Ideals and Peirce Decomposition of the Algebra of the Reduced Biquaternions
Introduction
This article reads the ideal theory and the Peirce decomposition of the multiplication $\breve Q_{1}\breve Q_{2}$, the general plain algebra of the reduced biquaternions. Because the multiplication is commutative, the three kinds of ideal — left, right and two-sided — coincide, and the theory is the theory of the ideals of a commutative ring. The algebra is the direct product $\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$ of two fields, so its ideal lattice is the shortest one a semisimple algebra can have:
$$ 0\subsetneq \mathbb{C}\Pi_+,\ \mathbb{C}\Pi_-\subsetneq \mathbb{B}_{\mathrm{r}}, $$
with the two middle ideals minimal and incomparable. The structure is semisimple but not simple; the failure of simplicity is the specificity of this block against the biquaternion block, whose general plain algebra is the simple algebra $M_{2}(\mathbb{C})$.
The Peirce decomposition is the other half of the article, and here it is diagonal. For the pair $\Pi_+,\Pi_-$ the four Peirce spaces are $\mathbb{B}_{\sigma\tau}=\{\breve Q:\Pi_\sigma \breve Q\Pi_\tau=\breve Q\}$ with $\sigma,\tau\in\{+,-\}$, but because the idempotents are central and orthogonal the two off-diagonal spaces vanish, $\mathbb{B}_{+-}=\Pi_+\mathbb{B}_{\mathrm{r}}\Pi_-=\mathbb{B}_{\mathrm{r}}\Pi_+\Pi_-=0$ and $\mathbb{B}_{-+}=0$. So the Peirce decomposition is the direct product decomposition $\mathbb{B}_{\mathrm{r}}=\mathbb{B}_{++}\oplus\mathbb{B}_{--}=\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$ and carries no off-diagonal corners; the biquaternion counterpart carries the two one-dimensional matrix-unit spaces $E_{12},E_{21}$ of $M_{2}(\mathbb{C})$, and the difference is that the reduced biquaternion idempotents are central.
The results. The two-sided ideals are $0$, $\mathbb{C}\Pi_+$, $\mathbb{C}\Pi_-$ and $\mathbb{B}_{\mathrm{r}}$; the two middle ones are the minimal ideals, each generated by its idempotent, and they are the simple modules of the algebra. The algebra is Artinian, semisimple, of length two as a module over itself, and not simple. Its idempotents are $0,1,\Pi_+,\Pi_-$; the family $\{\Pi_+,\Pi_-\}$ is complete ($\Pi_++\Pi_-=1$) and orthogonal ($\Pi_+\Pi_-=0$), and every idempotent is fixed by the bar. The Peirce decomposition along any nontrivial idempotent is the same direct product, the decomposition is a $\mathbb{Z}_{2}$-grading $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\cdot 1\oplus\mathbb{C}\cdot j$, and the radical is $0$.
| Question | Answer |
|---|---|
| Two-sided ideals | $0,\mathbb{C}\Pi_+,\mathbb{C}\Pi_-,\mathbb{B}_{\mathrm{r}}$ |
| Minimal and maximal ideals | $\mathbb{C}\Pi_+,\mathbb{C}\Pi_-$ |
| Simple? | no |
| Semisimple? | yes, $J(\mathbb{B}_{\mathrm{r}})=0$ |
| Module length | $2$ |
| Idempotents | $0,1,\Pi_+,\Pi_-$ |
| Peirce decomposition | $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$ |
| Ideal lattice | the four-element Boolean lattice |
Boundaries. The multiplication and its axioms are Introduction to the Algebra of the Reduced Biquaternions; the idempotents and the roots of one are Square Roots of One and the Idempotents of the Algebra of the Reduced Biquaternions; the zero divisors and the two lines are Zero Divisors of the Algebra of the Reduced Biquaternions; and the unit group is The Group of Units of the Algebra of the Reduced Biquaternions. The criterion that makes the ideal theory available — left alternativity, here associativity — is The Specificities of the Algebra of the Reduced Biquaternions, above this article. The multiplication operators and the enveloping algebra are The Multiplication Operators of the Algebra of the Reduced Biquaternions; the matrix model is The Algebra in the $2\times2$ Matrix Element Representation of the Reduced Biquaternions.
Conventions. The algebra is $\mathbb{B}_{\mathrm{r}}$ with product $\breve Q_{1}\breve Q_{2}$, basis $1,i,j,ij$, and general element $\breve Q=Q_{\blacktriangle}+jQ_{\blacktriangledown}$, $Q_{\blacktriangle},Q_{\blacktriangledown}\in\mathbb{C}$. The idempotents are $\Pi_\pm=\tfrac12(1\pm j)$, the idempotent coordinates are $Q_\pm=Q_{\blacktriangle}\pm Q_{\blacktriangledown}$, and $\breve Q=Q_+\Pi_++Q_-\Pi_-$. The left-regular model in the basis $1,j$ is written $\varphi$, and the Peirce-coordinate model is written $\varphi^{\Pi}$. The standard pair is $\Pi_+,\Pi_-$, with $\Pi_-=1-\Pi_+$. The involutions are the bar $\bar{\cdot}$, the tessarine sign ${}_{\mathrm{ts}}$, the star ${}^{*}=\bar{\cdot}\circ{}_{\mathrm{ts}}$ and the flat ${}^{\flat}=-{}^{*}$. The base field is $\mathbb{C}$ unless a statement names $\mathbb{R}$; the notation is that of Conventions in Mathematics.
