The Square, the Idempotents and the Jordan Inverse of the Symmetric Plain Algebra
Introduction
The symmetric plain algebra of $\mathbb{B}$ carries the 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 article reads the three structures a commutative unital Jordan algebra is built from: the square $\tilde Q_2\mapsto\tilde Q_2\bullet\tilde Q_2$ and the quadratic map it defines, the idempotents $\tilde Q_2\bullet\tilde Q_2=\tilde Q_2$, and the inverse of an element.
The results are collected here because they are one subject. The block is a Jordan algebra of degree two, and its degree is the statement that every element satisfies a quadratic identity with a generic trace $T(\tilde Q)=2Q_t$ and a generic norm $\langle\tilde Q\rangle_{\natural}=Q_t^2+Q_x^2+Q_y^2+Q_z^2$; the idempotents are read off the same identity at integral trace; the inverse is the companion root of the quadratic; and the isotropic cone $\{\langle\tilde Q\rangle_{\natural}=0\}$ is what makes the algebra not a division Jordan algebra.
The square of the block is the plain square, so the idempotents of the block are the idempotents of the plain product, classified in Square Roots of One and the Idempotents of the Biquaternion Algebra, and the norm is the biquaternion norm of Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra. What this article adds is the reading of those facts as the degree-two structure of the Jordan algebra, and the fuller theory of the Jordan inverse. The Common Theory articles of the block are Norm and Invertibility in the Symmetric Plain Algebra, Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, Square Roots of One and the Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, Square Roots of Zero and the Nilpotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, Square Roots of a General Element in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space and The Specificities of the Symmetric Plain Algebra; the trace form of the block is The Trace Form and the Invariance of the Symmetric Plain Algebra; the idempotents as minimal ideals of the parent are Ideals and Peirce Decompositions of the Biquaternion Algebra; and the placement of the block among the twelve operations is The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space.
The Square and the Quadratic Map
The Square
Proposition. The square of an element in the block is the plain square:
$$ \tilde Q\bullet\tilde Q=\tilde Q^2=\bigl[Q_t^2-(\vec Q,\vec Q)\bigr]e_t+2Q_t\vec Q. $$
Proof. $\tilde Q_2\bullet\tilde Q_2=\tfrac12(\tilde Q_2^2+\tilde Q_2^2)=\tilde Q_2^2$, and the plain square is the displayed form, read from the scalar–vector rule of The Biquaternion Algebra and Its Vector Spaces at $\tilde Q_1=\tilde Q_2$.
Remark (the block's square and the algebra's square are one). Because the antisymmetric part of the plain product is alternating, its diagonal vanishes, and the plain square is carried by the symmetric part alone. The square of the block and the square of the algebra are therefore the same map $\tilde Q\mapsto\tilde Q^2$, and the antisymmetric part carries none of it. The spectrum of the block is the spectrum of the algebra, and the minimal polynomial is read from this square.
The Polarisation of the Square
Proposition. The product is the polarisation of the square:
$$ \tilde Q_1\bullet\tilde Q_2=\tfrac12\bigl((\tilde Q_1+\tilde Q_2)\bullet(\tilde Q_1+\tilde Q_2)-\tilde Q_1\bullet\tilde Q_1-\tilde Q_2\bullet\tilde Q_2\bigr). $$
Proof. The product is $\mathbb{C}$-bilinear and commutative, so the square of a sum expands as the sum of the two squares and twice the product.
So the quadratic map $\tilde Q\mapsto\tilde Q\bullet\tilde Q$ and the linear structure determine the product, and the two are the data a Jordan algebra is generated by; this is $\sigma_2$ in the notation of The Symmetric and Antisymmetric Parts of an Algebra Product.
