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| | Hello! Permit me to start by saying my name - Leon my partner and i love doing it. Collecting marbles factor I really like doing. The job I've been occupying most desired is a production and planning officer and I do not think I'll change it anytime almost immediately. New Mexico exactly where her home is. I am running and maintaining weblog here: http://iwangun.tumblr.com |
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| In [[mathematics]], a '''bialgebra''' over a [[Field (mathematics)|field]] ''K'' is a [[vector space]] over ''K'' which is both a [[unital algebra|unital]] [[associative algebra]] and a [[coalgebra]], such that the algebraic- and coalgebraic structure satisfy certain compatibility relations. Specifically, the [[comultiplication]] and the [[counit]] are both unital algebra [[homomorphisms]], or equivalently, that the multiplication and the unit of the algebra both be [[Coalgebra#Further concepts and facts|coalgebra morphisms]]. These statements are equivalent in that they are expressed by ''the same [[commutative diagram]]s''. A bialgebra homomorphism is a [[linear map]] that is both an algebra and a coalgebra homomorphism.
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| As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is [[Dual (category theory)|self-dual]], so if one can define a [[Dual space|dual]] of ''B'' (which is always possible if ''B'' is finite-dimensional), then it is automatically a bialgebra.
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| {{Algebraic structures |Algebra}}
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| == Formal definition ==
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| '''(''B'', ∇, η, Δ, ε)''' is a '''bialgebra''' over ''K'' if it has the following properties:
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| * ''B'' is a vector space over ''K'';
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| * there are ''K''-[[linear map]]s (multiplication) ∇: ''B'' ⊗ ''B'' → ''B'' (equivalent to ''K''-[[multilinear map]] ∇: ''B'' × ''B'' → ''B'') and (unit) η: ''K'' → ''B'', such that (''B'', ∇, η) is a unital associative [[Algebra over a field|algebra]];
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| * there are ''K''-linear maps (comultiplication) Δ: ''B'' → ''B'' ⊗ ''B'' and (counit) ε: ''B'' → ''K'', such that (''B'', Δ, ε) is a (counital coassociative) [[coalgebra]];
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| * compatibility conditions expressed by the following [[commutative diagram]]s:
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| # Multiplication ∇ and comultiplication Δ <ref>Dăscălescu, Năstăsescu & Raianu (2001), {{Google books quote|id=pBJ6sbPHA0IC|page=147|text=is a morphism of coalgebras|p. 147 & 148}}</ref>
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| #::[[Image:Bialgebra2.svg|500px|Bialgebra commutative diagrams]]
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| #: where τ: ''B'' ⊗ ''B'' → ''B'' ⊗ ''B'' is the [[linear map]] defined by τ(''x'' ⊗ ''y'') = ''y'' ⊗ ''x'' for all ''x'' and ''y'' in ''B'',
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| # Multiplication ∇ and counit ε
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| #::[[Image:Bialgebra3.svg|310px|Bialgebra commutative diagrams]]
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| # Comultiplication Δ and unit η <ref>Dăscălescu, Năstăsescu & Raianu (2001), {{Google books quote|id=pBJ6sbPHA0IC|page=148|text=is a morphism of coalgebras|p. 148}}</ref>
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| #::[[Image:Bialgebra4a.svg|310px|Bialgebra commutative diagrams]]
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| # Unit η and counit ε
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| #::[[Image:Bialgebra1.svg|125px|Bialgebra commutative diagrams]]
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| ==Coassociativity and counit==
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| The [[multilinear map|''K''-linear map]] Δ: ''B'' → ''B'' ⊗ ''B'' is [[coalgebra|coassociative]] if <math>(\mathrm{id}_B \otimes \Delta) \circ \Delta = (\Delta \otimes \mathrm{id}_B) \circ \Delta</math>.