Ideals in an Algebra
An ideal is a subspace closed under multiplication by the whole algebra; in a commutative algebra the left, the right and the two-sided ideals coincide.
Definition. Let $\mathcal A$ be an algebra with product written by juxtaposition. A subspace $\mathrm A\subseteq\mathcal A$ is a left ideal when $\mathcal A\mathrm A\subseteq\mathrm A$, a right ideal when $\mathrm A\mathcal A\subseteq\mathrm A$, and a two-sided ideal when it is both. An ideal is proper when it is neither $0$ nor the whole algebra, and minimal when it is nonzero and contains no smaller nonzero ideal.
Proposition. In a commutative algebra every left ideal and every right ideal is two-sided.
Proof. If $\mathcal A$ is commutative and $\mathrm A$ is a left ideal then $\mathrm A\mathcal A=\mathcal A\mathrm A\subseteq\mathrm A$, so $\mathrm A$ is a right ideal and hence two-sided; the converse is the same argument. In particular the three notions define one set of ideals, which is the ideal lattice of the ring.
Corollary. Among the general products of the reduced biquaternion space only the general plain product is commutative, so it alone has a single notion of ideal; the other products are non-commutative — the general tessarine sesquilinear product is only symmetric up to the bar, $\breve Q_{2,\mathrm{ts}}\breve Q_{1}^{*}=\overline{\breve Q_{1,\mathrm{ts}}\breve Q_{2}^{*}}$, and not commutative — and their left and right ideals need not coincide. This article treats the general plain product, whose ideals are two-sided.
The Two-Sided Ideals
The two-sided ideals of the general plain algebra are $0$, the two idempotent lines and the whole algebra.
Theorem (the ideal lattice). The ideals of $(\mathbb{B}_{\mathrm{r}},\cdot)$ are $$ \boxed{\; 0,\qquad \mathbb{C}\Pi_+,\qquad \mathbb{C}\Pi_-,\qquad \mathbb{B}_{\mathrm{r}} \;}, $$ and there are no others. The two middle ones are minimal, they are generated by their idempotents, and their intersection is $0$.
| Ideal | Generator | Dimension over $\mathbb{C}$ | Minimal | Maximal | Quotient |
|---|---|---|---|---|---|
| $0$ | none | $0$ | no | no | $\mathbb{B}_{\mathrm{r}}$ |
| $\mathbb{C}\Pi_+$ | $\Pi_+$ | $1$ | yes | yes | $\mathbb{C}\Pi_-\cong\mathbb{C}$ |
| $\mathbb{C}\Pi_-$ | $\Pi_-$ | $1$ | yes | yes | $\mathbb{C}\Pi_+\cong\mathbb{C}$ |
| $\mathbb{B}_{\mathrm{r}}$ | $1$ | $2$ | no | no | $0$ |
Proof. Because the multiplication is componentwise in the idempotent coordinates $\breve Q=(Q_{+},Q_{-})$ (The Specificities of the Algebra of the Reduced Biquaternions), an ideal is a subset stable under multiplication by every pair $(Q_{2+},Q_{2-})$, that is a product $J_{1}\times J_{2}$ of an ideal $J_{1}$ of the first copy of $\mathbb{C}$ and an ideal $J_{2}$ of the second. A field has the two ideals $0$ and itself, so the ideals of $\mathbb{C}\times\mathbb{C}$ are the four products $\{0\}\times\{0\}$, $\mathbb{C}\times\{0\}$, $\{0\}\times\mathbb{C}$ and $\mathbb{C}\times\mathbb{C}$. In the algebra these are $0$, $\mathbb{C}\Pi_+$ (the first coordinate alone, which is $Q_{-}=0$), $\mathbb{C}\Pi_-$ (the second coordinate alone, $Q_{+}=0$) and $\mathbb{B}_{\mathrm{r}}$. The two middle ones are generated by $\Pi_+$ and $\Pi_-$, $\mathbb{C}\Pi_+=\mathbb{B}_{\mathrm{r}}\Pi_+$ and $\mathbb{C}\Pi_-=\mathbb{B}_{\mathrm{r}}\Pi_-$, and they meet in $0$ since an element of both has $Q_{+}=Q_{-}=0$.
Corollary (maximal ideals). The two proper nonzero ideals are the maximal ideals, and the intersection of the maximal ideals is the radical $0$ of §The Radical.
Artinian, Semisimple, and Length Two
The algebra is Artinian and semisimple, and it is not simple: that is the specificity of the block.
| Property | Result |
|---|---|
| Artinian | yes, the complex dimension is finite |
| Noetherian | yes, the complex dimension is finite |
| Semisimple | yes, $J(\mathbb{B}_{\mathrm{r}})=0$ |
| Simple | no |
| Length as a module over itself | $2$ |
| Simple modules | two, $\mathbb{C}\Pi_{+}$ and $\mathbb{C}\Pi_{-}$ |
| Radical | $0$ |
| Reduced | yes, no nonzero nilpotent |
Definition. An algebra is simple when it has no two-sided ideal other than $0$ and itself, and semisimple when it is a direct sum of simple pieces, equivalently when its radical vanishes.
