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An elegant construction of E7 is described in Adams' book on exceptional Lie algebras. Namely, E7 as the sum of sl(8) and where the second summand is the space of alternating 4-forms in an 8-dimensional vector space. The sl(8) piece is a subalgebra. To define the bracket on the second piece as well as between the two pieces, there are elegant short formulas using natural pairings. Somebody should add this description. Katzmik 11:28, 16 August 2007 (UTC)
Apologies if I'm missing something, but the Hasse diagram of the root poset here (and for several other root systems) does not make sense to me. I believe that, in general, the elements of height 2 should be in bijection with the edges, because the sum of 2 simple roots is a root if and only if they are connected by an edge. E7 has 6 edges but I only see 5 roots at height 2. An older version looks (more) correct to me, it at least has 6 elements at height 2. http://en.wikipedia.org/wiki/File:E7Hasse.svg
Any help understanding this would be appreciated! Kinser ( talk) 20:58, 29 May 2013 (UTC)
Hi I think that "E7 appears as a part of the gauge group of one the (unstable and non-supersymmetric) versions of the heterotic string." needs an of, as in "E7 appears as a part of the gauge group of one of the (unstable and non-supersymmetric) versions of the heterotic string." But it could just be that I'm not following the logic. Ϣere SpielChequers 22:05, 29 April 2021 (UTC)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
An elegant construction of E7 is described in Adams' book on exceptional Lie algebras. Namely, E7 as the sum of sl(8) and where the second summand is the space of alternating 4-forms in an 8-dimensional vector space. The sl(8) piece is a subalgebra. To define the bracket on the second piece as well as between the two pieces, there are elegant short formulas using natural pairings. Somebody should add this description. Katzmik 11:28, 16 August 2007 (UTC)
Apologies if I'm missing something, but the Hasse diagram of the root poset here (and for several other root systems) does not make sense to me. I believe that, in general, the elements of height 2 should be in bijection with the edges, because the sum of 2 simple roots is a root if and only if they are connected by an edge. E7 has 6 edges but I only see 5 roots at height 2. An older version looks (more) correct to me, it at least has 6 elements at height 2. http://en.wikipedia.org/wiki/File:E7Hasse.svg
Any help understanding this would be appreciated! Kinser ( talk) 20:58, 29 May 2013 (UTC)
Hi I think that "E7 appears as a part of the gauge group of one the (unstable and non-supersymmetric) versions of the heterotic string." needs an of, as in "E7 appears as a part of the gauge group of one of the (unstable and non-supersymmetric) versions of the heterotic string." But it could just be that I'm not following the logic. Ϣere SpielChequers 22:05, 29 April 2021 (UTC)