Bialgebra

Bialgebra

In mathematics, a bialgebra over a field "K" is a structure which is both a unital associative algebra and a coalgebra over "K", such that these structures are compatible.

Compatibility means that the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, that the multiplication and the unit of the algebra both be coalgebra morphisms: these statements are equivalent in that they are expressed by "the same diagrams".

As reflected in the symmetry of the diagrams, the definition of bialgebra is self-dual, so if one can define a dual of "B" (which is always possible if "B" is finite-dimensional), then it is automatically a bialgebra.

Diagrams

The compatibility conditions can be expressed by the following commutative diagrams, which can be read either as "comultiplication is a map of algebras" or "multiplication is a map of coalgebras" (similarly for the others):

Multiplication and comultiplication::

Multiplication and counit::

Comultiplication and unit::

Unit and counit::

Here ablacolon B otimes B o B is the algebra multiplication and etacolon K o B, is the unit of the algebra. Deltacolon B o B otimes B is the comultiplication and varepsiloncolon B o K, is the counit. aucolon B otimes B o B otimes B is the linear map defined by au(x otimes y) = yotimes x for all "x" and "y" in "B".

Formulas

In formulas, the bialgebra compatibility conditions look as follows (using the sumless Sweedler notation):

Multiplication and comultiplication::(ab)_{(1)}otimes (ab)_{(2)} = a_{(1)}b_{(1)} otimes a_{(2)}b_{(2)},

Multiplication and counit::varepsilon(ab)=varepsilon(a)varepsilon(b);

Comultiplication and unit::1_{(1)}otimes 1_{(2)} = 1 otimes 1 ,

Unit and counit::varepsilon(1)=1.;

Here we suppressed the algebra notation: we wrote the algebra multiplication abla as simple juxtaposition, and the unit eta via the multiplicative identity 1.

Examples

Examples of bialgebras include the Hopf algebras and the Lie bialgebras, which are bialgebras with certain additional structure. Additional examples are given in the article on coalgebras.

=See also=
*Quasi-bialgebra
*Hopf algebra


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Bialgebra — berührt die Spezialgebiete Mathematik Abstrakte Algebra Lineare Algebra Kommutative Algebra ist Spezialfall von Algebra Koalgebra umfasst als Spezialfälle …   Deutsch Wikipedia

  • bialgebra — noun A particular form of vector space which is a compatible form of two algebras …   Wiktionary

  • Quasi-bialgebra — In mathematics, quasi bialgebras are a generalization of bialgebras, which were defined by the Ukrainian mathematician Vladimir Drinfeld in 1990.A quasi bialgebra mathcal{B A} = (mathcal{A}, Delta, varepsilon, Phi) is an algebra mathcal{A} over a …   Wikipedia

  • Lie bialgebra — In mathematics, a Lie bialgebra is the Lie theoretic case of a bialgebra: its 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… …   Wikipedia

  • Hopfalgebra — berührt die Spezialgebiete Mathematik Abstrakte Algebra Lineare Algebra Kommutative Algebra ist Spezialfall von Bialgebra Eine Hopf Algebra – benannt nach dem Math …   Deutsch Wikipedia

  • Hopf algebra — In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra, a coalgebra, and has an antiautomorphism, with these structures compatible.Hopf algebras occur naturally in algebraic… …   Wikipedia

  • Exterior algebra — In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of… …   Wikipedia

  • Hopf-Algebra — Hopfalgebra berührt die Spezialgebiete Mathematik Abstrakte Algebra Lineare Algebra Kommutative Algebra ist Spezialfall von Bialgebra Eine Hopf Algebra – benannt …   Deutsch Wikipedia

  • Convolution — For the usage in formal language theory, see Convolution (computer science). Convolution of two square pulses: the resulting waveform is a triangular pulse. One of the functions (in this case g) is first reflected about τ = 0 and then offset by t …   Wikipedia

  • Quantum group — In mathematics and theoretical physics, quantum groups are certain noncommutative algebras that first appeared in the theory of quantum integrable systems, and which were then formalized by Vladimir Drinfel d and Michio Jimbo. There is no single …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”