Theorem. The general plain algebra is Artinian (it has finite complex dimension and every descending chain of ideals is stationary), it is semisimple, and it is not simple: the two minimal ideals $\mathbb{C}\Pi_+$ and $\mathbb{C}\Pi_-$ are proper.
Proof. Finite-dimensional algebras are Artinian. The algebra is the direct product $\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$ of two fields (Reduced Biquaternion Space), and a direct product of semisimple pieces is semisimple; it fails to be simple because each factor is a proper two-sided ideal.
Remark (the comparison with the biquaternion block). The general plain algebra of the biquaternion category is $\mathbb{B}\cong M_{2}(\mathbb{C})$, central simple, with no proper two-sided ideal and with a two-by-two matrix of corners. The reduced biquaternion general plain algebra is the other extreme: it is a product of two fields, it splits as an algebra, and the split is visible in its ideal lattice. The two blocks share the associativity and the criterion; they differ in the commutativity, and the whole of the ideal and Peirce theory follows from that one difference.
Idempotents and Orthogonal Idempotents
The idempotents of the algebra are $0,1,\Pi_+,\Pi_-$, and the pair $\{\Pi_+,\Pi_-\}$ is a complete orthogonal family of central idempotents.
Definition. Two idempotents $e,f$ are orthogonal when $ef=fe=0$. A finite family $\{e_{1},\dots,e_{n}\}$ of orthogonal idempotents is complete when $e_{1}+\dots+e_{n}=1$. An idempotent is central when it commutes with every element; in a commutative algebra every idempotent is central.
Theorem. The idempotents of the multiplication are $0$, $1$, $\Pi_+$ and $\Pi_-$, and the family $\{\Pi_+,\Pi_-\}$ is complete and orthogonal.
Proof. The idempotents are the solutions of $\breve Q^{2}=\breve Q$, computed in Square Roots of One and the Idempotents of the Algebra of the Reduced Biquaternions as $0,1,\Pi_+,\Pi_-$. The identities $\Pi_+\Pi_-=0$ and $\Pi_++\Pi_-=1$ are the orthogonality and the completeness.
| Idempotent | Form | Complement | Central | Primitive | Matrix model |
|---|---|---|---|---|---|
| $0$ | $0$ | $1$ | yes | no | $\operatorname{diag}(0,0)$ |
| $1$ | $1$ | $0$ | yes | no | $\mathrm I_{2}$ |
| $\Pi_{+}$ | $\tfrac12(1+j)$ | $\Pi_{-}$ | yes | yes | $\operatorname{diag}(1,0)$ |
| $\Pi_{-}$ | $\tfrac12(1-j)$ | $\Pi_{+}$ | yes | yes | $\operatorname{diag}(0,1)$ |
Corollary (the central splitting). Every element is a unique combination of the two complementary idempotents, $$ \breve Q=Q_{+}\Pi_++Q_{-}\Pi_-, $$ and the multiplication of two such combinations is componentwise. This is the content of the theorem in The Specificities of the Algebra of the Reduced Biquaternions, and it is the reason the Peirce decomposition below is diagonal.
The Peirce Decomposition
A nontrivial idempotent gives a fourfold decomposition of the space, the two diagonal spaces are the idempotent lines and the two off-diagonal spaces vanish.
One Idempotent
Let $\Pi$ be a nontrivial idempotent and write $\Pi'=1-\Pi$ for its complement. The sign $+$ names $\Pi$ and the sign $-$ names $\Pi'$. Every element expands as
$$ \breve Q=\Pi\breve Q\Pi+\Pi\breve Q\Pi'+\Pi'\breve Q\Pi+\Pi'\breve Q\Pi', $$
so the algebra splits into the four Peirce spaces
$$ \mathbb{B}_{\mathrm{r}}=\mathbb{B}_{++}\oplus\mathbb{B}_{+-}\oplus\mathbb{B}_{-+}\oplus\mathbb{B}_{--}, \qquad \mathbb{B}_{\sigma\tau}=\Pi_\sigma\mathbb{B}_{\mathrm{r}}\Pi_\tau, $$
with $\Pi_+=\Pi$ and $\Pi_-=\Pi'$. This is the Peirce decomposition of $\mathbb{B}_{\mathrm{r}}$ with respect to $\Pi$. It refines the two-term decomposition $\mathbb{B}_{\mathrm{r}}=\mathbb{B}_{\mathrm{r}}\Pi\oplus\mathbb{B}_{\mathrm{r}}\Pi'$.
Proposition (the Peirce multiplication). For $\sigma,\tau,\mu,\nu\in\{+,-\}$,
$$ \boxed{\; \mathbb{B}_{\sigma\tau}\mathbb{B}_{\mu\nu}\subseteq\delta_{\tau\mu}\mathbb{B}_{\sigma\nu} \;}. $$
Proof. An element of $\mathbb{B}_{\sigma\tau}$ is $\Pi_\sigma\breve Q\Pi_\tau$ and an element of $\mathbb{B}_{\mu\nu}$ is $\Pi_\mu\breve B\Pi_\nu$. The product is $\Pi_\sigma\breve Q\Pi_\tau\Pi_\mu\breve B\Pi_\nu$; the middle factor is $\Pi_\tau\Pi_\mu=\delta_{\tau\mu}\Pi_\tau$. The result is $\delta_{\tau\mu}\Pi_\sigma(\breve Q\Pi_\tau\breve B)\Pi_\nu\in\delta_{\tau\mu}\mathbb{B}_{\sigma\nu}$.
