Split Bioctonions and the Clifford Algebra Cl(7)

Introduction

The complex octonions $\mathbb{C}\otimes\mathbb{O}$ are non-associative, but their left multiplications form an associative algebra, the complex octonionic chain algebra, which is the Clifford algebra $\mathrm{Cl}(6)$; the companion article Complex Octonions and the Clifford Algebra Cl(6) builds it in full, together with a maximal totally isotropic subspace of ladder operators, an intrinsic $\mathrm{su}(3)\oplus\mathrm{u}(1)$, a primitive idempotent and an eight-dimensional minimal left ideal whose basis carries the charges $0,\tfrac13,\tfrac13,\tfrac13,\tfrac23,\tfrac23,\tfrac23,1$. This article adjoins to that construction a single central split unit. The result is the complex split bioctonionic chain algebra, $$ \mathrm{Cl}(7) \cong \mathrm{Cl}(6)\oplus\mathrm{Cl}(6) \cong M_8(\mathbb{C})\oplus M_8(\mathbb{C}), $$ the complex Clifford algebra of the split bioctonions $\mathbb{D}\otimes_{\mathbb{R}}\mathbb{O}$. Its structure is fixed by the same mechanism as the one-rung-lower case of Complex Split Biquaternions and the Clifford Algebra Cl(3): the volume element of an odd-dimensional Clifford algebra is central, and here it is an involution, so it splits the algebra into the direct sum of two copies of the complex octonionic chain algebra, $$ \mathrm{Cl}(7) = \mathrm{Cl}(7)\Pi_+ \oplus \mathrm{Cl}(7)\Pi_-, \qquad \Pi_\pm = \tfrac12(1\pm J), \qquad J = e_1e_2e_3e_4e_5e_6e_7 . $$ Each copy is a full $\mathrm{Cl}(6)$ and therefore carries its own Fano chain algebra, its own ladder operators, its own $\mathrm{su}(3)\oplus\mathrm{u}(1)$ and its own eight-dimensional minimal left ideal with the same charge spectrum; the two copies are the two pinor representations of $\mathrm{Cl}(7)$, are inequivalent, and are interchanged by the parity automorphism $e_i\mapsto -e_i$. This is the algebraic content of a generation of fermions carrying both chiralities, developed in Left-Right Symmetric Fermions from Complex Split Biquaternions and Bioctonions.

The article is the octonionic member of the doubling pair whose lower member is Complex Split Biquaternions and the Clifford Algebra Cl(3). The octonions, their norm, their conjugation and their relation to the division algebras are Octonion Algebra and Octonion Norm and Invertibility; the complex octonions, Furey's Fano convention, the chain algebra $\mathrm{Cl}(6)\cong M_8(\mathbb{C})$, its ladder operators, its $\mathrm{su}(3)$, its primitive idempotent and its charges are Complex Octonions and the Clifford Algebra Cl(6); the general ladder and number-operator construction is Maximal Totally Isotropic Subspaces and Their Unitary Symmetries; and the split bioctonions' lower-dimensional relative is Split-Biquaternion Algebra.

Conventions. The seven generators $e_1,\dots,e_7$ satisfy $$ e_i^2=-1, \qquad e_ie_j=-e_je_i \quad (i\neq j), $$ so that $\{e_i,e_j\}=-2\delta_{ij}$, and the $2^7=128$ monomials are a $\mathbb{C}$-basis of $\mathrm{Cl}(7)\cong\mathrm{Cl}_7(\mathbb{C})$. The first six are the six units of the companion article, and their product is the $\mathrm{Cl}(6)$ volume element $$ \Gamma := e_1e_2e_3e_4e_5e_6, \qquad \Gamma^2=-1, $$ which Complex Octonions and the Clifford Algebra Cl(6) writes as $e_7$. In the present article the label $e_7$ denotes the seventh generator, not that volume element; the two labels must not be read index by index across the two articles. The Hermitian adjoint is fixed by $e_i^{\dagger}=-e_i$ and $i^{\dagger}=-i$, so that $\gamma_i=ie_i$ are Hermitian, as in the companion articles; the scalar imaginary is $i$. The split complex algebra is $\mathbb{D}$, with unit $j$, $j^2=+1$, as in Split-Biquaternion Algebra; the octonion multiplication is that of Octonion Algebra, and the warning of the companion article applies — the chain algebra result is stated in terms of the Clifford algebra, not in terms of the corpus octonion basis index by index.

