nLab Albert algebra

Definition

The octonionic Albert algebra is the Jordan algebra of $3$-by-$3$ hermitian matrices over the octonions $\mathbb{O}$

(1)$$\mathfrak{h}_3(\mathbb{O}) \;\coloneqq\; \left\{ \left( \array{ (x_0 + x_1) & y & \psi_1 \\ y^\ast & (x_0 - x_1) & \psi_2 \\ \psi_1^\ast & \psi_2^\ast & \phi } \right) \;|\; \array{ x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O} \\ y, \psi_1, \psi_2 \in \mathbb{O} } \right\}$$

Similarly the split-octonionic Albert algebra is the algebra of $3$-by-$3$ hermitian matrices over the split-octonions.

The construction is due to (Albert 1934), originating in an algebraic approach to quantum mechanics.

Properties

Uniqueness

The octonionic and split-octonionic Albert algebras are (up to isomorphism) the only simple finite-dimensional formally real Jordan algebras over the real numbers that are not special, together comprising the real Albert algebras.

Their complexifications are isomorphic, the complex-octonionic Albert algebra, or simply the complex Albert algebra. Analogues exist over any field.

An exceptional Jordan algebra (over any field) is any Jordan algebra in which an Albert algebra appears as a direct summand. Every formally real Jordan algebra over the real numbers is either special or exceptional (so they all have excellent self-esteem). The exceptional Jordan algebras are related to the exceptional Lie algebras.

Relation to 10d super-Spacetime

The form of the $3 \times 3$-hermitian matrix in (1) makes it manifest that the exceptional Jordan algebra is naturally a linear direct sum of the form

$$\mathfrak{h}_3(\mathbb{O}) \;\simeq_{\mathbb{R}}\; \mathfrak{h}_2(\mathbb{O}) \oplus \mathbb{O}^2 \oplus \mathbb{R}$$

via

$$\underset{ \mathfrak{h}_3(\mathbb{O}) }{ \underbrace{ \left\{ \left( \array{ (x_0 + x_1) & y & \psi_1 \\ y^\ast & (x_0 - x_1) & \psi_2 \\ \psi_1^\ast & \psi_2^\ast & \phi } \right) \right\} }} \;\simeq\; \underset{ \mathfrak{h}_2(\mathbb{O}) }{ \underbrace{ \left\{ \left( \array{ (x_0 + x_1) & y & 0 \\ y^\ast & (x_0 - x_1) & 0 \\ 0 & 0 & 0 } \right) \right\} } } \oplus \underset{ \mathbb{O}^2 }{ \underbrace{ \left\{ \left( \array{ 0 & 0 & \psi_1 \\ 0 & 0 & \psi_2 \\ \psi_1^\ast & \psi_2^\ast & 0 } \right) \right\} } } \oplus \underset{ \mathbb{R} }{ \underbrace{ \left\{ \left( \array{ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \phi } \right) \right\} } }$$

with

$$\array{ x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O} \\ y, \psi_1, \psi_2 \in \mathbb{O} }$$

By the discussion at geometry of physics – supersymmetry in the section Real spinors in dimension 3,4,6,10 these summands may be further identified as follows:

• $\mathfrak{h}_2(\mathbb{O}) \simeq \mathbb{R}^{9,1}$ is the incarnation of 10-dimensional super-Minkowski spacetime via octonionic Pauli matrices;

• under this identification $\mathbb{O}^2 \simeq \mathbf{16}$ is the Majorana-Weyl spinor real spin representation of the spin group $Spin(9,1)$.

$$\mathfrak{h}_3(\mathbb{O}) \; \simeq_{\mathbb{R}} \; \underset{ dim_{\mathbb{R}} = 26 }{ \underbrace{ \mathbb{R}^{9,1} \oplus \mathbf{16} }} \oplus \mathbb{R} \,.$$

Under these identifications, $\phi \in \mathbb{R}$ looks like the size of $S^1/(\mathbb{Z}_2)$ in Horava-Witten theory.

This decomposition hence induces an action of the spin group $Spin(9,1)$ on the exceptional Jordan algebra. While only the subgroup $Spin(9) \hookrightarrow Spin(9,1)$ of that is an isomorphism of the Jordan algebra-structure itself, the full $Spin(9,1)$-action does preserve the determinant on $\mathfrak{h}_3(\mathbb{O})$.

Automorphisms and exceptional Lie groups

Proposition

(general linear group of $Mat_{3\times 3}^{herm}(\mathbb{O})$ preserving determinant is E6)

The group of determinant-preserving linear isomorphisms of the vector space underlying the octonionic Albert algebra is the exceptional Lie group E6${}_{(-26)}$.

(see e.g. (Manogue-Dray 09)).