The Quadratic Identity
Proposition (the quadratic identity). Every $\tilde Q$ satisfies
$$ \tilde Q\bullet\tilde Q-T(\tilde Q)\tilde Q+\langle\tilde Q\rangle_{\natural}e_t=0,\qquad T(\tilde Q)=2Q_t,\quad \langle\tilde Q\rangle_{\natural}=\sum_{\mu\in\{t,x,y,z\}}Q_\mu^2 . $$
Proof. On the square of the previous section, $\tilde Q\bullet\tilde Q=Q_t^2-(\vec Q,\vec Q)+2Q_t\vec Q$ and $T(\tilde Q)\tilde Q=2Q_t\cdot(Q_te_t+\vec Q)=2Q_t^2e_t+2Q_t\vec Q$. Subtracting the two leaves $\bigl(Q_t^2-(\vec Q,\vec Q)-2Q_t^2+\langle\tilde Q\rangle_{\natural}\bigr)e_t$, and the coefficient vanishes because $\langle\tilde Q\rangle_{\natural}=Q_t^2+(\vec Q,\vec Q)$.
Definition. The functional $T(\tilde Q)=2Q_t$ is the generic trace of the block and $\langle\tilde Q\rangle_{\natural}=\sum_\mu Q_\mu^2$ its generic norm. The quadratic identity is the generic minimal polynomial of the block, of degree two.
Remark (the block has degree two). The quadratic identity holds for every element, and for an element of the centre it degenerates: on $\tilde Q=Q_te_t$ the generic minimal polynomial, in the indeterminate $\lambda$, is $\lambda^2-2Q_t\lambda+Q_t^2=(\lambda-Q_t)^2$, a square. Generically the identity is of degree two, so the symmetric plain algebra is a Jordan algebra of degree two, and its elements are the roots of their own quadratic. The generic trace and the generic norm are the two symmetric functions of the roots; they are the Jordan-algebra structures with no analogue in the associative algebra, where the element has a degree-four generic polynomial inherited from the four-dimensional regular representation.
Remark (the discriminant). The discriminant of the quadratic identity is
$$ T(\tilde Q)^2-4\langle\tilde Q\rangle_{\natural}=4Q_t^2-4\sum_\mu Q_\mu^2=-4(\vec Q,\vec Q). $$
It vanishes exactly on the set $\{(\vec Q,\vec Q)=0\}$, the isotropic cone of the vector part, which contains the whole centre; there the two generic roots coincide, and off that set the element has two distinct generic roots. The set is strictly larger than the centre: the element $\vec e_x+i\vec e_y$ has a non-zero vector part and $(\vec Q,\vec Q)=1+i^2=0$, so its quadratic is $\lambda^2$ and its two roots coincide. The computed expression was checked on the coordinate rule.
The Idempotents of the Block
Definition. An idempotent of the block is an element $\tilde\Pi$ with
$$ \tilde\Pi\bullet\tilde\Pi=\tilde\Pi . $$
The idempotents of the block are the idempotents of the plain product, since the two squares agree, and their theory — the classification, the bijection with the roots of $+1$ through the halving map, the complementary idempotent $e_t-\tilde\Pi$, the isotropy, the cubic Peirce identity and the Jordan Peirce decomposition — is the Common Theory article Square Roots of One and the Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space. It is not repeated here, and this section keeps only the definition and the two facts that bear on the degree-two structure of the block.
Remark (the idempotents and the degree-two structure). An idempotent is read off the quadratic identity at integral trace: the trivial idempotents have $(T,N)=(0,0)$ and $(2,1)$, and the non-trivial ones $(T,N)=(1,0)$. The non-trivial idempotents are therefore isotropic, $\langle\tilde\Pi\rangle_{\natural}=0$, and hence are the non-invertible idempotents, the invertibility criterion being $N\neq0$; the full theory is Square Roots of One and the Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, and the isotropic idempotents as the scalar family of the zero divisors are Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space.
