En (Lie algebra)
{{DISPLAYTITLE:En (Lie algebra)}}
| align=right class=wikitable
|+ Dynkin diagrams |
| colspan=2|Finite |
|---|
| E3=A2A1
|{{Dynkin2|node_n1|3|node_n2|2|node_n3}} |
| E4=A4
|{{Dynkin2|node_n1|3|node_n2|3|branch}} |
| E5=D5
|{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4}} |
| E6
|{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5}} |
| E7
|{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6}} |
| E8
|{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7}} |
| colspan=2|Affine (Extended) |
| E9 or E{{su|b=8|p=(1)}} or E{{su|b=8|p=+}}
|{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7|3|nodeg_n8}} |
| colspan=2|Hyperbolic (Over-extended) |
| E10 or E{{su|b=8|p=(1)^}} or E{{su|b=8|p=++}}
|{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7|3|nodeg_n8|3|nodeg_n9}} |
| colspan=2|Lorentzian (Very-extended) |
| E11 or E{{su|b=8|p=+++}}
|{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7|3|nodeg_n8|3|nodeg_n9|3|nodeg_n10}} |
| colspan=2|Kac–Moody |
| E12 or E{{su|b=8|p=++++}}
|{{Dynkin2|node_n1|3|node_n2|3|branch|3|node_n4|3|node_n5|3|node_n6|3|node_n7|3|nodeg_n8|3|nodeg_n9|3|nodeg_n10|3|nodeg_n11}} |
| colspan=2|... |
In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and k, with {{nowrap|1=k = n − 4}}.
In some older books and papers, E2 and E4 are used as names for G2 and F4.
Finite-dimensional Lie algebras
The En group is similar to the An 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 En is {{nowrap|9 − n}}.
- E3 is another name for the Lie algebra A1A2 of dimension 11, with Cartan determinant 6.
- :
\begin{matrix}
2 & -1 & 0 \\
-1 & 2 & 0 \\
0 & 0 & 2
\end{matrix}\right ]
- E4 is another name for the Lie algebra A4 of dimension 24, with Cartan determinant 5.
- :
\begin{matrix}
2 & -1 & 0 & 0 \\
-1 & 2 & -1& 0 \\
0 & -1 & 2 & -1 \\
0 & 0 & -1 & 2
\end{matrix}\right ]
- E5 is another name for the Lie algebra D5 of dimension 45, with Cartan determinant 4.
- :
\begin{matrix}
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{matrix}\right ]
- E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
- :
\begin{matrix}
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{matrix}\right ]
- E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
- :
\begin{matrix}
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{matrix}\right ]
- E8 is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
- :
\begin{matrix}
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{matrix}\right ]
Infinite-dimensional Lie algebras
- E9 is another name for the infinite-dimensional affine Lie algebra Ẽ8 (also as E{{su|b=8|p=+}} or E{{su|b=8|p=(1)}} as a (one-node) extended E8) (or E8 lattice) corresponding to the Lie algebra of type E8. E9 has a Cartan matrix with determinant 0.
- :
\begin{matrix}
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{matrix}\right ]
- {{anchor|E10}} E10 (or E{{su|b=8|p=++}} or E{{su|b=8|p=(1)^}} as a (two-node) over-extended E8) 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. E10 has a Cartan matrix with determinant −1:
- :
\begin{matrix}
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{matrix}\right ]
- E11 (or E{{su|b=8|p=+++}} 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.
- En for {{nowrap|n ≥ 12}} is a family of infinite-dimensional Kac–Moody algebras that are not well studied.
Root lattice
The root lattice of En has determinant {{nowrap|9 − n}}, and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Zn,1 that are orthogonal to the vector {{nowrap|(1,1,1,1,...,1{{!}}3)}} of norm {{nowrap|n × 12 − 32}} = {{nowrap|n − 9}}.
E<sub>{{frac|7|1|2}}</sub>
{{main|E7½}}
Landsberg and Manivel extended the definition of En for integer n to include the case n = {{frac|7|1|2}}. They did this in order to fill the "hole" in dimension formulae for representations of the En series which was observed by Cvitanovic, Deligne, Cohen and de Man. E{{frac|7|1|2}} has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.
See also
References
- {{Cite book| 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 Academic Publishers Group | location=Dordrecht | series=NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. | mr=981374 | year=1988 | volume=250 | chapter=On E10 | pages=109–128 }}
Further reading
- {{cite journal |title=E11 and M Theory |year=2001 |last1=West | first1=P. |doi=10.1088/0264-9381/18/21/305 |journal=Classical and Quantum Gravity |volume=18 |issue=21 |pages=4443–4460 |arxiv=hep-th/0104081|bibcode=2001CQGra..18.4443W |s2cid=250872099 }} Class. Quantum Grav. 18 (2001) 4443-4460
- {{cite book|title=E10 for beginners |year=1994 |arxiv=hep-th/9411188 |last1=Gebert | first1=R. W. |last2=Nicolai | first2=H. |chapter=E 10 for beginners |doi= 10.1007/3-540-59163-X_269 |series=Lecture Notes in Physics |volume=447 |pages=197–210|isbn=978-3-540-59163-4 |s2cid=14570784 }} Guersey Memorial Conference Proceedings '94
- {{cite journal|first1=J. M.|last1=Landsberg|first2=L.|last2=Manivel|title=The sextonions and E7½|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, 2006 [https://arxiv.org/abs/0711.3498]
- A class of Lorentzian Kac-Moody algebras, Matthias R. Gaberdiel, David I. Olive and Peter C. West, 2002 [https://arxiv.org/abs/hep-th/0205068]