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E7½ (Lie Algebra)
In mathematics, the Lie algebra E7½ is a subalgebra of E8 (mathematics), E8 containing E7 (mathematics), E7 defined by Landsberg and Manivel in order to fill the "hole" in a dimension formula for the En (Lie algebra), exceptional series E''n'' of simple Lie algebras. This hole was observed by Predrag Cvitanović, Cvitanovic, Deligne, Cohen and de Man. E7½ has dimension 190, and is not simple: as a representation of its subalgebra E7, it splits as , where (56) is the 56-dimensional irreducible representation of E7. This representation has an invariant symplectic form, and this symplectic form equips with the structure of a Heisenberg algebra; this Heisenberg algebra is the Nilradical of a Lie algebra, nilradical in E7½. See also *Vogel plane References * A.M. Cohen, R. de Man, "Computational evidence for Deligne's conjecture regarding exceptional Lie groups", ''Comptes rendus de l'Académie des Sciences'', Série I 322 (1996) 427–432. * P. Deligne, "La série exceptionn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