The Jordan Inverse
The Inverse of a Jordan Algebra
Definition. In a commutative unital Jordan algebra an element $x$ is invertible when there exists $y$ with
$$ x\bullet y=e_t,\qquad x\bullet(y\bullet x)=x, $$
and $y$ is then the Jordan inverse $x^{-1}$. In a Jordan algebra of degree two with generic trace $T$ and generic norm $\langle\cdot\rangle_{\natural}$, an element with $\langle x\rangle_{\natural}\neq0$ is invertible and
$$ x^{-1}=\frac{T(x)e_t-x}{\langle x\rangle_{\natural}} . $$
The Criterion and the Formula in the Block
Theorem. An element $\tilde Q$ of the block is invertible exactly when $\langle\tilde Q\rangle_{\natural}\neq0$, and its Jordan inverse is
$$ \boxed{\; \tilde Q^{-1}=\frac{2Q_te_t-\tilde Q}{\langle\tilde Q\rangle_{\natural}}=\frac{\tilde Q^{\natural}}{\langle\tilde Q\rangle_{\natural}} \;}. $$
Proof. The quadratic identity of the block, divided by $\langle\tilde Q\rangle_{\natural}$, is
$$ \frac{\tilde Q\bullet\tilde Q}{\langle\tilde Q\rangle_{\natural}}-\frac{T(\tilde Q)}{\langle\tilde Q\rangle_{\natural}}\tilde Q+e_t=0, $$
that is, $\tilde Q\bullet\bigl((T(\tilde Q)e_t-\tilde Q)/\langle\tilde Q\rangle_{\natural}\bigr)=e_t$, so the displayed element is a Jordan inverse; the second equation $x\bullet(y\bullet x)=x$ follows, since $\bullet$ is commutative and associative with respect to the unit in the Jordan sense. The identity $T(\tilde Q)e_t-\tilde Q=2Q_te_t-\tilde Q=\tilde Q^{\natural}$ is the definition of the natural conjugate.
The Comparison with the Algebra Inverse
Theorem. The Jordan inverse of the block is the inverse of the algebra: for $\langle\tilde Q\rangle_{\natural}\neq0$,
$$ \boxed{\; \tilde Q^{-1}=\frac{\tilde Q^{\natural}}{\langle\tilde Q\rangle_{\natural}},\qquad \tilde Q\bullet\tilde Q^{-1}=\tilde Q\tilde Q^{-1}=e_t \;}. $$
Proof. The inverse formula of the algebra is $\tilde Q^{-1}=\tilde Q^{\natural}/\langle\tilde Q\rangle_{\natural}$ in Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, and $\langle\tilde Q\rangle_{\natural}=\langle\tilde Q,\tilde Q\rangle_{\natural}$. The two formulas are the same element; the Jordan inverse was computed above as the same quotient.
Remark (the inverse agrees because the block is special). In a general Jordan algebra the inverse is a new operation; here the block is the symmetrisation of an associative algebra, and the Jordan inverse coincides with the associative inverse and is therefore computed by the natural conjugate and the norm, exactly as in the algebra. The invertibility criterion is the same in the two readings: an element is invertible in the block precisely when it is invertible in $\mathbb{B}$, that is, precisely when it is not a zero divisor.
The Norm, the Isotropic Cone and the Elements of Square Zero
The Generic Norm and the Isotropic Cone
The generic norm $\langle\tilde Q\rangle_{\natural}=\sum_\mu Q_\mu^2$ is the biquaternion norm $\langle\tilde Q,\tilde Q\rangle_{\natural}$ of Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, a complex-valued multiplicative algebraic norm vanishing exactly on the isotropic cone $\{N=0\}$, the origin together with the zero divisors. Its polar form is the complex bilinear dot product $\langle\tilde Q_1,\tilde Q_2\rangle_{\natural}=\sum_\mu Q_{1,\mu} Q_{2,\mu}$, symmetric and non-degenerate, with Gram matrix the identity $\mathrm{I}_4$ in the coefficient basis.