Theorem (the off-diagonal vanishes). For the general plain algebra and a nontrivial idempotent $\Pi$,
$$ \mathbb{B}_{+-}=\Pi\,\mathbb{B}_{\mathrm{r}}\,\Pi'=0,\qquad \mathbb{B}_{-+}=0,\qquad \mathbb{B}_{++}=\mathbb{C}\Pi,\qquad \mathbb{B}_{--}=\mathbb{C}\Pi', $$
so the Peirce decomposition is the direct product $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi\oplus\mathbb{C}\Pi'$.
Proof. The idempotents are central, so $\Pi \breve Q\Pi'=\breve Q\Pi\Pi'=0$ for every $\breve Q$, which is the vanishing of $\mathbb{B}_{+-}$; the mirror gives $\mathbb{B}_{-+}=0$. The corner spaces are $\Pi\mathbb{B}_{\mathrm{r}}\Pi=\Pi^{2}\mathbb{B}_{\mathrm{r}}=\Pi\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi$ and likewise $\mathbb{C}\Pi'$, by the ideal theorem of §The Two-Sided Ideals.
Corollary (the two-term decomposition). The subspaces $\mathbb{B}_{\mathrm{r}}\Pi=\mathbb{B}_{++}$ and $\mathbb{B}_{\mathrm{r}}\Pi'=\mathbb{B}_{--}$ are the two minimal ideals, and $\mathbb{B}_{\mathrm{r}}=\mathbb{B}_{\mathrm{r}}\Pi\oplus\mathbb{B}_{\mathrm{r}}\Pi'$ is the direct product decomposition.
A Complete Orthogonal Family
For a complete orthogonal family $\Pi_{1},\dots,\Pi_{n}$ of central idempotents the same construction gives
$$ \mathbb{B}_{\mathrm{r}}=\bigoplus_{i,j=1}^{n}\Pi_i\mathbb{B}_{\mathrm{r}}\Pi_j, \qquad (\Pi_i\mathbb{B}_{\mathrm{r}}\Pi_j)(\Pi_k\mathbb{B}_{\mathrm{r}}\Pi_l)\subseteq\delta_{jk}\,\Pi_i\mathbb{B}_{\mathrm{r}}\Pi_l, $$
and the off-diagonal components vanish, because $\Pi_i\Pi_j=0$ for $i\neq j$. Each diagonal corner $\Pi_i\mathbb{B}_{\mathrm{r}}\Pi_i$ is an algebra with identity $\Pi_i$. For the family $\{\Pi_+,\Pi_-\}$ the diagonal spaces are $\mathbb{B}_{++}=\mathbb{C}\Pi_+$ and $\mathbb{B}_{--}=\mathbb{C}\Pi_-$ and the off-diagonal spaces vanish, so
$$ \mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-, $$
which is the same decomposition as in §One Idempotent with $\Pi=\Pi_+$, $\Pi'=\Pi_-$.
The Reduced Biquaternion Case
The four Peirce spaces of the family $\{\Pi_+,\Pi_-\}$ are two lines and two zero spaces.
| Corner | Definition | Basis | Dimension over $\mathbb{C}$ | Matrix unit |
|---|---|---|---|---|
| $\mathbb{B}_{++}$ | $\Pi_+\mathbb{B}_{\mathrm{r}}\Pi_+$ | $\Pi_+$ | $1$ | $E_{11}$ |
| $\mathbb{B}_{+-}$ | $\Pi_+\mathbb{B}_{\mathrm{r}}\Pi_-$ | none | $0$ | $0$ |
| $\mathbb{B}_{-+}$ | $\Pi_-\mathbb{B}_{\mathrm{r}}\Pi_+$ | none | $0$ | $0$ |
| $\mathbb{B}_{--}$ | $\Pi_-\mathbb{B}_{\mathrm{r}}\Pi_-$ | $\Pi_-$ | $1$ | $E_{22}$ |
The two zero spaces are the sharp contrast with the biquaternion table, whose two off-diagonal spaces are one-dimensional and carry the matrix units $E_{12},E_{21}$ with $E_{ij}E_{kl}=\delta_{jk}E_{il}$. Here there is no matrix unit off the diagonal, and the attempted system $E_{11}=\Pi_+$, $E_{22}=\Pi_-$ has no off-diagonal completion: the algebra is not a full matrix algebra. The real dimensions are twice the complex ones, so the total is $2+0+0+2=4$ over $\mathbb{R}$. With respect to the idempotent $0$ or $1$ the Peirce decomposition is trivial: for $\Pi=1$ the only nonzero component is the whole algebra, and for $\Pi=0$ it is the zero space.
From One Idempotent to Its Complement
The decomposition depends on the unordered pair of complementary idempotents, not on which member carries the sign $+$.
