Hopf algebra
bi-algebra, hyperalgebra
A graded module over an associative-commutative ring
with identity, equipped simultaneously with the structure of an associative graded algebra
with identity (unit element)
and the structure of an associative graded co-algebra
with co-identity (co-unit)
, satisfying the following conditions:
1) is a homomorphism of graded co-algebras;
2) is a homomorphism of graded algebras;
3) is a homomorphism of graded algebras.
Condition 3) is equivalent to:
3') is a homomorphism of graded co-algebras.
Sometimes the requirement that the co-multiplication is associative is discarded; such algebras are called quasi-Hopf algebras.
For any two Hopf algebras and
over
their tensor product
is endowed with the natural structure of a Hopf algebra. Let
be a Hopf algebra, where all the
are finitely-generated projective
-modules. Then
, where
is the module dual to
, endowed with the homomorphisms of graded modules
,
,
,
, is a Hopf algebra; it is said to be dual to
. An element
of a Hopf algebra
is called primitive if
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The primitive elements form a graded subalgebra in
under the operation
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If is connected (that is,
for
,
) and if
is a field of characteristic 0, then the subspace
generates the algebra
(with respect to multiplication) if and only if the co-multiplication is graded commutative [2].
Examples.
1) For any graded Lie algebra (that is, a graded algebra that is a Lie superalgebra under the natural
-grading) the universal enveloping algebra
becomes a Hopf algebra if one puts
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Here . If
is a field of characteristic 0, then any connected Hopf algebra
generated by primitive elements is naturally isomorphic to
(see [2]).
2) Similarly, the structure of a Hopf algebra (with a trivial grading) is defined in the group algebra of an arbitrary group
.
3) The algebra of regular functions on an affine algebraic group becomes a Hopf algebra (with trivial grading) if one defines the homomorphisms
and
by means of the multiplication
and the imbedding
, where
is the unit element of
(see [3]).
4) Suppose that is a path-connected
-space with multiplication
and unit element
and suppose that
,
,
are defined by the formulas
,
,
,
. If all cohomology modules
are projective and finitely generated, then the mappings
,
,
,
induced in the cohomology, turn
into a graded commutative quasi-Hopf algebra. If the multiplication
is homotopy-associative, then
is a Hopf algebra, and the Hopf algebra dual to it is the homology algebra
, equipped with the mappings
,
,
,
(the Pontryagin algebra). If
is a field of characteristic 0, then the Pontryagin algebra is generated by primitive elements and is isomorphic to
, where
is regarded as a graded Lie algebra under the Samelson product (see [2]).
The algebra in Example 4) was first considered by H. Hopf in [1], who showed that it is an exterior algebra with generators of odd degrees if
is a field of characteristic 0 and
is finite-dimensional. The structure of an arbitrary connected, graded, commutative quasi-Hopf algebra
subject to the condition
,
, over a perfect field
of characteristic
is described by the following theorem (see [4]). The algebra
splits into the tensor product of algebras with a single generator
and the relation
, where for
,
is a power of 2 or
, and for
,
is a power of
or
(
for
) if
has even degree, and
if the degree of
is odd. In particular, for
,
is the tensor product of an exterior algebra with generators of odd degree and an algebra of polynomials with generators of even degrees. On the other hand, every connected Hopf algebra
over a field
in which
for any element
of odd degree and in which all elements of odd degree and all elements of even degree are decomposable, is the exterior algebra
(see [2]). In particular, such are the cohomology algebra and the Pontryagin algebra of a connected compact Lie group over
.
References
[1] | H. Hopf, "Ueber die Topologie der Gruppenmannigfaltigkeiten und ihrer Verallgemeinerungen" Ann. of Math. (2) , 42 (1941) pp. 22–52 |
[2] | J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" Ann. of Math. (2) , 81 : 2 (1965) pp. 211–264 |
[3] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
[4] | A. Borel, "Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts" Ann. of Math. , 57 (1953) pp. 115–207 |
[5] | S. MacLane, "Homology" , Springer (1963) |
Comments
Terminology concerning Hopf algebras and bi-algebras is not yet quite standardized. However, the following nomenclature (and notation) seems to be on the way of being universally accepted.
A bi-algebra is a module over
equipped with module mappings
,
,
,
such that
i) is an associative algebra with unit;
ii) is a co-associative co-algebra with co-unit;
iii) is a homomorphism of co-algebras;
iv) is a homomorphism of algebras;
v) is a homomorphism of co-algebras.
This last condition is equivalent to:
v') is a homomorphism of algebras.
A grading is not assumed to be part of the definition. If there is a grading and every morphism under consideration is graded, then one speaks of a graded bi-algebra.
Let be a bi-algebra over
. An antipode for the bi-algebra is a module homomorphism
such that
vi) .
A bi-algebra with antipode is called a Hopf algebra. A graded Hopf algebra is a graded bi-algebra with antipode
which is a homomorphism of graded modules.
Given a co-algebra and an algebra
, the module
admits a convolution product, defined as follows
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In terms of this convolution product conditions vi) can be stated as
vi') ,
where is the identity morphism of the bi-algebra
.
An additional example of a Hopf algebra is the following. Let be a formal group. Let
. Identifying
with
, the
define a (continuous) algebra morphism
turning
into a bi-algebra. There is an antipode making
a Hopf algebra. It is called the contravariant bi-algebra or contravariant Hopf algebra of the formal group
. Note that here the completed tensor product is used.
Hopf algebras, under the name quantum groups, and related objects have also become important in physics; in particular in connection with the quantum inverse-scattering method [a3], [a4].
References
[a1] | E. Abe, "Hopf algebras" , Cambridge Univ. Press (1977) |
[a2] | M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) |
[a3] | V.G. Drinfel'd, "Quantum groups" , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , 1 , Amer. Math. Soc. (1987) pp. 798–820 |
[a4] | L.D. Faddeev, "Integrable models in (![]() |
Hopf algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_algebra&oldid=12897