Hopf algebra

bi-algebra, hyperalgebra

A graded module $ A $ over an associative-commutative ring $ K $ with identity, equipped simultaneously with the structure of an associative graded algebra $ \mu : \ A \otimes A \rightarrow A $ with identity (unit element) $ \iota : \ K \rightarrow A $ and the structure of an associative graded co-algebra $ \delta : \ A \rightarrow A \otimes A $ with co-identity (co-unit) $ \epsilon : \ A \rightarrow K $ , satisfying the following conditions:

1) $ \iota $ is a homomorphism of graded co-algebras;

2) $ \epsilon $ is a homomorphism of graded algebras;

3) $ \delta $ is a homomorphism of graded algebras.

Condition 3) is equivalent to:

3') $ \mu $ 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 $ A $ and $ B $ over $ K $ their tensor product $ A \otimes B $ is endowed with the natural structure of a Hopf algebra. Let $ A = \sum _ {n \in \mathbf Z} A _{n} $ be a Hopf algebra, where all the $ A _{n} $ are finitely-generated projective $ K $ - modules. Then $ A ^{*} = \sum _ {n \in \mathbf Z} A _{n} ^{*} $ , where $ A _{n} ^{*} $ is the module dual to $ A _{n} $ , endowed with the homomorphisms of graded modules $ \delta ^{*} : \ A ^{*} \otimes A ^{*} \rightarrow A ^{*} $ , $ \epsilon ^{*} : \ K \rightarrow A ^{*} $ , $ \mu ^{*} : \ A ^{*} \rightarrow A ^{*} \otimes A ^{*} $ , $ \iota ^{*} : \ A ^{*} \rightarrow K $ , is a Hopf algebra; it is said to be dual to $ A $ . An element $ x $ of a Hopf algebra $ A $ is called primitive if$$ \delta (x) = x \otimes 1 + 1 \otimes x. $$ The primitive elements form a graded subalgebra $ P _{A} $ in $ A $ under the operation$$ [x,\ y] = xy - (-1) ^{pq} yx, x \in A _{p} , y \in A _{q} . $$ If $ A $ is connected (that is, $ A _{n} = 0 $ for $ n < 0 $ , $ A _{0} = K \ $ ) and if $ K $ is a field of characteristic 0, then the subspace $ P _{A} $ generates the algebra $ A $ ( with respect to multiplication) if and only if the co-multiplication is graded commutative [2].

Examples.

1) For any graded Lie algebra $ \mathfrak g $ ( that is, a graded algebra that is a Lie superalgebra under the natural $ \mathbf Z _{2} $ - grading) the universal enveloping algebra $ U ( \mathfrak g ) $ becomes a Hopf algebra if one puts$$ \epsilon (x) = 0, \delta (x) = x \otimes 1 + 1 \otimes x, x \in \mathfrak g . $$ Here $ P _ {U ( \mathfrak g )} = \mathfrak g $ . If $ K $ is a field of characteristic 0, then any connected Hopf algebra $ A $ generated by primitive elements is naturally isomorphic to $ U (P _{A} ) $ ( see [2]).

2) Similarly, the structure of a Hopf algebra (with a trivial grading) is defined in the group algebra $ K [G] $ of an arbitrary group $ G $ .

3) The algebra of regular functions on an affine algebraic group $ G $ becomes a Hopf algebra (with trivial grading) if one defines the homomorphisms $ \delta $ and $ \epsilon $ by means of the multiplication $ G \times G \rightarrow G $ and the imbedding $ \{ e \} \rightarrow G $ , where $ e $ is the unit element of $ G $ ( see [3]).

4) Suppose that $ G $ is a path-connected $ H $ - space with multiplication $ m $ and unit element $ e $ and suppose that $ \Delta : \ G \rightarrow G \times G $ , $ \iota : \ \{ e \} \rightarrow G $ , $ p: \ G \rightarrow \{ e \} $ are defined by the formulas $ \Delta (a) = (a,\ a) $ , $ \iota (e) = e $ , $ p (a) = e $ , $ a \in G $ . If all cohomology modules $ H ^{n} (G,\ K) $ are projective and finitely generated, then the mappings $ \mu = \Delta ^{*} $ , $ \iota = p ^{*} $ , $ \delta = m ^{*} $ , $ \epsilon = \iota ^{*} $ induced in the cohomology, turn $ H ^{*} (G,\ K) $ into a graded commutative quasi-Hopf algebra. If the multiplication $ m $ is homotopy-associative, then $ H ^{*} (G ,\ K) $ is a Hopf algebra, and the Hopf algebra dual to it is the homology algebra $ H _{*} (G,\ K) $ , equipped with the mappings $ m _{*} $ , $ \iota _{*} $ , $ \Delta _{*} $ , $ p _{*} $ ( the Pontryagin algebra). If $ K $ is a field of characteristic 0, then the Pontryagin algebra is generated by primitive elements and is isomorphic to $ U ( \pi (G,\ K)) $ , where $ \pi (G,\ K) = \sum _ {i = 0} ^ \infty \pi _{i} (G) \otimes K $ is regarded as a graded Lie algebra under the Samelson product (see [2]).

