In
mathematics, especially in
Lie theory, E
''n'' is the
Kac–Moody algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...
whose
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
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 In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Ki ...
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''.
*E
3 is another name for the Lie algebra ''A''
1''A''
2 of dimension 11, with Cartan determinant 6.
*: