En (Lie algebra)

En (Lie algebra)

In mathematics, especially in Lie theory, E"n" is the Kac–Moody algebra whose Dynkin diagram is a line of "n"-1 points with an extra point attached to the third point from the end.

Finite dimensional Lie algebras

*E3 is another name for the Lie algebra "A"1"A"2 of dimension 11.
*E4 is another name for the Lie algebra "A"4 of dimension 24.
*E5 is another name for the Lie algebra "D"5 of dimension 45.
*E6 is the exceptional Lie algebra of dimension 78.
*E7 is the exceptional Lie algebra of dimension 133.
*E8 is the exceptional Lie algebra of dimension 248.

Infinite dimensional Lie algebras

*E9 is another name for the infinite dimensional affine Lie algebra E8(1) (or E8 lattice) corresponding to the Lie algebra of type E8
*E10 is an infinite dimensional Kac–Moody algebra whose root lattice is the even Lorentzian unimodular lattice II9,1 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.
*E11 is an infinite dimensional Kac–Moody algebra 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" = 7½. 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. E7½ has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.

See also

* Semiregular k21 polytope - Set of Semiregular polytopes based on En Lie algebras.

References

*cite arXiv|author=R.W. Gebert, H. Nicolai|title=E10 for beginners|year=1994|version=|eprint=hep-th/9411188 Guersey Memorial Conference Proceedings '94
*Citation | last1=Kac | first1=Victor G | last2=Moody | first2=R. V. | last3=Wakimoto | first3=M. | title=Differential geometrical methods in theoretical physics (Como, 1987) | publisher=Kluwer Acad. Publ. | location=Dordrecht | series=NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. | id=MathSciNet | id = 981374 | year=1988 | volume=250 | chapter=On E10 | pages=109–128
*cite arXiv|author=P.C. West|title=E11 and M Theory|year=2001|version=|eprint=hep-th/0104081 Class.Quant.Grav. 18 (2001) 4443-4460
* Landsberg, J. M. Manivel, L. [http://arxiv.org/abs/math.RT/0402157" The sextonions and E7½"] . Adv. Math. 201 (2006), no. 1, 143-179.


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