Proposition

(Jordan algebra automorphism group of $Mat_{3\times 3}^{herm}(\mathbb{O})$ is F4)

The group of automorphism with respect to the Jordan algebra structure $\circ$ on the octonionic Albert algebra is the exceptional Lie group F4:

$$Aut\left( Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4 \,.$$

(e.g. Yokota 09, section 2.2)

Proposition

(Jordan algebra automorphism group of $Mat_{3 \times 3}^{herm}(\mathbb{O})$ fixing an imaginary octonion)

Fix an imaginary octonion $i \in \mathbb{O}$, hence a $\mathbb{R}$-linear direct sum decomposition

$$\mathbb{O} \;\simeq_{\mathbb{R}}\; \mathbb{C} \oplus V \phantom{AA}\text{with}\phantom{AA} V \simeq_{\mathbb{R}} \mathbb{C}^3 \,,$$

and let

(2)$$\array{ Mat_{3 \times 3}^{herm}(\mathbb{O}) &\overset{w}{\longrightarrow}& Mat_{3 \times 3}^{herm}(\mathbb{O}) }$$

be given componentwise by the identity on $\mathbb{C}$ and by multiplication with some fixed non-vanishing number on $V$.

Then the subgroup of the Jordan algebra automorphism group $Aut\left(Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4$ (Prop. ) of elements that commute with $w$ (2)

$$F_4^w \;\coloneqq\; \left\{ \alpha \in F_4 \;\vert\; w \alpha = \alpha w \right\}$$

is

$$F_4^w \;\simeq\; \big( SU(3) \times SU(3) \big)/ \mathbb{Z}_3 \,,$$

where every element in the direct product group of SU(3) with itself

$$(A, B) \in SU(3) \times SU(3)$$

acts on an element

$$\underset{ \in Mat_{3\times 3}^{herm}(\mathbb{O}) }{\underbrace{\;X\;}} \;\simeq\; \underset{ \in Mat_{3\times 3}^{herm}(\mathbb{C}) }{\underbrace{\;X_{\mathbb{C}}\;}} \;+\; \underset{ \in Mat_{3 \times 3}(\mathbb{C}) }{\underbrace{X_{V}}}$$

via matrix multiplication as

(3)$$X_{\mathbb{C}} + X_{\mathbb{V}} \;\mapsto\; A X_{\mathbb{C}} A^\dagger \;+\; B X_{V} A^\dagger$$

(with $(-)^\dagger$ being the conjugate transpose matrix, hence the inverse matrix for the unitary matrices under consideration) and where the quotient is by the cyclic subgroup

$$\mathbb{Z}_3 \;\subset\; SU(3) \times SU(3)$$

which is generated by the pair of diagonal matrices

(4)$$\left( e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3 \;,\; e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3 \right)/ \;\in \; SU(3) \times SU(3) \,.$$

(Yokota 09, theorem 2.12.2)

Proposition

The further subgroup of $F_4^w \simeq \big( SU(3) \times SU(3) \big) / \mathbb{Z}_3 \;\subset\; F_4$ (Prop. ) which fixes a subspace

$$Mat_{2 \times 2}^{herm}(\mathbb{C}) \;\subset\; \underset{ Mat_{3 \times 3}^{herm}( \mathbb{O} ) }{ \underbrace{ Mat_{3 \times 3}^{herm}(\mathbb{C}) \;\oplus\; Mat_{3 \times 3}(V) }}$$

(hence, by the above, a 4d Minkowski spacetime (incarnated via its Pauli matrices) inside the 10d Minkowski spacetime inside the octonionic Albert algebra)

is

$$\big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6 \,,$$

where the quotient is by the cyclic subgroup which is generated by the element

$$\left( \exp\left(2 \pi i \tfrac{1}{6}\right)\;,\; \exp\left(2 \pi i \tfrac{1}{2}\right) \mathbf{1}_2\;,\; \exp\left(2 \pi i \tfrac{1}{3}\right) \mathbf{1}_3 \right) \;\in\; U(1) \times SU(2) \times SU(3) \,.$$

(Hence this group happens to coincide with the exact gauge group of the standard model of particle physics, see there).

This was claimed without proof in Dubois-Violette & Todorov 18. See also Krasnov 19.