The idempotent $\Pi'=1-\Pi$ gives the same decomposition with the two signs exchanged:
$$ \Pi'\mathbb{B}_{\mathrm{r}}\Pi'=\mathbb{B}_{--},\qquad \Pi'\mathbb{B}_{\mathrm{r}}\Pi=\mathbb{B}_{-+},\qquad \Pi\mathbb{B}_{\mathrm{r}}\Pi'=\mathbb{B}_{+-},\qquad \Pi\mathbb{B}_{\mathrm{r}}\Pi=\mathbb{B}_{++}. $$
The two ordered readings differ by the swap $+\leftrightarrow-$. For the standard pair the two readings are the same up to the exchange of $\Pi_+$ and $\Pi_-$, and since the only nontrivial idempotents are $\Pi_+$ and $\Pi_-$, every nontrivial idempotent gives the same decomposition, $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$.
The Peirce Table
The four spaces multiply by the matrix-unit rule.
The entry in row $S$ and column $T$ is the space containing the products of an element of $S$ with an element of $T$:
| $\cdot$ | $\mathbb{B}_{++}$ | $\mathbb{B}_{+-}$ | $\mathbb{B}_{-+}$ | $\mathbb{B}_{--}$ |
|---|---|---|---|---|
| $\mathbb{B}_{++}$ | $\mathbb{B}_{++}$ | $0$ | $0$ | $0$ |
| $\mathbb{B}_{+-}$ | $0$ | $0$ | $0$ | $0$ |
| $\mathbb{B}_{-+}$ | $0$ | $0$ | $0$ | $0$ |
| $\mathbb{B}_{--}$ | $0$ | $0$ | $0$ | $\mathbb{B}_{--}$ |
For the standard pair only the two diagonal entries survive, all other products zero. The table is the specialisation of the rule $\delta_{\tau\mu}$ to the vanishing off-diagonal spaces.
The Peirce Coordinates
In the Peirce coordinates an element is a pair of complex numbers, and the multiplication is the componentwise product.
An element $\breve Q$ is written $\breve Q=\Pi_+\breve Q\Pi_++\Pi_-\breve Q\Pi_-=Q_{+}\Pi_++Q_{-}\Pi_-$ with $Q_{+},Q_{-}\in\mathbb{C}$, its Peirce coordinates, which are the idempotent coordinates $Q_{+}$ and $Q_{-}$ of Reduced Biquaternions as a Vector Space over $\mathbb{R}$ and $\mathbb{C}$. Two elements multiply coordinatewise, $(Q_{1+},Q_{1-})(Q_{2+},Q_{2-})=(Q_{1+}Q_{2+},Q_{1-}Q_{2-})$, so the algebra is $\mathbb{C}\times\mathbb{C}$ and the Peirce coordinates are the coordinates of the product ring.
The Decomposition as a Grading
The splitting $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$ is an algebra decomposition, and it is equivalent to the $\mathbb{Z}_{2}$-grading whose even part is the scalar line and whose odd part is the $j$-line.
Definition. A $\mathbb{Z}_{2}$-grading of an algebra is a direct sum $\mathcal A=\mathcal A_{0}\oplus\mathcal A_{1}$ of subspaces with $\mathcal A_{\epsilon}\mathcal A_{\delta}\subseteq\mathcal A_{\epsilon+\delta}$ for the sum mod $2$.
Theorem. With
$$ \mathcal A_{0}=\mathbb{C}\cdot 1,\qquad \mathcal A_{1}=\mathbb{C}\cdot j, $$
the algebra is $\mathbb{Z}_{2}$-graded: $1\cdot1=1$, $1\cdot j=j$ and $j\cdot j=1$ put the even part in itself, the even part times the odd part in the odd part, and the odd part times itself in the even part.
Proof. The multiplication table of the basis $1,j$ is $1\cdot 1=1$, $1\cdot j=j\cdot 1=j$, $j\cdot j=1$, which is exactly the grading condition; the two spaces are independent and span $\mathbb{B}_{\mathrm{r}}$ over $\mathbb{C}$.
Remark (the two readings). The central splitting into the two idempotent lines and the $\mathbb{Z}_{2}$-grading into the scalar and $j$-lines are two presentations of the same algebra decomposition: the idempotent coordinates are the diagonalising coordinates of the grading. Both are algebra decompositions, and both are available exactly for the general plain product; the other seven structures do not respect the split.
The Involutions on the Peirce Spaces
The bar fixes the two idempotents and preserves each Peirce space; the tessarine sign and the star exchange the two diagonal spaces; the flat exchanges them with a sign.
