In mathematics, a Lie bialgebra is the Lie-theoretic case of a
bialgebra
In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. ...
: it is a set with a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
and a
Lie coalgebra In mathematics a Lie coalgebra is the dual structure to a Lie algebra.
In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally has the structure of a Lie coalgebra, and conversely.
Definition
Let ''E'' be ...
structure which are compatible.
It is a
bialgebra
In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. ...
where the multiplication is
skew-symmetric and satisfies a dual
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
, so that the dual vector space is a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, 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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of a
Poisson–Lie group In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold.
The infinitesimal counterpart of a Poisson–Lie group is a L ...
.
Lie bialgebras occur naturally in the study of the
Yang–Baxter equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve thei ...
s.
Definition
A vector space
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
and the Lie algebra structure on
is given by a Lie
bracket
.
Then the map dual to
is called the cocommutator,
and the compatibility condition is the following cocycle relation:
:
where