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...

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

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

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

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

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

Euclidean group E(3) of motions in 3 dimensions. Hopf algebra Lie bialgebra Poisson–Lie group Quantum affine algebra Schwiebert, Christian (1994), Generalized...

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

trivially holds at orders ℏ 0 , ℏ {\displaystyle \hbar ^{0},\hbar } ). Lie bialgebra Yangian Reidemeister move Quasitriangular Hopf algebra H.-D. Doebner...

into a sum of solitons and a decaying dispersive part. Lax pair Lie bialgebra Poisson–Lie group Krüger, Helge; Teschl, Gerald (2009), "Long-time asymptotics...

satisfying these conditions. This is the motivation for the definition of a bialgebra, where Δ is called the comultiplication and ε is called the counit. In...

coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain...

comultiplication if it satisfies certain axioms. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the...

include the tensor algebra, the exterior algebra, Hopf algebras and Lie bialgebras. Unlike the polynomial case above, none of these are commutative. Therefore...

does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended...

N=4 Supersymmetric Yang–Mills". YouTube. 30 November 2021. "Classical Lie Bialgebras for AdS/CFT Integrability by Contraction and Reduction by Niklas Beisert"...

operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean...

applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. Generally, the index set of a graded ring is assumed to be the set...

Its formal group ring (also called its hyperalgebra or its covariant bialgebra) is a cocommutative Hopf algebra H constructed as follows. As an R-module...

application to the quantum Yang–Baxter equation, the quantisation of Lie bialgebras and quantum Lévy area. Hudson, R. L.; Ion, P. D. F.; K. R. Parthasarathy...

operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra v t e...

⁠. The exterior algebra (as well as the symmetric algebra) inherits a bialgebra structure, and, indeed, a Hopf algebra structure, from the tensor algebra...

operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra v t e...

operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra v t e...

"quantum group" in reference to Hopf algebras that are deformations of simple Lie algebras, and connected them to the study of the Yang–Baxter equation, which...

Hilbert basis theorem. The enveloping algebra U of a finite-dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} is a both left and right Noetherian...

operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra v t e...

operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra v t e...