Definition. The isotropic cone of the block is the set
$$ \{\tilde Q\in\mathbb{B}:\langle\tilde Q\rangle_{\natural}=0\}. $$
Remark (the isotropic cone is the zero-divisor cone). $\langle\tilde Q\rangle_{\natural}=0$ holds for $\tilde Q=0$ and for the zero divisors of Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors, and for no other element. The isotropic cone of the block is the union of the origin and the zero-divisor cone of the algebra, so the symmetric plain algebra possesses non-zero isotropic elements and is not a division Jordan algebra. This is the Jordan reading of the failure of $\mathbb{B}$ to be a division algebra: the block is a Jordan algebra whose norm form is non-degenerate but isotropic, of realified signature $(4,4)$; it is the Hermitian form, and not the norm form, that is definite.
An element is isotropic exactly when it fails to be invertible, so the invertible elements of the block are its non-isotropic elements, the group of units of Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra.
The Elements of Square Zero
Theorem. The elements of square zero, $\tilde Q\bullet\tilde Q=0$, are exactly the pure isotropic vectors:
$$ \boxed{\; \tilde Q\bullet\tilde Q=0\iff Q_t=0\ \text{and}\ (\vec Q,\vec Q)=0 \;}, $$
that is, exactly the elements of the vector subspace $\mathrm{Vect}(\mathbb{B})$ on which the norm vanishes. They are a proper subset of the isotropic cone.
Proof. By the square of the block, $\tilde Q\bullet\tilde Q=\bigl(Q_t^2-(\vec Q,\vec Q)\bigr)e_t+2Q_t\vec Q$. Its vanishing gives the two scalar conditions $Q_t^2=(\vec Q,\vec Q)$ and $Q_t\vec Q=0$. If $Q_t\neq0$ the second gives $\vec Q=0$, and the first gives $Q_t=0$, a contradiction; so $Q_t=0$, and the first condition becomes $(\vec Q,\vec Q)=0$. Conversely a pure element with $(\vec Q,\vec Q)=0$ has zero square.
Remark (square zero is stronger than isotropic). The isotropic cone is $\{\langle\tilde Q\rangle_{\natural}=0\}$, which for a pure element is the same condition $(\vec Q,\vec Q)=0$; but for $Q_t\neq0$ the equation $\langle\tilde Q\rangle_{\natural}=0$ has the solutions $(\vec Q,\vec Q)=-Q_t^2$, whose square has the non-zero vector part $2Q_t\vec Q$ since $\vec Q\neq0$. The element $e_t+i\vec e_x$ is the witness: $\langle e_t+i\vec e_x\rangle_{\natural}=1+i^2=0$, while $(e_t+i\vec e_x)\bullet(e_t+i\vec e_x)=2e_t+2i\vec e_x\neq0$. The elements of square zero are the pure part of the isotropic cone, and the isotropic cone is strictly larger, so the two notions must not be identified.
The Isotopy with the Two by Two Matrices
Remark (degree two and the matrix model). The block is a special Jordan algebra of degree two, and it is isotopic to the symmetrised matrix algebra $M_2(\mathbb{C})^{+}$: the isomorphism $\mathsf{M}_2$ of The General Plain Algebra in the $2\times2$ Matrix Element Representation $M_2(\mathbb{C})$ carries the symmetric plain product to the symmetrised matrix product, $\mathsf{M}_2(\tilde Q\bullet\tilde{Q}_1)=\tfrac12(\mathsf{M}_2(\tilde Q)\mathsf{M}_2(\tilde{Q}_1)+\mathsf{M}_2(\tilde{Q}_1)\mathsf{M}_2(\tilde Q))$, and carries the generic trace to the matrix trace and the generic norm to the determinant,
$$ T(\tilde Q)=2Q_t=\operatorname{Tr}\mathsf{M}_2(\tilde Q),\qquad \langle\tilde Q\rangle_{\natural}=\sum_\mu Q_\mu^2=\det\mathsf{M}_2(\tilde Q). $$
Two Jordan algebras of degree two whose norm forms are carried to one another by an invertible linear map are isotopic, so the block is isotopic to the degree-two matrix algebra, and its quadratic identity is the Cayley–Hamilton identity of the $2\times2$ matrices. The degree-two structure of the block is the degree-two structure of the matrices, and the correspondence is read in full in Introduction to the Symmetric Plain Algebra.
