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...

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...

counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups. Many quantum groups...

algebra Index of a Lie algebra Lie algebra cohomology Lie algebra extension Lie algebra representation Lie bialgebra Lie coalgebra Lie operad Particle physics...

Lie algebra Lie-* algebra Lie algebra bundle Lie algebra cohomology Lie algebra representation Lie algebroid Lie bialgebra Lie coalgebra Lie conformal algebra...

is a complex reductive Lie algebra. If (g, p, q) is a finite-dimensional Manin triple then p can be made into a Lie bialgebra by letting the cocommutator...

vector bundles. Lie bialgebroids are the vector bundle version of Lie bialgebras. A Lie algebroid consists of a bilinear skew-symmetric operation [ ⋅ , ⋅...

× V → F. Bialgebra: an associative algebra with a compatible coalgebra structure. Lie bialgebra: a Lie algebra with a compatible bialgebra structure...

)} is a pre-Lie algebra. Lie coalgebra Lie bialgebra Lie algebra cohomology Frobenius algebra Quasi-Frobenius ring Jacobson, Nathan, Lie algebras, Republication...

the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids. A Courant algebroid consists...