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. E 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 E"] . Adv. Math. 201 (2006), no. 1, 143-179.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Lie algebra representation — Lie groups …   Wikipedia

  • Lie-Algebra —   [nach M. S. Lie], liesche Algebra, ein hyperkomplexes System (Algebra) LR, für das die Multiplikation nicht assoziativ ist. Stattdessen sollen folgende Gesetzmäßigkeiten gelten (das Produkt von …   Universal-Lexikon

  • Lie algebra — In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term… …   Wikipedia

  • Lie-Algebra — Eine Lie Algebra, benannt nach Sophus Lie, ist eine algebraische Struktur, die hauptsächlich zum Studium geometrischer Objekte wie Lie Gruppen und differenzierbarer Mannigfaltigkeiten eingesetzt wird. Inhaltsverzeichnis 1 Definition 2 Beispiele 2 …   Deutsch Wikipedia

  • Lie algebra cohomology — In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by Chevalley and Eilenberg (1948) in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie …   Wikipedia

  • Lie algebra bundle — In Mathematics, a weak Lie algebra bundle : xi=(xi, p, X, heta), is a vector bundle xi, over a base space X together with a morphism : heta : xi oplus xi ightarrow xi which induces a Lie algebra structure on each fibre xi x, .A Lie algebra bundle …   Wikipedia

  • Lie-Algebra sl — In der Mathematik ist die Lie Algebra sl(2,C) der Prototyp einer einfachen Lie Algebra. Die sl(2,C) ist eine dreidimensionale, komplexe, einfache Lie Algebra. Durch diese Eigenschaften ist sie als Lie Algebra bereits eindeutig identifiziert. Bei… …   Deutsch Wikipedia

  • Lie-Algebra sl(2,C) — In der Mathematik ist die Lie Algebra der Prototyp einer einfachen Lie Algebra. Die ist eine dreidimensionale, komplexe, einfache Lie Algebra. Durch diese Eigenschaften ist sie als Lie Algebra bereits eindeutig identifiziert. Die ist die… …   Deutsch Wikipedia

  • lie algebra — ¦lē noun Usage: usually capitalized L Etymology: after Sophus Lie died 1899 Norwegian mathematician : a linear algebra which has the multiplicative operation denoted by [, ] and is bilinear such that [aA + bB,C] = a[A,C] + b[B,C] and [A,aB + bC] …   Useful english dictionary

  • Abelsche Lie-Algebra — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia

  • Auflösbare Lie-Algebra — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”