E11 (mathematics)

In mathematics, especially in Lie theory, E_''n'' is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and ''k'', with ''k'' = ''n'' − 4.

In some older books and papers, ''E''₂ and ''E''₄ are used as names for ''G''₂ and ''F''₄.

Finite-dimensional Lie algebras

The E_''n'' group is similar to the A_''n'' group, except the nth node is connected to the 3rd node. So the Cartan matrix appears similar, -1 above and below the diagonal, except for the last row and column, have −1 in the third row and column. The determinant of the Cartan matrix for E_''n'' is 9 − ''n''.

*E₃ is another name for the Lie algebra ''A''₁''A''₂ of dimension 11, with Cartan determinant 6.
*:\left \begin
2 & -1 & 0 \\
-1 & 2 & 0 \\
0 & 0 & 2
\end\right /math>
*E₄ is another name for the Lie algebra ''A''₄ of dimension 24, with Cartan determinant 5.
*:\left \begin
2 & -1 & 0 & 0 \\
-1 & 2 & -1& 0 \\
0 & -1 & 2 & -1 \\
0 & 0 & -1 & 2
\end\right /math>
*E₅ is another name for the Lie algebra ''D''₅ of dimension 45, with Cartan determinant 4.
*:\left \begin
2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 \\
0 & -1 & 2 & -1 & -1 \\
0 & 0 & -1 & 2 & 0 \\
0 & 0 & -1 & 0 & 2
\end\right /math>
* E₆ is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
*:$\left [ \begin 2 & -1 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 2 \end\right ]$
*E7 (mathematics), E₇ is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
*:$\left [ \begin 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 2 \end\right ]$
* E8 (mathematics), E₈ is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
*:$\left [ \begin 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 \end\right ]$

Infinite-dimensional Lie algebras

*E₉ is another name for the infinite-dimensional affine Lie algebra $_8$ (also as E₈⁺ or E₈⁽¹⁾ as a (one-node) extended E₈) (or E8 lattice) corresponding to the Lie algebra of type E8 (mathematics), E₈. E₉ has a Cartan matrix with determinant 0.
*:$\left [ \begin 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 2 \end\right ]$
* E₁₀ (or E₈⁺⁺ or E₈^{(1)^} as a (two-node) over-extended E₈) is an infinite-dimensional Kac–Moody algebra whose root lattice is the even Lorentzian unimodular lattice II_9,1 of dimension 10. Some of its root multiplicities have been calculated; for small roots the multiplicities seem to be well behaved, but for larger roots the observed patterns break down. E₁₀ has a Cartan matrix with determinant −1:
*:$\left [ \begin 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \end\right ]$
*E₁₁ (or E₈⁺⁺⁺ as a (three-node) very-extended E₈) is a Lorentzian algebra, containing one time-like imaginary dimension, that has been conjectured to generate the symmetry "group" of M-theory.
*E_''n'' for ''n''≥12 is an infinite-dimensional Kac–Moody algebra that has not been studied much.

Root lattice

The root lattice of E_''n'' has determinant 9 − ''n'', and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Z_''n'',1 that are orthogonal to the vector (1,1,1,1,...,1, 3) of norm ''n'' × 1² − 3² = ''n'' − 9.

E7½

Landsberg and Manivel extended the definition of E_''n'' for integer ''n'' to include the case ''n'' = 7. They did this in order to fill the "hole" in dimension formulae for representations of the E_''n'' series which was observed by Cvitanovic, Deligne, Cohen and de Man. E₇ has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.

See also

* k₂₁, 2_k1, 1_k2 polytopes based on E_n Lie algebras.

References

*

Further reading

* Class. Quantum Grav. 18 (2001) 4443-4460
* Guersey Memorial Conference Proceedings '94
*{{cite journal, first1=J. M., last1=Landsberg, first2=L., last2=Manivel, title=The sextonions and E_7½, journal=Advances in Mathematics, year=2006, volume=201, issue=1, pages=143–179, arxiv=math.RT/0402157, doi=10.1016/j.aim.2005.02.001, doi-access=free
* ''Connections between Kac-Moody algebras and M-theory'', Paul P. Cook, 200

* ''A class of Lorentzian Kac-Moody algebras'', Matthias R. Gaberdiel, David I. Olive and Peter C. West, 200


Lie groups