the trace form and its invariance under the operators of the Jordan product": the operators are the left multiplications $L^{\bullet}_{\tilde q}$ and the inner derivations they generate, and the invariance is the pair of identities $\tau(L^{\bullet}_{\tilde q_1}\tilde q_2,\tilde q_3)=\tau(\tilde q_2,L^{\bullet}_{\tilde q_1}\tilde q_3)$ and $\tau(D\tilde q_1,\tilde q_2)+\tau(\tilde q_1,D\tilde q_2)=0$. The self-adjointness is what makes the form the trace form, and the skewness of the derivations is what makes it the invariant form of the block.
Summary
The trace form of the block is $\tau(\tilde q_1,\tilde q_2)=\mathrm{Sc}(\tilde q_1\bullet\tilde q_2)$, symmetric and bilinear, equal to the plain form $q_{1t}q_{2t}-\vec q_1\cdot\vec q_2$ and polarising the quadratic trace $\mathrm{Sc}(\tilde q^2)=q_t^2-|\vec q|^2$. Its Gram matrix in the standard basis is $\operatorname{diag}(1,-1,-1,-1)$, so its signature is $(1,3)$ and it is indefinite and Lorentzian; it is non-degenerate with radical $0$, and it is the indefinite companion of the positive definite norm. It is associative, $\tau(\tilde q_1\bullet\tilde q_2,\tilde q_3)=\tau(\tilde q_1,\tilde q_2\bullet\tilde q_3)$, so every multiplication operator is self-adjoint for it, and every derivation is skew for it; these two invariances are the form content of the block. Over $\mathbb{C}$ the trace form has the same matrix shape and the same invariances, the form part of the degree-two Common Theory; the field changes only the reading of the isotropy.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $\tau(\tilde q_1,\tilde q_2)=\mathrm{Sc}(\tilde q_1\bullet\tilde q_2)=q_{1t}q_{2t}-\vec q_1\cdot\vec q_2$ | the trace form |
| $\operatorname{diag}(1,-1,-1,-1)$ | the Gram matrix, signature $(1,3)$ |
| $\operatorname{rad}\tau=0$ | the radical |
| $\tau(\tilde q_1\bullet\tilde q_2,\tilde q_3)=\tau(\tilde q_1,\tilde q_2\bullet\tilde q_3)$ | associativity |
| $\tau(L^{\bullet}_{\tilde q_1}\tilde q_2,\tilde q_3)=\tau(\tilde q_2,L^{\bullet}_{\tilde q_1}\tilde q_3)$ | self-adjointness of the operators |
| $\tau(D\tilde q_1,\tilde q_2)+\tau(\tilde q_1,D\tilde q_2)=0$ | skewness of the derivations |
Further Reading
- Introduction to the Symmetric Plain Algebra, for the product.
- Norm and Invertibility in the Symmetric Plain Algebra, for the norm, the quaternion form and the contrast with the trace form.
- Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the isotropic cone of the trace form against the zero-divisor set.
- The Multiplication Operators of the Symmetric Plain Algebra, below this article in this category, for the operators and the derivations that are self-adjoint and skew.
- The 2 Forms over the Quaternioid $\mathbb{R}$ Space, for the plain and quaternionic forms.
- Jordan Algebras, for the trace form of a Jordan algebra and its invariance.