The algebra $ H ^{*} (G,\ K) $ 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 $ K $ is a field of characteristic 0 and $ H ^{*} (G,\ K) $ is finite-dimensional. The structure of an arbitrary connected, graded, commutative quasi-Hopf algebra $ A $ subject to the condition $ \mathop{\rm dim}\nolimits \ A _{n} < \infty $ , $ n \in \mathbf Z $ , over a perfect field $ K $ of characteristic $ p $ is described by the following theorem (see [4]). The algebra $ A $ splits into the tensor product of algebras with a single generator $ x $ and the relation $ x ^{s} = 0 $ , where for $ p = 2 $ , $ s $ is a power of 2 or $ \infty $ , and for $ p \neq 2 $ , $ s $ is a power of $ p $ or $ \infty $ ( $ \infty $ for $ p = 0 $ ) if $ x $ has even degree, and $ s = 2 $ if the degree of $ x $ is odd. In particular, for $ p = 0 $ , $ A $ 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 $ A $ over a field $ K $ in which $ x ^{2} = 0 $ for any element $ x $ of odd degree and in which all elements of odd degree and all elements of even degree are decomposable, is the exterior algebra $ A = \land P _{A} $ ( see [2]). In particular, such are the cohomology algebra and the Pontryagin algebra of a connected compact Lie group over $ \mathbf R $ .

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 MR0174052 Zbl 0163.28202
[3]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[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 MR0051508 Zbl 0052.40001
[5]	S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009

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 $ A $ over $ K $ equipped with module mappings $ m: \ A \otimes A \rightarrow A $ , $ e : \ K \rightarrow A $ , $ \mu : \ A \rightarrow A \otimes A $ , $ \epsilon : \ A \rightarrow K $ such that

i) $ ( A ,\ m ,\ e ) $ is an associative algebra with unit;

ii) $ ( A ,\ \mu ,\ \epsilon ) $ is a co-associative co-algebra with co-unit;

iii) $ e $ is a homomorphism of co-algebras;

iv) $ \epsilon $ is a homomorphism of algebras;

v) $ m $ is a homomorphism of co-algebras.

This last condition is equivalent to:

v') $ \mu $ 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 $ ( A ,\ m ,\ e ,\ \mu ,\ \epsilon ) $ be a bi-algebra over $ K $ . An antipode for the bi-algebra is a module homomorphism $ \iota : \ A \rightarrow A $ such that

vi) $ m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon $ .

A bi-algebra with antipode $ \iota $ is called a Hopf algebra. A graded Hopf algebra is a graded bi-algebra with antipode $ \iota $ which is a homomorphism of graded modules.

Given a co-algebra $ ( C ,\ \mu _{C} ,\ \epsilon _{C} ) $ and an algebra $ ( A ,\ m _{A} ,\ e _{A} ) $ , the module $ \mathop{\rm Mod}\nolimits _{K} ( C ,\ A ) $ admits a convolution product, defined as follows$$ f \star g = m _{A} \circ ( f \otimes g ) \circ \mu _{C} . $$ In terms of this convolution product conditions vi) can be stated as

vi') $ \iota \star \mathop{\rm id}\nolimits = \mathop{\rm id}\nolimits \star \iota = e \circ \epsilon $ ,

where $ \mathop{\rm id}\nolimits : \ A \rightarrow A $ is the identity morphism of the bi-algebra $ A $ .

An additional example of a Hopf algebra is the following. Let $ F _{1} ( X ; \ Y ) \dots F _{n} ( X ; \ Y ) \in K [ [ X _{1} \dots X _{n} ; \ Y _{1} \dots Y _{n} ] ] $ be a formal group. Let $ A = K [ [ X _{1} \dots X _{n} ] ] $ . Identifying $ Y _{i} $ with $ 1 \otimes X _{i} \in A \widehat \otimes A $ , the $ F _{1} \dots F _{n} $ define a (continuous) algebra morphism $ \mu : \ A \rightarrow A \widehat \otimes A $ turning $ A $ into a bi-algebra. There is an antipode making $ A $ a Hopf algebra. It is called the contravariant bi-algebra or contravariant Hopf algebra of the formal group $ F $ . 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) 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 (h047970148.png)-dimensional quantum field theory (Les Houches, 1982)" , Elsevier (1984) MR782509