[Axiom-mail] Re: Hopf Algebras

[Bertfried Fauser]

On Mon, 11 Nov 2002, root wrote:

> When Axiom comes out we should have a further discussion about
> possible contributions.

Dear Tim,

please keep me informed, but I guess it will be posted on the axiom-mail
group?

> I'm unfamiliar with Hopf Algebras but
> that is no surprise. Is there a paper or reference on the subject
> you can recommend?

The idea of a Hopf algebra came out of topology in the early 40ies, the
name was given by Milnor and Moore (see refs below) and refers to Heinz
Hopf`s pioneering work. However, they survived in the theory of
combinatorics for some decades. In physics, Hopf algebras became famous in
a special flavor called quantum groups in the late 80ies. You can look at
a Hopf algebra as a sort of generalized group (yes, group not algebra)
which has a pseudo inverse called antipode and allows to transport much
reasoning of group theory to the a general setting.

Technically there is one new operation:

-- Algebra   ==   Module + morphism A \otimes A --> A  ++ as common
-- Coalgebra == coModule + morphism A --> A \otimes A  ++ new

In computer science, you have already coproducts, if you think of anything
which generates a tree of possibilities. It is possible to build a
coalgebra theory on an equal footing with an algebra theory, there are
comodules, corepresentation, coideals, etc which however have sometimes
different properties than modules, ideals, representations of algebras.

-- Hopf algebra == Algebra + Coalgebra + compatibility laws -> Antipode

A Hopf algebra is hence in the same time an algebra and coalgebra, where
the algebra product is a coalgebra homomorphism and the coalgebra
coproduct is an algebra homomorphism and the existence of a map
S : module -> module is required, which is called antipode and is
a generalized inverse.

-- Convolution

Today it seems to be most promising to start with a convolution. That is
with a product say m : V \otimes V -> V and a coproduct say \Delta : U
--> U \otimes U. Given morphisms f,g : U ---> V, you can define an

-- convolution algebra with product * on morphisms

via the formula

f * g(x) := m ° ( f \otimes g) ° \Delta(x)  +++ You can drop the x savely!

One can prove that if product and coproduct are (co)associative and
(co)unital and the identity morphism has a convolutive inverse, than this
inverse is the antipode called S and the convolution is already a Hopf
algebra, i.e. Id^(-1) (w.r.t. *) = S

-- Quantum groups

in physics are Hopf algebras having special `nice` features, called
(quasi) triangular structures, which can be recast in categorial terms to
be based on naturalness and coherence of certain categorial maps. (In
physicist's terms, they have a sort of integrability condition which
guarantee that if you reach a result by different way's of reasoning in a
category then the result does not depend on the way you got it, which is
convenient and that you can identify several tensor modules)

-- Example: A group is a (trivial) Hopf algebra

let G be a group, * the group multiplication and define \Delta to be the
`diagonal map` \Delta : G --> G \otimes G, \Delta(g) = g \otimes g (just
doubling the element). One can show that the antipode S is an
anti-homomorphisms and is given in the group case by the inverse
S : G --> G, S(g) = g^(-1). A group action may be given as (using a switch
sw : G \otimes G --> G \otimes G, sw(g \otimes h ) = h \otimes g)

g > h =  m ° (m \otimes Id) ° (Id\otimes S) ° \Delta(g) \otimes h
      = g * h * g^(-1)

--> generalization. \Delta(g) = \sum g_[1] \otimes g_[2] which is no
longer diagonal.

-- Example: Universal enveloping Lie algebras are Hopf algebras

(Its generalizations are called sometimes 'matrix pseudo groups',
you end up with matrices which have non-commuting entries)
The product is the matrix product in a suitable representation (fundamental
representation) and the coproduct is given on generators x as
\Delta(x) = x \otimes 1 + 1 \otimes x and extended to the whole (graded)
module via the homomorphism properties. The antipode is given on
generators by S(x)=-x and also extended in a recursive way.

The extension of Lie symmetries was the main motivation in physics to study
this type of objects.

Given this short explanation you might judge yourself if you really need
to look for such mathematics. Some references are:

-- Classical texts

[1] Milnor, Moore, "On the structure of Hopf algebras", Ann. of Math.
81:211---264, 1965

[2] M.E. Sweedler, "Hopf algebras", W.A. Benjamin, INC, New York, 1996

--

if you are interested in combinatorics, you have to seek for texts of
Gian-Calro Rota and collaborators  or on Umbral calculus.

-- more recent

[3] S. Montgomery, "Hopf Algebras and their Actions on Rings", AMS
conference series CBMS, No. 82, 1993

[4] my habilitation, see xxx.lanl.gov math.QA/0202059 (partly physics)
    (gives also refs to geometry and Rota's work)

-- physics

[5] Shahn Majid, "Foundations of quantum group theory", Cambridge Univ.
Press, Cambridge, 1995


this is a subjective selection, but you might scan references of these
refs. to find more suitable reading.

greetings
BF.

% Bertfried Fauser          Fachbereich Physik    Fach M 678
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