Manin triple

In mathematics, a Manin triple (g, p, q) consists of a Lie algebra g with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras p and q such that g is the direct sum of p and q as a vector space.

Manin triples were introduced by Drinfeld (1987, p.802), who named them after Yuri Manin.

Delorme (2001) classified the Manin triples where g is a complex reductive Lie algebra.

Manin triples and Lie bialgebras

If (g, p, q) is a finite-dimensional Manin triple then p can be made into a Lie bialgebra by letting the cocommutator map p → p ⊗ p be dual to the map q ⊗ q → q (using the fact that the symmetric bilinear form on g identifies q with the dual of p).

Conversely if p is a Lie bialgebra then one can construct a Manin triple from it by letting q be the dual of p and defining the commutator of p and q to make the bilinear form on g = p ⊕ q invariant.

Examples

• Suppose that a is a complex semisimple Lie algebra with invariant symmetric bilinear form (,). Then there is a Manin triple (g,p,q) with g = a⊕a, with the scalar product on g given by ((w,x),(y,z)) = (w,y) – (x,z). The subalgebra p is the space of diagonal elements (x,x), and the subalgebra q is the space of elements (x,y) with x in a fixed Borel subalgebra containing a Cartan subalgebra h, y in the opposite Borel subalgebra, and where x and y have the same component in h.

References

• Delorme, Patrick (2001), "Classification des triples de Manin pour les algèbres de Lie réductives complexes", Journal of Algebra, 246 (1): 97–174, doi:10.1006/jabr.2001.8887, ISSN 0021-8693, MR 1872615 
• Drinfeld, V. G. (1987), "Quantum groups", Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), 1, Providence, R.I.: American Mathematical Society, pp. 798–820, ISBN 978-0-8218-0110-9, MR 934283