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
The primitive elements form a graded subalgebra in under the operation
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 .
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
Here . If is a field of characteristic 0, then any connected Hopf algebra generated by primitive elements is naturally isomorphic to (see ).
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 ).
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 ).
The algebra in Example 4) was first considered by H. Hopf in , 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 ). 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 ). In particular, such are the cohomology algebra and the Pontryagin algebra of a connected compact Lie group over .
|||H. Hopf, "Ueber die Topologie der Gruppenmannigfaltigkeiten und ihrer Verallgemeinerungen" Ann. of Math. (2) , 42 (1941) pp. 22–52|
|||J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" Ann. of Math. (2) , 81 : 2 (1965) pp. 211–264 MR0174052 Zbl 0163.28202|
|||A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201|
|||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 MR0051508 Zbl 0052.40001|
|||S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009|
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
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
In terms of this convolution product conditions vi) can be stated as
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.
|[a1]||E. Abe, "Hopf algebras" , Cambridge Univ. Press (1977) MR1857062 MR0594432 MR0321962 Zbl 0476.16008 Zbl 0236.14021|
|[a2]||M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) MR0506881 MR0463184 Zbl 0454.14020|
|[a3]||V.G. Drinfel'd, "Quantum groups" , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , 1 , Amer. Math. Soc. (1987) pp. 798–820 Zbl 0667.16003|
|[a4]||L.D. Faddeev, "Integrable models in ()-dimensional quantum field theory (Les Houches, 1982)" , Elsevier (1984) MR782509|
Hopf algebra. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hopf_algebra&oldid=21981