# Category: Coalgebras

Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can
Quasi-Frobenius Lie algebra
In mathematics, a quasi-Frobenius Lie algebra over a field is a Lie algebra equipped with a nondegenerate skew-symmetric bilinear form , which is a Lie algebra 2- of with values in . In other words, f
Lie bialgebra
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
Cofree coalgebra
In algebra, the cofree coalgebra of a vector space or module is a coalgebra analog of the free algebra of a vector space. The cofree coalgebra of any vector space over a field exists, though it is mor
Quasi-Hopf algebra
A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989. A quasi-Hopf algebra is a quasi-bialgebra for which there exist an
F-coalgebra
In mathematics, specifically in category theory, an -coalgebra is a structure defined according to a functor , with specific properties as defined below. For both algebras and coalgebras, a functor is
Primitive element (co-algebra)
In algebra, a primitive element of a co-algebra C (over an element g) is an element x that satisfies where is the co-multiplication and g is an element of C that maps to the multiplicative identity 1
Measuring coalgebra
In algebra, a measuring coalgebra of two algebras A and B is a coalgebra enrichment of the set of homomorphisms from A to B. In other words, if coalgebras are thought of as a sort of linear analogue o
Quasi-triangular quasi-Hopf algebra
A quasi-triangular quasi-Hopf algebra is a specialized form of a quasi-Hopf algebra defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. It is also a generalized form of a quasi-triangula
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
Comodule
In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebr
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 coalgeb
Quasi-bialgebra
In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by hav