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| The ''K''-linear map ε: ''B'' → ''K'' is a counit if <math>(\mathrm{id}_B \otimes \epsilon) \circ \Delta = \mathrm{id}_B = (\epsilon \otimes \mathrm{id}_B) \circ \Delta</math>. | |
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| Coassociativy and counit are expressed by the [[commutative diagram|commutativity]] of the following two diagrams with ''B'' in place of ''C'' (they are the duals of the diagrams expressing associativity and unit of an algebra):
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| [[Image:coalg.png|center|800px]]
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| == Compatibility conditions ==
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| The four commutative diagrams can be read either as "comultiplication and counit are [[homomorphism]]s of algebras" or, equivalently, "multiplication and unit are [[homomorphism]]s of coalgebras".
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| These statements are meaningful once we explain the natural structures of algebra and coalgebra in all the vector spaces involved besides ''B'': (''K'', ∇<sub>0</sub>, η<sub>0</sub>) is a unital associative algebra in an obvious way and (''B'' ⊗ ''B'', ∇<sub>2</sub>, η<sub>2</sub>) is a unital associative algebra with unit and multiplication
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| :<math>\eta_2 := (\eta \otimes \eta) : K \otimes K \equiv K \to (B \otimes B) </math>
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| :<math>\nabla_2 := (\nabla \otimes \nabla) \circ (id \otimes \tau \otimes id) : (B \otimes B) \otimes (B \otimes B) \to (B \otimes B) </math>,
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| so that <math>\nabla_2 ( (x_1 \otimes x_2) \otimes (y_1 \otimes y_2) ) = \nabla(x_1 \otimes y_1) \otimes \nabla(x_2 \otimes y_2) </math> or, omitting ∇ and writing multiplication as juxtaposition, <math>(x_1 \otimes x_2)(y_1 \otimes y_2) = x_1 y_1 \otimes x_2 y_2 </math>;
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| similarly, (''K'', Δ<sub>0</sub>, ε<sub>0</sub>) is a coalgebra in an obvious way and ''B'' ⊗ ''B'' is a coalgebra with counit and comultiplication
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| :<math>\epsilon_2 := (\epsilon \otimes \epsilon) : (B \otimes B) \to K \otimes K \equiv K</math>
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| :<math>\Delta_2 := (id \otimes \tau \otimes id) \circ (\Delta \otimes \Delta) : (B \otimes B) \to (B \otimes B) \otimes (B \otimes B)</math>.
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| Then, diagrams 1 and 3 say that Δ: ''B'' → ''B'' ⊗ ''B'' is a homomorphism of unital (associative) algebras (''B'', ∇, η) and (''B'' ⊗ ''B'', ∇<sub>2</sub>, η<sub>2</sub>)
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| :<math>\Delta \circ \nabla = \nabla_2 \circ (\Delta \otimes \Delta) : (B \otimes B) \to (B \otimes B)</math>, or simply Δ(''xy'') = Δ(''x'') Δ(''y''),
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| :<math>\Delta \circ \eta = \eta_2 : K \to (B \otimes B)</math>, or simply Δ(1<sub>''B''</sub>) = 1<sub>''B'' ⊗ ''B''</sub>;
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| diagrams 2 and 4 say that ε: ''B'' → ''K'' is a homomorphism of unital (associative) algebras (''B'', ∇, η) and (''K'', ∇<sub>0</sub>, η<sub>0</sub>):
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| :<math>\epsilon \circ \nabla = \nabla_0 \circ (\epsilon \otimes \epsilon) : (B \otimes B) \to K</math>, or simply ε(''xy'') = ε(''x'') ε(''y'')
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| :<math>\epsilon \circ \eta = \eta_0 : K \to K</math>, or simply ε(1<sub>''B''</sub>) = 1<sub>''K''</sub>.