The bar fixes $\Pi_+$ and $\Pi_-$, so it preserves each of the four Peirce spaces. The tessarine sign and the star exchange the two idempotents, hence exchange $\mathbb{B}_{++}$ with $\mathbb{B}_{--}$; the two off-diagonal spaces are zero. The flat is not multiplicative, but as a linear map it exchanges the two diagonal lines with a sign. The action is collected in the table.
| Involution | $\Pi_{+}$ | $\Pi_{-}$ | $\mathbb{B}_{++}$ | $\mathbb{B}_{--}$ | $\mathbb{B}_{+-}$ | $\mathbb{B}_{-+}$ |
|---|---|---|---|---|---|---|
| bar $\bar{\cdot}$ | $\Pi_{+}$ | $\Pi_{-}$ | $\mathbb{B}_{++}$ | $\mathbb{B}_{--}$ | $0$ | $0$ |
| tessarine sign ${}_{\mathrm{ts}}$ | $\Pi_{-}$ | $\Pi_{+}$ | $\mathbb{B}_{--}$ | $\mathbb{B}_{++}$ | $0$ | $0$ |
| star ${}^{*}$ | $\Pi_{-}$ | $\Pi_{+}$ | $\mathbb{B}_{--}$ | $\mathbb{B}_{++}$ | $0$ | $0$ |
| flat ${}^{\flat}$ (linear action) | $-\Pi_{-}$ | $-\Pi_{+}$ | $\mathbb{B}_{--}$ | $\mathbb{B}_{++}$ | $0$ | $0$ |
The bar is $\mathbb{C}$-antilinear and preserves the ordered Peirce decomposition; the tessarine sign is $\mathbb{C}$-linear and reverses it; the star is $\mathbb{C}$-antilinear and reverses it. The bar and the star are real algebra automorphisms, and the tessarine sign is a $\mathbb{C}$-algebra automorphism. The flat is the negative of the star, so it is not an automorphism of the algebra even though its square is the identity; it is recorded here only for its action on the two lines.
The Peirce Coordinates and the Matrix Model
In the idempotent basis the Peirce decomposition is the diagonal decomposition of the matrix model; the two projectors are the diagonal matrix units, and the scalar coefficients of the Peirce coordinates are the two eigenvalues.
In the Peirce-coordinate model the element is the diagonal matrix
$$ \varphi^{\Pi}(\breve Q)=\operatorname{diag}(Q_{+},Q_{-}) =\begin{pmatrix}Q_{\blacktriangle}+Q_{\blacktriangledown}&0\\0&Q_{\blacktriangle}-Q_{\blacktriangledown}\end{pmatrix} $$
(The Algebra in the $2\times2$ Matrix Element Representation of the Reduced Biquaternions). The Peirce components are the two eigenspaces:
$$ \mathbb{B}_{++}=\mathbb{C}\Pi_{+},\qquad \mathbb{B}_{--}=\mathbb{C}\Pi_{-}. $$
The projectors are
$$ P_{+}=\operatorname{diag}(1,0)=E_{11},\qquad P_{-}=\operatorname{diag}(0,1)=E_{22}, $$
with $P_{+}+P_{-}=\mathrm I_{2}$ and $P_{+}P_{-}=0$. The off-diagonal Peirce components, which would be the matrix units $E_{12}$ and $E_{21}$, are absent: the image is the subalgebra of diagonal matrices in $M_{2}(\mathbb{C})$, not the full matrix algebra. In particular, $E_{12}$ and $E_{21}$ are square-zero in $M_2(\mathbb{C})$, but no nonzero nilpotent matrix lies in the image: a diagonal matrix with square zero is zero. This is the matrix form of the reducedness of $\mathbb{B}_{\mathrm{r}}$.
Two matrix models. The diagonal matrix $\varphi^{\Pi}(\breve Q)=\operatorname{diag}(Q_+,Q_-)$ is the Peirce-coordinate model: its two entries are the Peirce coordinates, and in this model the two Peirce components are the two matrix units $E_{11}$ and $E_{22}$. The left-regular model in the basis $(1,j)$ is the full matrix $$ \boxed{\; \varphi(\breve Q)=\begin{pmatrix}Q_{\blacktriangle}&Q_{\blacktriangledown}\\Q_{\blacktriangledown}&Q_{\blacktriangle}\end{pmatrix} \;}. $$ The two are related by the change of basis $$ \mathsf{C}=\frac12\begin{pmatrix}1&1\\1&-1\end{pmatrix},\qquad \mathsf{C}^{-1}=\begin{pmatrix}1&1\\1&-1\end{pmatrix},\qquad \varphi^{\Pi}(\breve Q)=\mathsf{C}^{-1}\varphi(\breve Q)\mathsf{C}. $$ The columns of $\mathsf{C}$ are the coordinate vectors of $\Pi_+$ and $\Pi_-$ in the basis $(1,j)$, and they are eigenvectors of $\varphi(\breve Q)$ with eigenvalues $Q_+$ and $Q_-$. The idempotent basis $(\Pi_+,\Pi_-)$ is a complex eigenbasis of $\breve Q$. In the left-regular model the matrices of the two idempotents are $$ \varphi(\Pi_+)=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad \varphi(\Pi_-)=\frac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}. $$ In the scalar case $Q_{\blacktriangledown}=0$, $\varphi(\breve Q)=Q_{\blacktriangle}\mathrm I_2$, every nonzero complex vector is an eigenvector, and every complex basis is an eigenbasis. The eigenbasis is developed in Eigenvalue Decomposition and Eigenbasis of a Reduced Biquaternion, §The Eigenbasis; the two models are developed in Eigenbasis of a Reduced Biquaternion in Matrix Elements Representations.
The Decomposition into Minimal Ideals
The two idempotent lines are the minimal left ideals and the minimal right ideals, and they are the two simple modules.
