“Manin triple”的意思、由来-开放百科全书

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Manin triple

释义

Manin triples and Lie bialgebras
Examples
References

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 {{harvs|txt|last=Drinfeld|authorlink=Vladimir Drinfeld|year=1987|loc=p.802}}, who named them after Yuri Manin.

{{harvtxt|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

{{Citation | last1=Delorme | first1=Patrick | title=Classification des triples de Manin pour les algèbres de Lie réductives complexes | doi=10.1006/jabr.2001.8887 |mr=1872615 | year=2001 | journal=Journal of Algebra | issn=0021-8693 | volume=246 | issue=1 | pages=97–174}}
{{Citation | last1=Drinfeld | first1=V. G. | title=Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986) | chapter-url=http://www.mathunion.org/ICM/ICM1986.1/ | publisher=American Mathematical Society | location=Providence, R.I. | isbn= 978-0-8218-0110-9 |mr=934283 | year=1987 | volume=1 | chapter=Quantum groups | pages=798–820}}

1 : Lie algebras

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