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Like most other mathematicians, I am not an expert on the mathematical physics literature related to Lie algebras. But the E series has led further into Kac-Moody algebras: affine, hyperbolic, ... The hyperbolic Kac-Moody algebra $E_{10}$ shows up quite a bit in the literature (and there is even $E_{11}$). For instance, a random search of MathSciNet turns up such papers as:

MR1354261 (96g:17028) 17B67 (17B69 17B81 81R10). Gebert, R.W. (D-HAMB-2P); Nicolai, H. (D-HAMB-2P), On E10 and the DDF construction. Comm. Math. Phys. 172 (1995), no. 3, 571–622.

MR1894911 (2003i:83107) 83E50 (17B81 81R10 81T30 83E30). West, P. (4-LNDKC), E11 and M theory. Classical Quantum Gravity 18 (2001), no. 21, 4443–4460.

Mathematicians like Victor Kac and Ed Frenkel have been active in some of this work involving Lie theory and physics. As others have pointed out, the exceptional finite dimensional Lie algebras don't arise directly from quadratic forms and such in the way classical Lie algebras do, but they do come up in "symmetry" questions related to physics (at least theoretically) and do have interesting characterizations in terms of exceptional algebraic structures like octonions and the 27-dimensional special Jordan algebra. Similarly, some of the infinite dimensional Kac-Moody analogues are natural for symmetry questions in modern physics.

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Like most other mathematicians, I am not an expert on the mathematical physics literature related to Lie algebras. But the E series has led further into Kac-Moody algebras: affine, hyperbolic, ... The hyperbolic Kac-Moody algebra $E_{10}$ shows up quite a bit in the literature (and there is even $E_{11}$). For instance, a random search of MathSciNet turns up such papers as:

MR1354261 (96g:17028) 17B67 (17B69 17B81 81R10). Gebert, R.W. (D-HAMB-2P); Nicolai, H. (D-HAMB-2P), On E10 and the DDF construction. Comm. Math. Phys. 172 (1995), no. 3, 571–622.

MR1894911 (2003i:83107) 83E50 (17B81 81R10 81T30 83E30). West, P. (4-LNDKC), E11 and M theory. Classical Quantum Gravity 18 (2001), no. 21, 4443–4460.

Mathematicians like Victor Kac and Ed Frenkel have been active in some of this work involving Lie theory and physics. As others have pointed out, the exceptional finite dimensional Lie algebras don't arise directly from quadratic forms and such in the way classical Lie algebras do, but they do come up in "symmetry" questions related to physics (at least theoretically) and do have interesting characterizations. Similarly, the Kac-Moody analogues are natural for symmetry questions in modern physics.