Theorem. The minimal left ideals of $(\mathbb{B}_{\mathrm{r}},\cdot)$ are $\mathbb{C}\Pi_+$ and $\mathbb{C}\Pi_-$, and they are also the minimal right ideals, the minimal two-sided ideals and the simple left modules. They are the two spinor spaces of Eigenvalue Decomposition and Eigenbasis of a Reduced Biquaternion, §The Spinor Spaces: each is a complex line, and a spinor is a single complex coefficient on one of them.
Proof. Every left ideal is two-sided by commutativity, and the only proper nonzero two-sided ideals are the two lines by the ideal theorem; a left ideal contained in a line is $0$ or the line, so the line is minimal. The left modules $\mathbb{B}_{\mathrm{r}}\Pi_+$ and $\mathbb{B}_{\mathrm{r}}\Pi_-$ are simple because they have no proper nonzero submodules, the submodules being the left ideals inside them.
Corollary (length two). As a module over itself the algebra is the direct sum of the two simple modules, $$ {}_{\mathbb{B}_{\mathrm{r}}}\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-, $$ so its length is two and it is the direct sum of two non-isomorphic simples; the algebra is its own basic algebra, with two simple modules.
The Lattice of Ideals
The four ideals form the four-element Boolean lattice $\mathcal{P}(\{+,-\})$, with meet intersection and join sum.
The four ideals of $\mathbb{B}_{\mathrm{r}}$ are $0$, $\mathbb{C}\Pi_+$, $\mathbb{C}\Pi_-$ and $\mathbb{B}_{\mathrm{r}}$, with the inclusions
$$ 0\subset\mathbb{C}\Pi_+\subset\mathbb{B}_{\mathrm{r}},\qquad 0\subset\mathbb{C}\Pi_-\subset\mathbb{B}_{\mathrm{r}}, $$
and no inclusion between $\mathbb{C}\Pi_+$ and $\mathbb{C}\Pi_-$ (they are independent, and their intersection is $0$).
| Meet | Value | Join | Value |
|---|---|---|---|
| $\mathbb{C}\Pi_+\cap\mathbb{C}\Pi_-$ | $0$ | $\mathbb{C}\Pi_++\mathbb{C}\Pi_-$ | $\mathbb{B}_{\mathrm{r}}$ |
| $\mathbb{C}\Pi_+\cap\mathbb{B}_{\mathrm{r}}$ | $\mathbb{C}\Pi_+$ | $\mathbb{C}\Pi_++\mathbb{B}_{\mathrm{r}}$ | $\mathbb{B}_{\mathrm{r}}$ |
| $0\cap I$ | $0$ | $0+I$ | $I$ |
Theorem. The lattice $\mathrm{Id}(\mathbb{B}_{\mathrm{r}})$ is a Boolean lattice with four elements, isomorphic to the power set of a two-point set:
$$ \mathrm{Id}(\mathbb{B}_{\mathrm{r}})\cong\mathcal{P}(\{+,-\}),\qquad 0\leftrightarrow\varnothing,\quad \mathbb{C}\Pi_+\leftrightarrow\{+\},\quad \mathbb{C}\Pi_-\leftrightarrow\{-\},\quad \mathbb{B}_{\mathrm{r}}\leftrightarrow\{+,-\}. $$
Proof. The map is the ideal generated by the indicator subset under the isomorphism $\mathbb{B}_{\mathrm{r}}\cong\mathbb{C}\oplus\mathbb{C}$: the ideal of $\mathbb{C}\oplus\mathbb{C}$ generated by the indicator vector of a subset $S$ is the set of pairs supported on $S$, and these four sets are exactly the ideals. Under this identification the complement of the ideal associated with $S$ is the ideal associated with the complementary subset $S^{\mathrm{c}}$, meets are intersections and joins are unions, so the lattice is the power-set lattice $\mathcal{P}(\{+,-\})$, which is Boolean.
The lattice is finite, distributive and complemented. The left, right and two-sided ideals coincide, so there is a single finite lattice, and it is the Boolean one above.
The Radical
The radical of the algebra is $0$.
Definition. The Jacobson radical of a finite-dimensional algebra is the intersection of its maximal left ideals, equivalently the largest nilpotent two-sided ideal.
Theorem. The Jacobson radical of the general plain algebra is $0$; the algebra is semisimple, and its only nilpotent element is $0$.
| Radical | Value |
|---|---|
| Jacobson radical | $0$ |
| Nilradical | $0$ |
| Prime radical | $0$ |
Proof. The maximal ideals are the two minimal ideals of §The Two-Sided Ideals, whose intersection is $\mathbb{C}\Pi_+\cap\mathbb{C}\Pi_-=0$. The algebra is reduced by Square Roots of Zero and the Nilpotents of the Algebra of the Reduced Biquaternions, so its largest nilpotent ideal is $0$ as well; the two descriptions agree. The prime radical of a commutative ring is its nilradical, hence it is zero too.
The Real Structure
Read over the real scalars the algebra is the direct product of two copies of $\mathbb{C}$ as a real algebra, and the ideals and the Peirce decomposition are the realification of the complex ones.