Summary
The symmetric plain algebra is a commutative unital Jordan algebra of degree two. Its square is the plain square, $\tilde Q\bullet\tilde Q=Q_t^2-(\vec Q,\vec Q)+2Q_t\vec Q$, and it satisfies the quadratic identity $\tilde Q\bullet\tilde Q-T(\tilde Q)\tilde Q+\langle\tilde Q\rangle_{\natural}e_t=0$ with generic trace $T(\tilde Q)=2Q_t$ and generic norm $\langle\tilde Q\rangle_{\natural}=\sum_\mu Q_\mu^2$, of discriminant $-4(\vec Q,\vec Q)$. Its idempotents are the idempotents of the plain product — the trivial elements $0$ and $e_t$ and the isotropic elements $\tfrac12(e_t+\vec q)$ with $\vec q^2=e_t$ pure — and their theory is the Common Theory article Square Roots of One and the Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space. Its inverse is $\tilde Q^{-1}=(2Q_te_t-\tilde Q)/\langle\tilde Q\rangle_{\natural}=\tilde Q^{\natural}/\langle\tilde Q\rangle_{\natural}$, which is the inverse of the algebra, and the invertibility criterion is $\langle\tilde Q\rangle_{\natural}\neq0$. The isotropic cone is $\{\langle\tilde Q\rangle_{\natural}=0\}$, the union of the origin and the zero-divisor cone, so the block is not a division Jordan algebra; the elements of square zero are the pure isotropic vectors, a proper subset of that cone, and $e_t+i\vec e_x$ separates the two. The block is a degree-two Jordan algebra isotopic to the symmetrised $2\times2$ matrices, with the generic trace as the matrix trace and the generic norm as the determinant.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $\tilde Q\bullet\tilde Q=\tilde Q^2$ | the square of the block, the plain square |
| $T(\tilde Q)=2Q_t$ | the generic trace |
| $\langle\tilde Q\rangle_{\natural}=\sum_{\mu\in\{t,x,y,z\}}Q_\mu^2=\langle\tilde Q,\tilde Q\rangle_{\natural}$ | the generic norm, the biquaternion norm |
| $\tilde Q\bullet\tilde Q-T(\tilde Q)\tilde Q+\langle\tilde Q\rangle_{\natural}e_t=0$ | the quadratic identity, the generic minimal polynomial |
| $\tfrac12(e_t+\vec q)$, $\vec q^2=e_t$ pure | the non-trivial idempotents (Square Roots of One and the Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space) |
| $\tilde Q^{-1}=\tilde Q^{\natural}/\langle\tilde Q\rangle_{\natural}$ | the Jordan inverse, the algebra inverse |
| $\{\langle\tilde Q\rangle_{\natural}=0\}$ | the isotropic cone, the zero-divisor cone with the origin |
| $Q_t=0,\ (\vec Q,\vec Q)=0$ | the elements of square zero, the pure isotropic vectors |
| $\mathsf{M}_2$ | the $2\times2$ matrix model, carrying $T$ to $\operatorname{Tr}$ and $\langle\cdot\rangle_{\natural}$ to $\det$ |
Further Reading
- Norm and Invertibility in the Symmetric Plain Algebra, Zero Divisors in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, Square Roots of One and the Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space and Square Roots of Zero and the Nilpotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the Common Theory of the block.
- Square Roots of One and the Idempotents of the Biquaternion Algebra, for the classification of the roots of $+1$ and of the idempotents, the projections and the minimal left ideals.
- Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, for the biquaternion norm, its multiplicativity and the invertibility criterion.
- Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, for the zero divisors and the structure of the isotropic set.
- The Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the degree-two reading and the placement of the block.
- Introduction to the Symmetric Plain Algebra, for the matrix model and the isotopy with the symmetrised matrices.