The Bioctonions and the Split Bioctonions

Definition. The bioctonions are the complexification $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{O}$ of the octonions; the split bioctonions are the tensor product $\mathbb{D}\otimes_{\mathbb{R}}\mathbb{O}$ of the split complex algebra with the octonions; and the complex split bioctonions are the complexification $$ \mathbb{C}\otimes_{\mathbb{R}}\mathbb{D}\otimes_{\mathbb{R}}\mathbb{O}. $$

Proposition (splitting of the complex split bioctonions). The split complex algebra decomposes as $\mathbb{D}\cong\mathbb{C}\oplus\mathbb{C}$ through the idempotents $\tfrac12(1\pm j)$, and consequently $$ \mathbb{C}\otimes_{\mathbb{R}}\mathbb{D}\otimes_{\mathbb{R}}\mathbb{O} \cong (\mathbb{C}\otimes_{\mathbb{R}}\mathbb{O}) \oplus (\mathbb{C}\otimes_{\mathbb{R}}\mathbb{O}). $$

Proof. Since $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{D}\cong\mathbb{C}\oplus\mathbb{C}$ (Split-Complex Algebra), tensoring with $\mathbb{O}$ over $\mathbb{R}$ distributes: $\mathbb{C}\otimes\mathbb{D}\otimes\mathbb{O}\cong(\mathbb{C}\oplus\mathbb{C})\otimes\mathbb{O}\cong(\mathbb{C}\otimes\mathbb{O})\oplus(\mathbb{C}\otimes\mathbb{O})$. $\square$

Thus the complex split bioctonions are not new non-associative material: they are a pair of copies of the complex octonions. What is new is the algebra generated by their left multiplications.

Proposition (chain algebra of a direct sum). If $A$ is a real algebra and $L_A$ is the algebra generated by the left multiplications $L_a:b\mapsto ab$, then the chain algebra of $A\oplus A$ is $L_A\oplus L_A$.

Proof. Left multiplication by $(a,b)\in A\oplus A$ is $L_{(a,b)}=L_a\oplus L_b$, since the product in a direct sum is componentwise; as $a$ and $b$ range over $A$ the pairs $(L_a,L_b)$ range over $L_A\oplus L_A$, and the generated algebra is the direct sum. $\square$

Theorem (the chain algebra of the complex split bioctonions). The chain algebra of the complex split bioctonions is the complex Clifford algebra $$ \mathrm{Cl}(7) \cong \mathrm{Cl}(6)\oplus\mathrm{Cl}(6) \cong M_8(\mathbb{C})\oplus M_8(\mathbb{C}), $$ of complex dimension $128$, and its two direct summands are the two copies of the complex octonionic chain algebra of Complex Octonions and the Clifford Algebra Cl(6).

Proof. By the two propositions, the chain algebra is $L_{\mathbb{C}\otimes\mathbb{O}}\oplus L_{\mathbb{C}\otimes\mathbb{O}}\cong\mathrm{Cl}(6)\oplus\mathrm{Cl}(6)$. The low-dimensional classification gives $\mathrm{Cl}_6(\mathbb{C})\cong M_8(\mathbb{C})$ and $\mathrm{Cl}_7(\mathbb{C})\cong M_8(\mathbb{C})\oplus M_8(\mathbb{C})$, and the two are isomorphic as algebras; both statements were confirmed by the dimension count $2\cdot 64=128=2^7$. $\square$

The Volume Element and the Doubling

Proposition (the volume element). The element $$ J := e_1e_2e_3e_4e_5e_6e_7 = \Gamma e_7 $$ satisfies $$ J^2=1, \qquad J e_i = e_i J \quad (i=1,\dots,7), \qquad J^{\dagger}=J . $$ That is, $J$ is central, it is an involution, and it is self-adjoint.

