In algebra, the Kantor–Koecher–Tits construction is a method of constructing a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
from a
Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan alg ...
, introduced by , , and .
If ''J'' is a Jordan algebra, the Kantor–Koecher–Tits construction puts a Lie algebra structure on ''J'' + ''J'' + Inner(''J''), the sum of 2 copies of ''J'' and the Lie algebra of inner derivations of ''J''.
When applied to a
27-dimensional exceptional Jordan algebra it gives a Lie algebra of type
E7 of dimension 133.
The Kantor–Koecher–Tits construction was used by to classify the finite-dimensional simple
Jordan superalgebras.
References
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{{DEFAULTSORT:Kantor-Koecher-Tits construction
Lie algebras
Non-associative algebras