Kantor–Koecher–Tits construction
Encyclopedia
In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra
from a Jordan algebra
, introduced by , , and .
If J is a Jordan algebra, the Kantor–Koecher–Tits construction puts a Lie algebra structure on J + J + J + Inner(J), the sum of 3 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.
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
from a Jordan algebra
Jordan algebra
In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
, introduced by , , and .
If J is a Jordan algebra, the Kantor–Koecher–Tits construction puts a Lie algebra structure on J + J + J + Inner(J), the sum of 3 copies of J and the Lie algebra of inner derivations of J.
When applied to a 27-dimensional exceptional Jordan algebra
Albert algebra
In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are two such Jordan algebras up to isomorphism...
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.