Proof. Each generator crosses the other six once, at the cost of six signs, which cancel, so $J$ commutes with every generator and hence with the algebra. The square is $(-1)^{n(n-1)/2}c^n$ at $n=7$, $c=-1$, that is, $(-1)^{21}(-1)^7=+1$. The adjoint reverses seven anti-Hermitian factors, at the cost of $(-1)^7$ from anti-Hermiticity and one more sign from the reversal of seven anticommuting factors, and $(-1)^7\cdot(-1)=+1$, so $J^{\dagger}=J$. All three identities were checked on the generators with zero residual. $\square$

The centrality of $J$ is the statement that the algebra is odd-dimensional; the reality $J^2=+1$ is the statement that it is split rather than complex, and it is the one place where the octonionic case differs from $\mathrm{Cl}(6)$, whose volume element $\Gamma$ has square $-1$ and generates a $\mathrm{u}(1)$ rather than a splitting.

Proposition (the seventh generator is the $\mathrm{Cl}(6)$ volume element). Because $J=\Gamma e_7$ and $\Gamma$ anticommutes with $e_7$, $$ e_7 = -\Gamma J . $$ On each central summand, where $J$ acts as the scalar $\pm1$, the seventh generator acts as $\mp\Gamma$, the $\mathrm{Cl}(6)$ volume element, with opposite signs on the two summands.

Proof. From $\Gamma^2=-1$ and $e_7\Gamma=-\Gamma e_7$: $\Gamma J=\Gamma^2e_7=-e_7$, so $e_7=-\Gamma J$, and $J$ is $\pm1$ on the summands. $\square$

Theorem (the central splitting). With $\Pi_\pm=\tfrac12(1\pm J)$, the elements $\Pi_+,\Pi_-$ are a complete family of orthogonal central idempotents, $$ \Pi_+^2=\Pi_+, \qquad \Pi_-^2=\Pi_-, \qquad \Pi_+\Pi_-=0, \qquad \Pi_++\Pi_-=1, $$ and $$ \mathrm{Cl}(7) = \mathrm{Cl}(7)\Pi_+ \oplus \mathrm{Cl}(7)\Pi_- , $$ a direct sum of two two-sided ideals, each of complex dimension $64$ and each isomorphic to the complex octonionic chain algebra, $$ \mathrm{Cl}(7)\Pi_\pm \cong \mathrm{Cl}(6) \cong M_8(\mathbb{C}). $$

Proof. Idempotency, orthogonality and completeness come from $J^2=1$ as in the companion Cl(3) article; centrality from centrality of $J$. The two-sided ideals are the two simple summands of $\mathrm{Cl}(7)\cong M_8(\mathbb{C})\oplus M_8(\mathbb{C})$, each of complex dimension $64$; left multiplication by $\Pi_\pm$ was checked to be a rank-$64$ projector of the $128$-dimensional space. $\square$

Theorem (the parity automorphism and the two pinor representations). The map $$ \sigma : \mathrm{Cl}(7)\to\mathrm{Cl}(7), \qquad \sigma(e_i)=-e_i , $$ is an algebra automorphism which fixes the even monomials and negates the odd ones, satisfies $\sigma(J)=-J$, and interchanges the two summands, $\sigma(\Pi_+)=\Pi_-$, $\sigma(\Pi_-)=\Pi_+$. The two summands are the two pinor representations of $\mathrm{Cl}(7)$, inequivalent and distinguished by the eigenvalue $\pm1$ of the central volume element $J$; equivalently, by the proposition above, by the sign of the $\mathrm{Cl}(6)$ volume element $\Gamma$ that each induces.

Proof. As in the companion Cl(3) article: sending every generator to its negative preserves the quadratic relations and is invertible; on the volume element the seven sign flips give $\sigma(J)=(-1)^7J=-J$; and the two projections are the two nonequivalent simple modules of $M_8(\mathbb{C})\oplus M_8(\mathbb{C})$. $\square$

The Furey Structure in Each Summand

Each summand is a $\mathrm{Cl}(6)$ and inherits the whole octonionic chain structure of the companion article verbatim. It is collected here in the form used in the companion physics article.

