E8 (mathematics)

Therefore, the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article.

There are two non-isomorphic irreducible representations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000 (sequence A181746 in the OEIS)).

The most difficult case (for exceptional groups) is the split real form of E8 (see above), where the largest matrix is of size 453060×453060.

The Lusztig–Vogan polynomials for all other exceptional simple groups have been known for some time; the calculation for the split form of E8 is far longer than any other case.

The announcement of the result in March 2007 received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.

This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits (Landsberg & Manivel 2001).

Explicitly, there are 112 roots with integer entries obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots with half-integer entries obtained from by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).

Specifically, the entries of the Cartan matrix are given by where ( , ) is the Euclidean inner product and αi are the simple roots.

The 120 roots of negative height relative to the same set of simple roots can be adequately represented by a second copy of the Hasse diagram with the arrows reversed; but it is less straightforward to connect these two diagrams via a basis for the eight-dimensional Cartan subalgebra.

and can have only one upward arrow, connected to a node in the height 0 layer representing the element of the Cartan subalgebra given by

— or any eight linearly independent, mutually commuting Lie derivations on any manifold with E8 structure — would have served just as well.

can be represented schematically as circles and arrows, but this simply breaks down on the chosen Cartan subalgebra.

It transforms under E6×SU(3) as a sum of tensor product representations, which may be labelled as a pair of dimensions as (78,1) + (1,8) + (27,3) + (27,3).

In this description, The 248-dimensional adjoint representation of E8, when similarly restricted to the second maximal subgroup, transforms under E7×SU(2) as: (133,1) + (1,3) + (56,2).

A complete exposition of this construction may be found in standard texts on Jordan algebras such as Jacobson 1968 or McCrimmon 2004.

What matters is that the kernel of the Lie bracket with the generator chosen as the "grade operator" be an

For details on how this asymmetric structure works starting from a general 3-graded algebra, see references at Frölicher–Nijenhuis bracket.

The relevance of this observation to E8 is simply that E7 and E8 are their own clusters of structures, distinguished as exceptional simple Lie groups/algebras, and that any particular reconstruction of them using representations of their subgroups/subalgebras will have extensions beyond the motivating case.

Varying conventions of sign, scale, and conjugate relationship in the literature are due not just to inaccuracies but also to the directions in which the authors seek to extend their constructions.)

), each subspace may be given a quite particular non-associative (nor even power-associative) product operation, resulting in two copies of Brown's 56-dimensional structurable algebra.

See Distler and Garibaldi 2009 for discussion of the mathematical obstacles to constructing a chiral gauge theory based on E8.

The same may be said of connections to Jordan and Heisenberg algebras, whose historical origins are intertwined with the development of quantum mechanics.

The finite quasisimple groups that can embed in (the compact form of) E8 were found by Griess & Ryba (1999).

The E8 Lie group has applications in theoretical physics and especially in string theory and supergravity.

One way to incorporate the standard model of particle physics into heterotic string theory is the symmetry breaking of E8 to its maximal subalgebra SU(3)×E6.

In 1982, Michael Freedman used the E8 lattice to construct an example of a topological 4-manifold, the E8 manifold, which has no smooth structure.

Antony Garrett Lisi's incomplete "An Exceptionally Simple Theory of Everything" attempts to describe all known fundamental interactions in physics as part of the E8 Lie algebra.

[7][8] R. Coldea, D. A. Tennant, and E. M. Wheeler et al. (2010) reported an experiment where the electron spins of a cobalt-niobium crystal exhibited, under certain conditions, two of the eight peaks related to E8 that were predicted by Zamolodchikov (1989).

Cartan determined that a complex simple Lie algebra of type E8 admits three real forms.

Chevalley (1955) introduced algebraic groups and Lie algebras of type E8 over other fields: for example, in the case of finite fields they lead to an infinite family of finite simple groups of Lie type.