Lie bialgebra

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible. In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the comultiplication 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. A vector space g {displaystyle {mathfrak {g}}} is a Lie bialgebra if it is a Lie algebra,and there is the structure of Lie algebra also on the dual vector space g ∗ {displaystyle {mathfrak {g}}^{*}} which is compatible.More precisely the Lie algebra structure on g {displaystyle {mathfrak {g}}} is given by a Lie bracket [   ,   ] : g ⊗ g → g {displaystyle :{mathfrak {g}}otimes {mathfrak {g}} o {mathfrak {g}}} and the Lie algebra structure on g ∗ {displaystyle {mathfrak {g}}^{*}} is given by a Liebracket δ ∗ : g ∗ ⊗ g ∗ → g ∗ {displaystyle delta ^{*}:{mathfrak {g}}^{*}otimes {mathfrak {g}}^{*} o {mathfrak {g}}^{*}} .Then the map dual to δ ∗ {displaystyle delta ^{*}} is called the cocommutator, δ : g → g ⊗ g {displaystyle delta :{mathfrak {g}} o {mathfrak {g}}otimes {mathfrak {g}}} and the compatibility condition is the following cocycle relation: where ad X ⁡ Y = [ X , Y ] {displaystyle operatorname {ad} _{X}Y=} is the adjoint.Note that this definition is symmetric and g ∗ {displaystyle {mathfrak {g}}^{*}} is also a Lie bialgebra, the dual Lie bialgebra. Let g {displaystyle {mathfrak {g}}} 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 t ⊂ g {displaystyle {mathfrak {t}}subset {mathfrak {g}}} and a choice of positive roots. Let b ± ⊂ g {displaystyle {mathfrak {b}}_{pm }subset {mathfrak {g}}} be the corresponding opposite Borel subalgebras, so that t = b − ∩ b + {displaystyle {mathfrak {t}}={mathfrak {b}}_{-}cap {mathfrak {b}}_{+}} and there is a natural projection π : b ± → t {displaystyle pi :{mathfrak {b}}_{pm } o {mathfrak {t}}} .Then define a Lie algebra which is a subalgebra of the product b − × b + {displaystyle {mathfrak {b}}_{-} imes {mathfrak {b}}_{+}} , and has the same dimension as g {displaystyle {mathfrak {g}}} .Now identify g ′ {displaystyle {mathfrak {g'}}} with dual of g {displaystyle {mathfrak {g}}} via the pairing where Y ∈ g {displaystyle Yin {mathfrak {g}}} and K {displaystyle K} is the Killing form.This defines a Lie bialgebra structure on g {displaystyle {mathfrak {g}}} , and is the 'standard' example: it underlies the Drinfeld-Jimbo quantum group.Note that g ′ {displaystyle {mathfrak {g'}}} is solvable, whereas g {displaystyle {mathfrak {g}}} is semisimple.