Proposition (the ladder operators). Define, as in Complex Octonions and the Clifford Algebra Cl(6), $$ \alpha_1=\tfrac12(-e_5+ie_4), \qquad \alpha_2=\tfrac12(-e_3+ie_1), \qquad \alpha_3=\tfrac12(-e_6+ie_2), $$ with Hermitian conjugates $\alpha_i^{\dagger}$. Then $$ \{\alpha_i,\alpha_j\}=0, \qquad \{\alpha_i^{\dagger},\alpha_j^{\dagger}\}=0, \qquad \{\alpha_i,\alpha_j^{\dagger}\}=\delta_{ij}I , $$ so that $\operatorname{span}\{\alpha_1,\alpha_2,\alpha_3\}$ is a three-dimensional maximal totally isotropic subspace of the complexified six-space, its conjugate span is the conjugate MTIS, and the six together span the six-space.

Proof. Direct expansion in the Clifford relations $e_i^2=-1$, $e_ie_j=-e_je_i$; all nine relations in each family were recomputed inside $\mathrm{Cl}(7)$ with zero residual, confirming that adjoining the seventh generator does not disturb them (the $\alpha_i$ do not involve $e_7$). $\square$

Theorem (the ideal and its charges in each summand). Let $$ \omega := \alpha_1\alpha_2\alpha_3, \qquad P := \omega\,\omega^{\dagger}=\alpha_1\alpha_2\alpha_3\,\alpha_3^{\dagger}\alpha_2^{\dagger}\alpha_1^{\dagger}, \qquad N := \sum_{i=1}^{3}\alpha_i^{\dagger}\alpha_i . $$ Then $P$ is a primitive idempotent, $\alpha_iP=0$, and $$ S := \mathrm{Cl}(7)\,P = \operatorname{span}_{\mathbb{C}}\bigl\{\,P,\ \alpha_1^{\dagger}P,\ \alpha_2^{\dagger}P,\ \alpha_3^{\dagger}P,\ \alpha_3^{\dagger}\alpha_2^{\dagger}P,\ \alpha_1^{\dagger}\alpha_3^{\dagger}P,\ \alpha_2^{\dagger}\alpha_1^{\dagger}P,\ \alpha_3^{\dagger}\alpha_2^{\dagger}\alpha_1^{\dagger}P\,\bigr\} $$ is an eight-dimensional minimal left ideal on which $N$ takes the values $0,1,1,1,2,2,2,3$, so that the charge operator $$ Q=\tfrac13 N $$ takes the values $$ 0,\ \tfrac13,\ \tfrac13,\ \tfrac13,\ \tfrac23,\ \tfrac23,\ \tfrac23,\ 1 . $$ The intrinsic symmetry algebra on $S$ is $\mathrm{su}(3)\oplus\mathrm{u}(1)$; the $\mathrm{su}(3)$ is generated by the eight chains $\Lambda_1,\dots,\Lambda_8$ of the companion article, and the $\mathrm{u}(1)$ by $N$.

Proof. The construction is that of Complex Octonions and the Clifford Algebra Cl(6) carried out on the generators $e_1,\dots,e_6$ inside $\mathrm{Cl}(7)$, which satisfy the same relations. The rank of $P$, the independence of the eight elements, the eigenvalues of $N$ on them and the commutation of $\Lambda_1$ with $N$ were recomputed inside $\mathrm{Cl}(7)$ with zero residual; the $\mathrm{su}(3)$ structure constants and the weight and Casimir computations are the companion article's and are not repeated. $\square$

Because the $\alpha_i$, $P$, $N$ and $\Lambda_a$ involve only $e_1,\dots,e_6$, they commute with the seventh generator and hence with $J$; the ideal $S$ therefore lies inside a single summand, and its image under the parity automorphism is the corresponding ideal of the other summand. Explicitly, $$ [\Lambda_a,J]=0, \qquad [N,J]=0, \qquad [\Lambda_a,N]=0 $$ (all residuals zero), so the two summands carry the same $\mathrm{su}(3)\oplus\mathrm{u}(1)$ and the same charge spectrum, and differ only in chirality: the eigenvalue of $J$, equivalently the sign of $\Gamma$.

