Representation theory of Hopf algebras

In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra "H" over a field "K" is a "K"-vector space "V" with an action "H" × "V" → "V" usually denoted by juxtaposition (that is, the image of ("h","v") is written "hv"). The vector space "V" is called an "H"-module.


The module structure of a representation of a Hopf algebra "H" is simply its structure as a module for the underlying associative algebra. The main use of considering the additional structure of a Hopf algebra is when considering all "H"-modules as a category. The additional structure is also used to define invariant elements of an "H"-module "V". An element "v" in "V" is invariant under "H" if for all "h" in "H", hv=varepsilon(h)v, where ε is the counit of "H".The subset of all invariant elements of "V" forms a submodule of "V".

Categories of representations as a motivation for Hopf algebras

For an associative algebra "H", the tensor product V_1otimes V_2 of two "H"-modules "V"1 and "V"2 is a vector space, but not necessarily an "H"-module. For the tensor product to be a functorial product operation on "H"-modules, there must be a linear binary operation Delta:H ightarrow Hotimes H, such that for any "v" in V_1otimes V_2 and any "h" in "H",

:hv=Delta h(v_{(1)}otimes v_{(2)})=h_{(1)}v_{(1)}otimes h_{(2)}v_{(2)},,

and for any "v" in V_1otimes V_2 and "a" and "b" in "H",

:Delta(ab)(v_{(1)}otimes v_{(2)})=(ab)v=a [b [v] =Delta a [Delta b(v_{(1)}otimes v_{(2)})] =(Delta a )(Delta b)(v_{(1)}otimes v_{(2)}).,

using sumless Sweedler's notation, which is kind of like an index free form of Einstein's summation convention. This is satisfied if there is such a Δ such that Delta(ab)=Delta(a)Delta(b) for all "a" and "b" in "H".

For the category of "H"-modules to be a strict monoidal category with respect to otimes, V_1otimes(V_2otimes V_3) and (V_1otimes V_2)otimes V_3 must be equivalent and there must be unit object varepsilon_H, called the trivial module, such that varepsilon_Hotimes V, "V" and Votimes varepsilon_H are equivalent.

This means that for any "v" in V_1otimes(V_2otimes V_3)=(V_1otimes V_2)otimes V_3 and "h" in "H",:((operatorname{id}otimes Delta)Delta h)(v_{(1)}otimes v_{(2)}otimes v_{(3)})=h_{(1)}v_{(1)}otimes h_{(2)(1)}v_{(2)}otimes h_{(2)(2)}v_{(3)}=hv=((Deltaotimes operatorname{id})Delta h)(v_{(1)}otimes v_{(2)}otimes v_{(3)}).This will hold for any three "H"-modules if Delta satisfies (operatorname{id}otimes Delta)Delta A=(Delta otimes operatorname{id})Delta A.

The trivial module must be one dimensional, and so an algebra homomorphism varepsilon:H ightarrow F may be defined such that hv=varepsilon(h)v for all "v" in varepsilon_H. The trivial module may be identified with "F", with 1 being the element such that 1otimes v=v=votimes 1 for all "v". It follows that for any "v" in any "H"-module "V", any "c" in varepsilon_H and any "h" in "H",:(varepsilon(h_{(1)})h_{(2)})cv=h_{(1)}cotimes h_{(2)}v=h(cotimes v)=h(cv)=(h_{(1)}varepsilon(h_{(2)}))cv.The existence of an algebra homomorphism ε satisfying varepsilon(h_{(1)})h_{(2)} = h = h_{(1)}varepsilon(h_{(2)}) is a sufficient condition for the existence of the trivial module.

It follows that in order for the category of "H"-modules to be a monoidal category with respect to the tensor product, it is sufficient for "H" to have maps Delta and varepsilon satisfying these conditions. This is the motivation for the definition of a bialgebra, where Delta is called the comultiplication and varepsilon is called the counit.

In order for each "H"-module "V" to have a dual representation "V*" such that the underlying vector spaces are dual and the operation * is functorial over the monoidal category of "H"-modules, there must be a linear map S:H ightarrow H such that for any "h" in "H", "x" in "V" and "y" in "V"*,

:langle y, S(h)x angle = langle hy, x angle.

where langlecdot,cdot angle is the usual pairing of dual vector spaces. If the map varphi:Votimes V^* ightarrow varepsilon_H induced by the pairing is to be an "H"-homomorphism, then for any "h" in "H", "x" in "V" and "y" in "V"*,:varphileft(h(xotimes y) ight)=varphileft(xotimes S(h_{(1)})h_{(2)}y ight)=varphileft(S(h_{(2)})h_{(1)}xotimes y ight)=hvarphi(xotimes y)=varepsilon(h)varphi(xotimes y),which is satisfied if S(h_{(1)})h_{(2)}=varepsilon(h)=h_{(1)}S(h_{(2)}) for all "h" in "H".

If there is such a map "S", then it is called an "antipode", and "H" is a Hopf algebra. The desire for a monoidal category of modules with functorial tensor products and dual representations is therefore one motivation for the concept of a Hopf algebra.

Representations on an algebra

A Hopf algebra also has representations which carry additional structure, namely they are algebras.

Let "H" be a Hopf algebra. If "A" is an algebra with the product operation mu:Aotimes A ightarrow A, and ho:Hotimes A ightarrow A is a representation of "H" on "A", then "ρ" is said to be a representation of "H" on an algebra if "μ" is "H"-equivariant. As special cases, Lie algebras, Lie superalgebras and groups can also have representations on an algebra.

ee also

*Tannaka-Krein reconstruction theorem

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Representation theory — This article is about the theory of representations of algebraic structures by linear transformations and matrices. For the more general notion of representations throughout mathematics, see representation (mathematics). Representation theory is… …   Wikipedia

  • List of representation theory topics — This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics, which is more directed towards the mathematical analysis aspects of representation theory. Contents 1 General representation theory 2… …   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

  • Representation of a Lie group — In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the… …   Wikipedia

  • Representation of a Lie superalgebra — In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z2 graded vector space V , such that if A and B are any two pure elements of L and X and Y are any two pure… …   Wikipedia

  • Algebra representation — In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint… …   Wikipedia

  • Dual representation — In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation ρ is defined over the dual vector space V as follows[1]: ρ(g) is the transpose of ρ(g−1) for all g in G. Then ρ is also a… …   Wikipedia

  • Restricted representation — In mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand …   Wikipedia

  • automata theory — Body of physical and logical principles underlying the operation of any electromechanical device (an automaton) that converts information input in one form into another, or into some action, according to an algorithm. Norbert Wiener and Alan M.… …   Universalium

  • List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this …   Wikipedia

Share the article and excerpts

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

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.