The exceptional Jordan algebra | Complex Projective 4-Space

In the punctual 1930s, Pascual Jordan finisheavored to establishalise the algebraic properties of Hermitian matrices. In particular:

Now, this symmetascfinishd product is commutative by definition, and is also (bi)licforfeit: . What other algebraic properties must this product prent? The beginant ones are:

Power-associativity: the transmition does not depfinish on the parenthesisation.
Formal fact: a sum of squares is zero if and only if all of the summands are zero.

The second of these conditions uncomardents that we can say that an element of the Jordan algebra is ‘nonadverse’ if it can be transmited as a sum of squares. (In the understandn context of authentic symmetric matrices, this coincides with the property of the matrix being likeable-semidefinite.) The nonadverse elements establish a ‘cone’ seald under multiplication by likeable authentic scalars and compriseition.

Jordan, von Neumann, and Wigner carry oned to sort all of the finite-stupidensional algebras of this establish (understandn as establishassociate authentic Jordan algebras). They showed that every such algebra is a honest sum of ‘basic’ algebras, each of which is isomorphic to [at least] one of the folloprosperg:

the authentic symmetric matrices of stupidension n (for any likeable integer n) with the aforealludeed symmetascfinishd product;
the complicated Hermitian matrices of stupidension n;
the quaternionic Hermitian matrices of stupidension n;
the octonionic Hermitian matrices of stupidension n (where n ≤ 3);
the algebras with the product , understandn as ‘spin factors’. As John Baez alludes, these can be identified with Minkowski space, and the nonadverse elements are exactly the ‘future cone’ of the origin.

The qualification ‘at least’ is becaemploy there are some isomorphisms here:

Simple establishassociate authentic Jordan algebras, shoprosperg the four infinite families and the exceptional Jordan algebra

Exactly one of these basic establishassociate authentic Jordan algebras flunks to fit into any of the four infinite families. This exceptional Jordan algebra is , the 3-by-3 self-adunitet octonionic matrices finishowed with the symmetascfinishd product. Viewed as a authentic vector space, it is 27-stupidensional: an arbitrary element can be portrayd distinctly by determineing the three diagonal elements (which must be authentic) and three drop off-diagonal elements (which can be arbitrary octonions); the three upper off-diagonal elements are then determined.

Projective spaces from Jordan algebras

Given a establishassociate authentic Jordan algebra, we can ponder the idempotent elements prenting . For the Jordan algebras built from n-by-n authentic, complicated, or quaternionic matrices, these are the matrices with eigenappreciates 0 and 1.

We get a fragmentary order on these ‘projection’ matrices: A ‘comprises’ B if and only if . This partiassociate-ordered set can be identified with the stratified collection of subspaces in the (n−1)-stupidensional projective space over the base field:

the zero matrix correacts to the vacant space;
the rank-1 projection matrices correact to points;
the rank-2 projection matrices correact to lines;
…
the rank-(n−1) projection matrices correact to hyperset upes;
the identity matrix correacts to the filled projective space itself.

The exceptional Jordan algebra donates us the octonionic projective set upe, finded by Ruth Moufang. We can’t get any higher-stupidensional octonionic projective spaces becaemploy Desdisputes’ theorem is deceptive in the octonionic projective set upe, whereas it’s genuine in any set upe that can be embedded in a 3-stupidensional projective space. We alludeed this seven years ago.

This hints at why 4-by-4 and higher octonionic matrices have no hope of establishing a establishassociate authentic Jordan algebra: we’d be able to clarify an octonionic projective 3-space, which is impossible.

What about the spin factors? The idempotents in are:

the zero element (0, 0), correacting to the ’vacant space’;
the identity element (0, 1), correacting to the ‘filled space’;
the points (x, ½) where x is an arbitrary vector of length ½.

In other words, these correact to spheres! Recall that the authentic, complicated, quaternionic, and octonionic projective lines are (as toporeasonable manifageders) equitable the 1-, 2-, 4-, and 8-spheres, esteemively; we can leank of the spin factors as ‘projective lines’ built from arbitrary-stupidensional spheres.

As for non-basic establishassociate authentic Jordan algebras, the correacting ‘projective spaces’ are equitable Cartesian products of the ‘projective spaces’ correacting to the honest summands.

An exotic spacetime

In August of this year, Blake Stacey posted the folloprosperg comment on John Baez’s essay:

Now for some context: it is possible to clarify the determinant of a 3-by-3 octonionic Hermitian matrix, and the group of licforfeit operators (seeing as a 27-stupidensional authentic vector space) that carry ons the determinant is a noncompact authentic establish of the Lie group E6.

This group E6 is transitive on the likeable-definite matrices of determinant 1. The subgroup mending any one of these (without loss of vagueity, the identity matrix) is the compact authentic Lie group F4, which also carry ons the Jordan product. This uncomardents that it maps idempotents to idempotents, so can be seen as acting on the octonionic projective set upe as its group of projective alterations.

This group F4 is transitive on the rank-1 idempotent matrices, and the subgroup mending any one of these is Spin(9). (As a result, we can portray the octonionic projective set upe as the quotient F4 / Spin(9). Elie Cartan showd that all compact Riemannian symmetric spaces are quotients of compact Lie groups.)

What’s the analogy for understandn (3+1)-stupidensional Minkowski spacetime?

the filled group (analogous to E6) is the proper orthochronous Lorentz group;
the subgroup mending a timeappreciate vector (analogous to F4) is the rotation group SO(3);
the subgroup compriseitionassociate mending a weightlessappreciate vector (analogous to Spin(9)) is the rotation group SO(2);
the symmetric space (analogous to the octonionic projective set upe) is the quotient SO(3) / SO(2), which is equitable the understandn 2-sphere.

A lattice in this exotic spacetime

It is authentic to ponder the ‘integer points’ in this spacetime, namely the octonionic Hermitian matrices where the off-diagonal elements are Cayley integers and the diagonal elements are frequent integers. John Baez alludes that this is the distinct integral unimodular lattice in (26+1)-stupidensional spacetime, and it can be seen as the honest sum of the exceptional Lorentzian lattice with a duplicate of the integers.

This lattice was thocimpolitely dispenseigated in a marvellous paper by Noam Elkies and Benedict Gross. Possibly the most unanticipateed findy in this paper is that whilst E6 acts transitively on the likeable-definite matrices of determinant 1, this no extfinisheder hageders when you ‘discretise’! More exactly, the likeable-definite ‘integer points’ of determinant 1 establish two distinct orbits under the discrete subgroup of E6 that carry ons the lattice.

One of these orbits comprises the identity matrix; the other comprises the circulant matrix with elements {2, η, η*} where . [Note: there’s a 6-dimensional sphere of octonionic square-roots of -7. You’ll need to choose one that results in η being a Cayley integer.] If you employ this other matrix E as your quadratic establish instead of the identity matrix I, this directs to a very authentic erection of the Leech lattice.

Specificassociate, as shown in the Elkies-Gross paper, triples of Cayley integers with the norm establish an isometric duplicate of the Leech lattice! By contrast, the common inner product using the identity matrix as the quadratic establish donates the honest sum — aacquire an even unimodular lattice in 24 stupidensions, but not as exceptional or pretty or effective as the Leech lattice.