Summary

The split bioctonions are $\mathbb{D}\otimes_{\mathbb{R}}\mathbb{O}$, and the complex split bioctonions are $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{D}\otimes_{\mathbb{R}}\mathbb{O}\cong(\mathbb{C}\otimes_{\mathbb{R}}\mathbb{O})\oplus(\mathbb{C}\otimes_{\mathbb{R}}\mathbb{O})$. Their chain algebra, the algebra generated by the left multiplications, is the complex Clifford algebra $$ \mathrm{Cl}(7) \cong \mathrm{Cl}(6)\oplus\mathrm{Cl}(6) \cong M_8(\mathbb{C})\oplus M_8(\mathbb{C}), $$ of complex dimension $128$. Its volume element $J=e_1e_2e_3e_4e_5e_6e_7$ is central, self-adjoint and an involution, $J^2=1$, and through the idempotents $\Pi_\pm=\tfrac12(1\pm J)$ it splits the algebra into two copies of the complex octonionic chain algebra $\mathrm{Cl}(6)\cong M_8(\mathbb{C})$, each of complex dimension $64$. The seventh generator acts on the two copies as the $\mathrm{Cl}(6)$ volume element $\Gamma=e_1\cdots e_6$ with opposite signs, $e_7=-\Gamma J$. The parity automorphism $e_i\mapsto-e_i$ negates $J$ and interchanges the two copies, which are the two inequivalent pinor representations of $\mathrm{Cl}(7)$, distinguished by the eigenvalue of $J$; this is the doubling of the octonionic chain algebra into the two chiralities of a generation. Each copy carries the Furey ladder operators with $\{\alpha_i,\alpha_j^{\dagger}\}=\delta_{ij}$, an eight-dimensional minimal left ideal with charges $0,\tfrac13,\tfrac13,\tfrac13,\tfrac23,\tfrac23,\tfrac23,1$ under $Q=\tfrac13N$, and the same intrinsic $\mathrm{su}(3)\oplus\mathrm{u}(1)$, which commutes with $J$: the two copies differ only in chirality. The construction is the six-dimensional case of the general statement that the chain algebra of a split normed algebra is $\mathrm{Cl}(2n)\oplus\mathrm{Cl}(2n)\cong\mathrm{Cl}(2n+1)$, of which the quaternionic case is Complex Split Biquaternions and the Clifford Algebra Cl(3).

Summary of Notation

Further Reading

• V. Vaibhav and T. P. Singh, "Left-Right Symmetric Fermions and Sterile Neutrinos from Complex Split Biquaternions and Bioctonions," arXiv:2108.01858, for the construction of this article's algebra and its physics reading.
• C. Furey, "Standard model physics from an algebra?" (2016), arXiv:1611.09182, for the octonionic chain algebra, its ladder operators, its $\mathrm{su}(3)$ and its minimal left ideal, which the two summands reproduce.
• P. Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the classification $\mathrm{Cl}_7(\mathbb{C})\cong M_8(\mathbb{C})\oplus M_8(\mathbb{C})$, the centrality of the volume element of an odd complex Clifford algebra, and the two pinor representations.
• H. B. Lawson and M.-L. Michelsohn, Spin Geometry (Princeton, 1989), for the volume element, the chirality grading and the passage $\mathrm{Cl}(2n)\to\mathrm{Cl}(2n+1)$.
• J. C. Baez, "The octonions," Bulletin of the American Mathematical Society 39 (2002) 145–205, for the octonions, the complex octonions and the split octonions.
• Companion article Complex Octonions and the Clifford Algebra Cl(6), for the chain algebra, the ladder operators, the $\mathrm{su}(3)$, the idempotent and the charges of each summand, and the warning about the two bases.
• Companion article Complex Split Biquaternions and the Clifford Algebra Cl(3), for the same doubling in the quaternionic case.
• Companion article Maximal Totally Isotropic Subspaces and Their Unitary Symmetries, for the general ladder, number-operator and charge construction.