In mathematics, 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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a Kac-Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. ... See also Simple Lie group. ...

E_7½ is a name for a certain Lie algebra of dimension 190.

Examples

• E₃ is another name for the Lie algebra A₁A₂ of dimension 11.
• E₄ is another name for the Lie algebra A₄ of dimension 24.
• E₅ is another name for the Lie algebra D₅ 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.
• E₉ is another name for the infinite dimensional affine Lie algebra E₈⁽¹⁾ 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 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.
• E_n for n≥11 is an infinite dimensional Kac-Moody algebra that has not been studied much.

In mathematics, E6 is the name of a Lie group and also its Lie algebra . ... In mathematics, E7 is the name of a Lie group and also its Lie algebra . ... In mathematics, E8 is the name of a Lie group and also its Lie algebra . ... In mathematics, an affine Lie algebra is a Kac-Moody algebra whose generalized Cartan matrix is positive semi definite (and which has no finite-dimensional direct summands). ... In mathematics, E8 is the name of a Lie group and also its Lie algebra . ... In mathematics, a Kac-Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. ... In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1. ... In mathematics, a Kac-Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. ...

References

• E₁₀ for beginners R.W. Gebert, H. Nicolai