- Source: En (Lie algebra)
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 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 9 − n.
E3 is another name for the Lie algebra A1A2 of dimension 11, with Cartan determinant 6.
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{\displaystyle \left[{\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.
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{\displaystyle \left[{\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.
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{\displaystyle \left[{\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.
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{\displaystyle \left[{\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.
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{\displaystyle \left[{\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.
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{\displaystyle \left[{\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+8 or E(1)8 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.
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{\displaystyle \left[{\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]}
E10 (or E++8 or E(1)^8 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:
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{\displaystyle \left[{\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+++8 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 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 9 − n, and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Zn,1 that are orthogonal to the vector (1,1,1,1,...,1|3) of norm n × 12 − 32 = n − 9.
E7+1⁄2
Landsberg and Manivel extended the definition of En for integer n to include the case n = 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. E7+1⁄2 has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.
See also
k21, 2k1, 1k2 polytopes based on En Lie algebras.
References
Kac, Victor G; Moody, R. V.; Wakimoto, M. (1988). "On E10". Differential geometrical methods in theoretical physics (Como, 1987). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 250. Dordrecht: Kluwer Academic Publishers Group. pp. 109–128. MR 0981374.
Further reading
West, P. (2001). "E11 and M Theory". Classical and Quantum Gravity. 18 (21): 4443–4460. arXiv:hep-th/0104081. Bibcode:2001CQGra..18.4443W. doi:10.1088/0264-9381/18/21/305. S2CID 250872099. Class. Quantum Grav. 18 (2001) 4443-4460
Gebert, R. W.; Nicolai, H. (1994). "E 10 for beginners". E10 for beginners. Lecture Notes in Physics. Vol. 447. pp. 197–210. arXiv:hep-th/9411188. doi:10.1007/3-540-59163-X_269. ISBN 978-3-540-59163-4. S2CID 14570784. Guersey Memorial Conference Proceedings '94
Landsberg, J. M.; Manivel, L. (2006). "The sextonions and E7½". Advances in Mathematics. 201 (1): 143–179. arXiv:math.RT/0402157. doi:10.1016/j.aim.2005.02.001.
Connections between Kac-Moody algebras and M-theory, Paul P. Cook, 2006 [1]
A class of Lorentzian Kac-Moody algebras, Matthias R. Gaberdiel, David I. Olive and Peter C. West, 2002 [2]
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