Proof

By Prop. (3) it is clear that the subgroup in question is that represented by those pairs $(A,B) \in SU(3) \times SU(3)$ for which $A$ is $(1 + 2)$-block diagonal. Such matrices $A$ form the subgroup of SU(3) of matrices that may be written in the form

$$diag\left( c^2, c^{-1} \mathbf{\sigma} \right)$$

for $c \in U(1)$ and $\mathbf{\sigma} \in$ SU(2). The kernel of the group homomorphism

(5)$$\array{ U(1) \times SU(2) &\longrightarrow& SU(3) \\ (c,\mathbf{\sigma}) &\mapsto& diag\left( c^{2}, c^{-1} \mathbf{\sigma} \right) }$$

is clearly the cyclic group

(6)$$\left\{ (1,\mathbf{1}_2)\;,\; \left( e^{2\pi i \tfrac{1}{2}},e^{2\pi i \tfrac{1}{2}}\mathbf{1}_2 \right) \right\} \;\simeq\; \mathbb{Z}_2 \,.$$

Hence the subgroup in question is

$$\begin{aligned} \Big( \big( U(1) \times SU(2) \big)/ \mathbb{Z}_2 \;\times\; SU(3) \Big)/ \mathbb{Z}_3 & \simeq \Big( \Big( \big( U(1) \times SU(2) \big) \;\times\; SU(3) \Big) / \mathbb{Z}_2 \Big) / \mathbb{Z}_3 \\ &\simeq \Big( \big( U(1) \times SU(2) \big) \;\times\; SU(3) \Big) / \mathbb{Z}_6 \,, \end{aligned}$$

where in the first step we extended the $\mathbb{Z}_2$-action as the trivial action on the $SU(3)$-factor, and in the second step we used the evident isomorphism $\mathbb{Z}_2 \times \mathbb{Z}_3 \simeq \mathbb{Z}_6$ (an application of the “fundamental theorem of cyclic groups”, if you wish).

It remains to see that the action of $\mathbb{Z}_6$ is as claimed. By the above identification $\mathbb{Z}_6 \simeq \mathbb{Z}_2 \times \mathbb{Z}_3$, it is generated by the joint action of that of the generators of $\mathbb{Z}_3$ and of $\mathbb{Z}_2$, which, by (4) and (6), is

$$\underset{ \text{generator of}\, \mathbb{Z}_3 }{ \underbrace{ \Big( e^{2\pi i \tfrac{1}{3}} \;,\; e^{2\pi i \tfrac{1}{3}} \mathbf{1}_2\;,\; e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3 \Big) } } \underset{ \text{generator of} \, \mathbb{Z}_2 }{ \underbrace{ \Big( 1 , (e^{2 \pi i \tfrac{1}{2}}) (e^{2 \pi i \tfrac{1}{2}}\mathbf{1}_2), \mathbf{1}_3 \Big) } } \;=\; \left( e^{2\pi i \tfrac{1}{3}} \;,\; \underset{ = e^{2\pi i \tfrac{-1}{6}} }{ \underbrace{ e^{2\pi i \tfrac{1}{2}} e^{2\pi i \tfrac{1}{3}} }} \; ( e^{2 \pi i \tfrac{1}{2}} \mathbf{1}_2) \;,\; e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3 \right)$$

as an element in $SU(3) \times SU(3)$, hence is

$$\Big( e^{2\pi i \tfrac{1}{6}} \;,\; e^{2\pi i \tfrac{1}{2}}\mathbf{1}_2 \;,\; e^{2\pi i \tfrac{1}{3}}\mathbf{1}_3 \;,\; \Big) \;\in\; U(1) \times SU(2) \times SU(3)$$

under the lift through (5).

Related concepts

• Kantor-Koecher-Tits construction$\rightarrow$ E7

References

General

The original article is

• Abraham Adrian Albert, On a Certain Algebra of Quantum Mechanics, Annals of Mathematics, Second Series 35 (1): 65–73, (1934)(doi:10.2307/1968118, JSTOR).

A textbook account is in

• Tonny Springer, Ferdinand Veldkamp, chapter 5 of Octonions, Jordan Algebras, and Exceptional Groups, Springer Monographs in Mathematics, 2000

Further discussion:

• John Baez, section 3.4 $\mathbb{O}P^2$ and the Exceptional Jordan Algebra of The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. (web)

• Ichiro Yokota, Exceptional Lie groups (arXiv:0902.0431)

See also

• Wikipedia, Albert algebra

Possible relation to color gauge structure

Attempts to identify aspects of the color gauge group of the standard model of particle physics within the exceptional Jordan algebra:

• Corinne Manogue, Tevian Dray, Octonions, $E_6$, and Particle Physics, J. Phys. Conf.Ser.254:012005,2010 (arXiv:0911.2253)

• Michel Dubois-Violette, Ivan Todorov, Exceptional quantum geometry and particle physics II (arXiv:1808.08110)

• Ivan Todorov, Exceptional quantum algebra for the standard model of particle physics (arXiv:1911.13124)

• Kirill Krasnov, $SO(9)$ characterisation of the Standard Model gauge group (arXiv:1912.11282)

• Latham Boyle, The Standard Model, The Exceptional Jordan Algebra, and Triality (arXiv:2006.16265)