Lie bialgebra explained
In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary.
They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group.
Lie bialgebras occur naturally in the study of the Yang–Baxter equations.
Definition
is a Lie bialgebra if it is a Lie algebra,and there is the structure of Lie algebra also on the dual vector space
which is compatible.More precisely the Lie algebra structure on
is given by a Lie bracket
[ , ]:ak{g} ⊗ ak{g}\toak{g}
and the Lie algebra structure on
is given by a Liebracket
\delta*:ak{g}* ⊗ ak{g}*\toak{g}*
.Then the map dual to
is called the cocommutator,
\delta:ak{g}\toak{g} ⊗ ak{g}
and the compatibility condition is the following cocycle relation:
\delta([X,Y])=\left(\operatorname{ad}X ⊗ 1+1 ⊗ \operatorname{ad}X\right)\delta(Y)-\left(\operatorname{ad}Y ⊗ 1+1 ⊗ \operatorname{ad}Y\right)\delta(X)
where
\operatorname{ad}XY=[X,Y]
is the adjoint.Note that this definition is symmetric and
is also a Lie bialgebra, the dual Lie bialgebra.
Example
Let
be any
semisimple Lie algebra. To specify a Lie bialgebra structure we thus need to specify a compatible Lie algebra structure on the dual vector space. Choose a Cartan subalgebra
and a choice of positive roots. Let
be the corresponding opposite Borel subalgebras, so that
and there is a natural projection
.Then define a Lie algebra
ak{g'}:=\{(X-,X+)\inak{b}- x ak{b}+ l\vert \pi(X-)+\pi(X+)=0\}
which is a subalgebra of the product
, and has the same dimension as
.Now identify
with dual of
via the pairing
\langle(X-,X+),Y\rangle:=K(X+-X-,Y)
where
and
is the
Killing form.This defines a Lie bialgebra structure on
, and is the "standard" example: it underlies the Drinfeld-Jimbo
quantum group.Note that
is
solvable, whereas
is semisimple.
Relation to Poisson–Lie groups
The Lie algebra
of a Poisson–Lie group
G has a natural structure of Lie bialgebra.In brief the
Lie group structure gives the Lie bracket on
as usual, and the linearisation of the Poisson structure on
G gives the Lie bracket on
(recalling that a linear Poisson structure on a vector space is the same thing as a Lie bracket on the dual vector space). In more detail, let
G be a Poisson–Lie group, with
being two smooth functions on the group manifold. Let
be the differential at the identity element. Clearly,
. The
Poisson structure on the group then induces a bracket on
, as
[\xi1,\xi2]=(d\{f1,f2\})e
where
is the
Poisson bracket. Given
be the
Poisson bivector on the manifold, define
to be the right-translate of the bivector to the identity element in
G. Then one has that
The cocommutator is then the tangent map:
so that
is the dual of the cocommutator.
See also
References
- H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG, 1989, Springer-Verlag Berlin, .
- Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge .
- Beisert . N. . Spill . F. . 2009 . The classical r-matrix of AdS/CFT and its Lie bialgebra structure . Communications in Mathematical Physics . 285 . 2. 537–565 . 10.1007/s00220-008-0578-2 . 0708.1762 . 2009CMaPh.285..537B . 8946457 .