The real algebra has real dimension four, with the real basis $1,i,j,ij$ of Reduced Biquaternions as a Vector Space over $\mathbb{R}$ and $\mathbb{C}$; the two complex lines $\mathbb{C}\Pi_+$ and $\mathbb{C}\Pi_-$ become two real planes of real dimension two, and the scalar line becomes the real plane $\mathbb{R}\{1,i\}$. The ideal lattice is unchanged, since the two factors are the fields $\mathbb{C}$ read over $\mathbb{R}$; the Peirce decomposition is likewise the realification, the off-diagonal remaining $0$. No new ideal appears and none is lost: the ideal theory is a statement over the coefficient field and the realification does not alter the lattice, because $\mathbb{C}$ is a field over $\mathbb{R}$ as well.
Summary
| Statement | Result |
|---|---|
| Algebra | $\mathbb{B}_{\mathrm{r}}$, commutative, of dimension $2$ over $\mathbb{C}$ |
| Two-sided ideals | $0,\ \mathbb{C}\Pi_{+},\ \mathbb{C}\Pi_{-},\ \mathbb{B}_{\mathrm{r}}$ |
| Simple? | no |
| Semisimple? | yes, $J(\mathbb{B}_{\mathrm{r}})=0$ |
| Length over itself | $2$ |
| Idempotents | $0,1,\Pi_{+},\Pi_{-}$ |
| Peirce decomposition | $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi_{+}\oplus\mathbb{C}\Pi_{-}$ |
| Grading | $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\cdot1\oplus\mathbb{C}\cdot j$ |
| Jacobson radical | $0$ |
The general plain algebra $(\mathbb{B}_{\mathrm{r}},\cdot)$ is commutative, so its left, right and two-sided ideals coincide, and it is the direct product $\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$ of two fields, so its ideal lattice is $0\subsetneq\mathbb{C}\Pi_+,\mathbb{C}\Pi_-\subsetneq\mathbb{B}_{\mathrm{r}}$ with the two middle ideals minimal, generated by the idempotents and meeting in $0$. The algebra is Artinian and semisimple, with radical $0$, and it is not simple — the point at which it differs from the biquaternion general plain algebra $M_{2}(\mathbb{C})$. Its idempotents are $0,1,\Pi_+,\Pi_-$, and $\{\Pi_+,\Pi_-\}$ is a complete orthogonal family of central idempotents. The Peirce decomposition along a nontrivial idempotent has vanishing off-diagonal spaces $\mathbb{B}_{+-}=\mathbb{B}_{-+}=0$ because the idempotents are central, so it reduces to the central splitting $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$; there is no off-diagonal matrix unit. The Peirce coordinates are the idempotent coordinates $(Q_{+},Q_{-})=(Q_{\blacktriangle}+Q_{\blacktriangledown},Q_{\blacktriangle}-Q_{\blacktriangledown})$, in which the multiplication is componentwise, and the same splitting is the $\mathbb{Z}_{2}$-grading with even part the scalar line $\mathbb{C}\cdot 1$ and odd part the $j$-line $\mathbb{C}\cdot j$. The two idempotent lines are the minimal left, right and two-sided ideals and the two simple modules, so the algebra has length two and is its own basic algebra. Over the reals the lattice and the Peirce decomposition are the realification of the complex ones, unchanged.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $\mathbb{B}_{\mathrm{r}}$ | Reduced biquaternion algebra, $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}$ |
| $\breve Q=Q_{\blacktriangle}+jQ_{\blacktriangledown}$ | General element, with idempotent coordinates $Q_{\pm}=Q_{\blacktriangle}\pm Q_{\blacktriangledown}$ |
| $\Pi_{\pm}=\tfrac12(1\pm j)$ | The two nontrivial central idempotents |
| $\Pi'=1-\Pi$ | The complement of an idempotent |
| $\mathbb{B}_{\sigma\tau}=\Pi_\sigma\mathbb{B}_{\mathrm{r}}\Pi_\tau$ | The Peirce spaces of the sign pair, with $\sigma,\tau\in\{+,-\}$ |
| $\mathbb{B}_{\mathrm{r}}=\mathbb{C}\Pi_+\oplus\mathbb{C}\Pi_-$ | The central splitting into two fields |
| $\mathcal A_{0}=\mathbb{C}\cdot 1,\ \mathcal A_{1}=\mathbb{C}\cdot j$ | The even and odd parts of the $\mathbb{Z}_{2}$-grading |
| $\mathbb{C}\Pi_+,\ \mathbb{C}\Pi_-$ | The two minimal ideals and simple modules |
| $\operatorname{rad}(\mathbb{B}_{\mathrm{r}})$ | Jacobson radical, $=0$ |
Further Reading
- Richard S. Pierce, Associative Algebras (Springer, 1982), for the Peirce decomposition, the corner subalgebras and the matrix units.
- Richard D. Schafer, An Introduction to Nonassociative Algebras (Academic Press, 1966), for the radical and the idempotents of an algebra.
- Ideals and Peirce Decompositions of the Biquaternion Algebra, for the simple companion with its matrix-unit corners.
- The Specificities of the Algebra of the Reduced Biquaternions, for the criterion that makes the theory available.
- Square Roots of One and the Idempotents of the Algebra of the Reduced Biquaternions, for the idempotents.
- The Group of Units of the Algebra of the Reduced Biquaternions, for the unit group complementing the ideals.
- The Multiplication Operators of the Algebra of the Reduced Biquaternions, for the operators the ideal projections carry.