Manin triple

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In mathematics, a Manin triple ( g , p , q ) {\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})} {\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})} consists of a Lie algebra g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}} and q {\displaystyle {\mathfrak {q}}} {\displaystyle {\mathfrak {q}}} such that g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} is the direct sum of p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}} and q {\displaystyle {\mathfrak {q}}} {\displaystyle {\mathfrak {q}}} as a vector space. A closely related concept is the (classical) Drinfeld double, which is an even dimensional Lie algebra which admits a Manin decomposition.

Manin triples were introduced by Vladimir Drinfeld in 1987, who named them after Yuri Manin.[1]

In 2001 Delorme classified Manin triples where g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} is a complex reductive Lie algebra.[2]

Manin triples and Lie bialgebras

There is an equivalence of categories between finite-dimensional Manin triples and finite-dimensional Lie bialgebras.

More precisely, if ( g , p , q ) {\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})} {\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})} is a finite-dimensional Manin triple, then p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}} can be made into a Lie bialgebra by letting the cocommutator map p → p ⊗ p {\displaystyle {\mathfrak {p}}\to {\mathfrak {p}}\otimes {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}\to {\mathfrak {p}}\otimes {\mathfrak {p}}} be the dual of the Lie bracket q ⊗ q → q {\displaystyle {\mathfrak {q}}\otimes {\mathfrak {q}}\to {\mathfrak {q}}} {\displaystyle {\mathfrak {q}}\otimes {\mathfrak {q}}\to {\mathfrak {q}}} (using the fact that the symmetric bilinear form on g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} identifies q {\displaystyle {\mathfrak {q}}} {\displaystyle {\mathfrak {q}}} with the dual of p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}}).

Conversely if p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}} is a Lie bialgebra then one can construct a Manin triple ( p ⊕ p ∗ , p , p ∗ ) {\displaystyle ({\mathfrak {p}}\oplus {\mathfrak {p}}^{*},{\mathfrak {p}},{\mathfrak {p}}^{*})} {\displaystyle ({\mathfrak {p}}\oplus {\mathfrak {p}}^{*},{\mathfrak {p}},{\mathfrak {p}}^{*})} by letting q {\displaystyle {\mathfrak {q}}} {\displaystyle {\mathfrak {q}}} be the dual of p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}} and defining the commutator of p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}} and q {\displaystyle {\mathfrak {q}}} {\displaystyle {\mathfrak {q}}} to make the bilinear form on g = p ⊕ q {\displaystyle {\mathfrak {g}}={\mathfrak {p}}\oplus {\mathfrak {q}}} {\displaystyle {\mathfrak {g}}={\mathfrak {p}}\oplus {\mathfrak {q}}} invariant.

Examples

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

References

  1. Drinfeld, V. G. (1987). Gleason, Andrew (ed.). "Quantum groups" (PDF). Proceedings of the International Congress of Mathematicians 1986. 1. Berkeley: American Mathematical Society: 798–820. ISBN 978-0-8218-0110-9. MR 0934283.
  2. Delorme, Patrick (2001-12-01). "Classification des triples de Manin pour les algèbres de Lie réductives complexes: Avec un appendice de Guillaume Macey". Journal of Algebra. 246 (1): 97–174. arXiv:math/0003123. doi:10.1006/jabr.2001.8887. ISSN 0021-8693. MR 1872615.