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Albert algebra

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Context

Exceptional structures

Algebra

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Definition

The octonionic Albert algebra is the Jordan algebra of 33-by-33 hermitian matrices over the octonions 𝕆\mathbb{O}

(1)𝔥 3(𝕆){((x 0+x 1) y ψ 1 y * (x 0x 1) ψ 2 ψ 1 * ψ 2 * ϕ)|x 0,x 1,ϕ𝕆 ψ 1,ψ 2𝕆} \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} \\ \psi_1, \psi_2 \in \mathbb{O} } \right\}

Similarly the split-octonionic Albert algebra is the algebra of 33-by-33 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 3×33 \times 3-hermitian matrix in (1) makes it manifest that the exceptional Jordan algebra is naturally a linear direct sum of the form

𝔥 3(𝕆) 𝔥 2(𝕆)𝕆 2 \mathfrak{h}_3(\mathbb{O}) \;\simeq_{\mathbb{R}}\; \mathfrak{h}_2(\mathbb{O}) \oplus \mathbb{O}^2 \oplus \mathbb{R}

via

{((x 0+x 1) y ψ 1 y * (x 0x 1) ψ 2 ψ 1 * ψ 2 * ϕ)}𝔥 3(𝕆){((x 0+x 1) y 0 y * (x 0x 1) 0 0 0 0)}𝔥 2(𝕆){(0 0 ψ 1 0 0 ψ 2 ψ 1 * ψ 2 * 0)}𝕆 2{(0 0 0 0 0 0 0 0 ϕ)} \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

x 0,x 1,ϕ𝕆 ψ 1,ψ 2𝕆 \array{ x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O} \\ \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:

𝔥 3(𝕆) 9,116dim =26. \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, ϕ\phi \in \mathbb{R} looks like the size of S 1/( 2)S^1/(\mathbb{Z}_2) in Horava-Witten theory.

This decomposition hence induces an action of the spin group Spin(9,1)Spin(9,1) on the exceptional Jordan algebra. While only the subgroup Spin(9)Spin(9,1)Spin(9) \hookrightarrow Spin(9,1) of that is an isomorphism of the Jordan algebra-structure itself, the full Spin(9,1)Spin(9,1)-action does preserve the determinant on 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}).

Automorphisms and exceptional Lie groups

Proposition

(general linear group of Mat 3×3 herm(𝕆)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){}_{(-26)}.

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

Proposition

(Jordan algebra automorphism group of Mat 3×3 herm(𝕆)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(Mat 3×3 herm(𝕆),)F 4. 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 3×3 herm(𝕆)Mat_{3 \times 3}^{herm}(\mathbb{O}) fixing an imaginary octonion)

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

𝕆 VAAwithAAV 3, \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)Mat 3×3 herm(𝕆) w Mat 3×3 herm(𝕆) \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 VV.

Then the subgroup of the Jordan algebra automorphism group Aut(Mat 3×3 herm(𝕆),)F 4Aut\left(Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4 (Prop. ) of elements that commute with ww (2)

F 4 w{αF 4|wα=αw} F_4^w \;\coloneqq\; \left\{ \alpha \in F_4 \;\vert\; w \alpha = \alpha w \right\}

is

F 4 w(SU(3)×SU(3))/ 3, 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)SU(3)×SU(3) (A, B) \in SU(3) \times SU(3)

acts on an element

XMat 3×3 herm(𝕆)X Mat 3×3 herm()+X VMat 3×3() \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 +X 𝕍AX A +BX VA 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

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

which is generated by the pair of diagonal matrices

(4)(e 2πi131 3,e 2πi131 3)/SU(3)×SU(3). \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(SU(3)×SU(3))/ 3F 4 F_4^w \simeq \big( SU(3) \times SU(3) \big) / \mathbb{Z}_3 \;\subset\; F_4 (Prop. ) which fixes a subspace

Mat 2×2 herm()Mat 3×3 herm()Mat 3×3(V)Mat 3×3 herm(𝕆) 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

(U(1)×SU(2)×SU(3))/ 6, \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

(exp(2πi16),exp(2πi12)1 2,exp(2πi13)1 3)U(1)×SU(2)×SU(3). \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.

Proof

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

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

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

(5)U(1)×SU(2) SU(3) (c,σ) diag(c 2,c 1σ) \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){(1,1 2),(e 2πi12,e 2πi121 2)} 2. \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

((U(1)×SU(2))/ 2×SU(3))/ 3 (((U(1)×SU(2))×SU(3))/ 2)/ 3 ((U(1)×SU(2))×SU(3))/ 6, \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 2\mathbb{Z}_2-action as the trivial action on the SU(3)SU(3)-factor, and in the second step we used the evident isomorphism 2× 3 6\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 6\mathbb{Z}_6 is as claimed. By the above identification 6 2× 3\mathbb{Z}_6 \simeq \mathbb{Z}_2 \times \mathbb{Z}_3, it is generated by the joint action of that of the generators of 3\mathbb{Z}_3 and of 2\mathbb{Z}_2, which, by (4) and (6), is

(e 2πi13,e 2πi131 2,e 2πi131 3)generator of 3(1,(e 2πi12)(e 2πi121 2),1 3)generator of 2=(e 2πi13,e 2πi12e 2πi13=e 2πi16(e 2πi121 2),e 2πi131 3) \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)×SU(3)SU(3) \times SU(3), hence is

(e 2πi16,e 2πi121 2,e 2πi131 3,)U(1)×SU(2)×SU(3) \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).

References

The original article is

A textbook account is in

Further discussion:

See also

Last revised on August 28, 2019 at 04:07:27. See the history of this page for a list of all contributions to it.