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| Equivalently, diagrams 1 and 2 say that ∇: ''B'' ⊗ ''B'' → ''B'' is a homomorphism of (counital coassociative) coalgebras (''B'' ⊗ ''B'', Δ<sub>2</sub>, ε<sub>2</sub>) and (''B'', Δ, ε):
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| :<math> \nabla \otimes \nabla \circ \Delta_2 = \Delta \circ \nabla : (B \otimes B) \to (B \otimes B),</math>
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| :<math>\epsilon \circ \nabla = \nabla_0 \circ \epsilon_2 : (B \otimes B) \to K</math>;
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| diagrams 3 and 4 say that η: ''K'' → ''B'' is a homomorphism of (counital coassociative) coalgebras (''K'', Δ<sub>0</sub>, ε<sub>0</sub>) and (''B'', Δ, ε):
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| :<math>\Delta \circ \eta = \eta_2 \circ \Delta_0: K \to (B \otimes B),</math>
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| :<math>\epsilon \circ \eta = \eta_0 \circ \epsilon_0 : K \to K</math>.
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| ==Examples==
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| A simple example of a bialgebra is the set of functions from a [[group (mathematics)|group]] ''G'' to <math>\mathbb R</math>, which we may represent as a vector space <math>\mathbb R^G</math> consisting of linear combinations of standard basis vectors '''e'''<sub>''g''</sub> for each ''g'' ∈ ''G'', which may represent a [[probability distribution]] over ''G'' in the case of vectors whose coefficients are all non-negative and sum to 1. An example of suitable comultiplication operators and counits which yield a counital coalgebra are
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| :<math>\Delta(\mathbf e_g) = \mathbf e_g \otimes \mathbf e_g \,,</math>
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| which represents making a copy of a [[random variable]] (which we extend to all <math>\mathbb R^G</math> by linearity), and
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| :<math>\varepsilon(\mathbf e_g) = 1 \,,</math>
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| (again extended linearly to all of <math> \mathbb R^G</math>) which represents "tracing out" a random variable — ''i.e.,'' forgetting the value of a random variable (represented by a single tensor factor) to obtain a [[marginal distribution]] on the remaining variables (the remaining tensor factors).
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| Given the interpretation of (Δ,ε) in terms of probability distributions as above, the bialgebra consistency conditions amount to constraints on (∇,η) as follows:
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| 4. η is an operator preparing a normalized probability distribution which is independent of all other random variables;
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| 3. The product ∇ maps a probability distribution on two variables to a probability distribution on one variable;
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| 2. Copying a random variable in the distribution given by η is equivalent to having two independent random variables in the distribution η;
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| 1. Taking the product of two random variables, and preparing a copy of the resulting random variable, has the same distribution as preparing copies of each random variable independently of one another, and multiplying them together in pairs.
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| A pair (∇,η) which satisfy these constraints are the [[convolution]] operator
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| :<math>\nabla\bigl(\mathbf e_g \otimes \mathbf e_h\bigr) = \mathbf e_{gh} \,,</math> | |
| again extended to all <math>\mathbb R^G \otimes \mathbb R^G</math> by linearity; this produces a normalized probability distribution from a distribution on two random variables, and has as a unit the delta-distribution <math> \eta = \mathbf e_{i} \;,</math> where ''i'' ∈ ''G'' denotes the identity element of the group ''G''.
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| Other examples of bialgebras include the [[Hopf algebra]]s.<ref>Dăscălescu, Năstăsescu & Raianu (2001), {{Google books quote|id=pBJ6sbPHA0IC|page=151|text=Hopf|p. 151}}</ref> Similar structures with different compatibility between the product and comultiplication, or different types of multiplication and comultiplication, include [[Lie bialgebra]]s and [[Frobenius algebra]]s. Additional examples are given in the article on [[coalgebra]]s.
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| ==See also==
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| *[[Quasi-bialgebra]]
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| *[[Frobenius algebra]]
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| *[[Hopf algebra]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{Citation| last1=Dăscălescu| first1=Sorin| last2=Năstăsescu| first2=Constantin| last3=Raianu| first3=Șerban| year=2001| title=Hopf Algebras| subtitle=An introduction| edition=1st| volume = 235| series=Pure and Applied Mathematics | publisher=Marcel Dekker| isbn = 0-8247-0481-9}}.
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| [[Category:Bialgebras]]
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| [[Category:Coalgebras]]
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| [[Category:Monoidal categories]]
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