Kantor–Koecher–Tits construction

In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra, introduced by Jacques Tits (1962), Kantor (1964), and Koecher (1967).

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 Kac (1977) to classify the finite-dimensional simple Jordan superalgebras.

References

Jacobson, Nathan (1968), Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, Providence, R.I.: American Mathematical Society, MR 0251099
Kac, Victor G (1977), "Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras", Communications in Algebra, 5 (13): 1375–1400, doi:10.1080/00927877708822224, ISSN 0092-7872, MR 0498755
Kantor, I. L. (1964), "Classification of irreducible transitive differential groups", Doklady Akademii Nauk SSSR, 158: 1271–1274, ISSN 0002-3264, MR 0175941
Koecher, Max (1967), "Imbedding of Jordan algebras into Lie algebras. I", American Journal of Mathematics, 89: 787–816, doi:10.2307/2373242, ISSN 0002-9327, JSTOR 2373242, MR 0214631
Tits, Jacques (1962), "Une classe d'algèbres de Lie en relation avec les algèbres de Jordan", Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indagationes Mathematicae, 24: 530–535, MR 0146231