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...

In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras...

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

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

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...

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...

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

algebra and the coalgebra structure are compatible, i.e. g {\displaystyle {\mathfrak {g}}} is a Lie bialgebra, The classical Lie group–Lie algebra correspondence...

Jordan identity. Coalgebra: a vector space with a "comultiplication" defined dually to that of associative algebras. Lie coalgebra: a vector space with...

also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated...

comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra. Let K be a field, and C be a coalgebra over K. A (right)...

The coproduct Δ is given by ΔD(n) = ΣD(i) ⊗ D(n−i) (so the dual of this coalgebra is just the ring of formal power series). The counit η is given by the...

structure of a coalgebra. There is also an abstract notion of F-coalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed...

simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that...

which is both a unital associative algebra and a counital coassociative coalgebra.: 46  The algebraic and coalgebraic structures are made compatible with...

unary involution operation Von Neumann algebra (or W*-algebra) See also coalgebra, the dual notion. A different class of "algebras" consists of objects...

\phi _{2}(X)} . This product can be recognized as the coproduct on a coalgebra. In general, the tensor product of irreducible representations is not...

integer multiple of another element in the lattice Primitive element (coalgebra), an element X on which the comultiplication Δ has the value Δ(X) = X⊗1...

differential graded cocommutative coalgebras. (The associated coalgebra is the rational homology of X as a coalgebra; the dual vector space is the rational...

algebra). If C is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual commutator bracket [ x , y ] = x y − y x {\displaystyle...

serve to encapsulate the induction principle, and the dual construction F-coalgebras. If C {\displaystyle C} is a category, and F : C → C {\displaystyle F:C\rightarrow...

abstract algebra, a dialgebra is the generalization of both algebra and coalgebra. The notion was originally introduced by Lambek as "subequalizers", and...

ISBN 0-521-24870-1 Steven Shnider and Shlomo Sternberg (1993) Quantum Groups. From Coalgebras to Drinfeld Algebras: A Guided Tour (Mathematical Physics Ser.) International...

of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra A is a square matrix v = ( v i j ) i , j = 1 , … , n {\displaystyle v=(v_{ij})_{i...

Differential graded algebra (DGA) Simplicial Lie algebra Homotopy Lie algebra Hinich, Vladimir (2001). "DG coalgebras as formal stacks". Journal of Pure and...

(mathematics) Category theory Monoidal category Groupoid Group object Coalgebra Bialgebra Hopf algebra Magma object Torsion (algebra) Symbolic mathematics...

chain complexes, groupoids (or even the category of categories itself), coalgebras, etc. Given a commutative ring R we consider the category R - M o d {\displaystyle...

the result is a coalgebra isomorphism, and not merely a K-module isomorphism, equipping both S(L) and U(L) with their natural coalgebra structures such...

sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensor algebra Free object Noncommutative ring Rational series Term algebra...

{\displaystyle {\textstyle \bigwedge }(V)} ⁠, giving the structure of a coalgebra. The coproduct is a linear function ⁠ Δ : ⋀ ( V ) → ⋀ ( V ) ⊗ ⋀ ( V )...