nLab complete Hopf algebra

Definition

A complete Hopf algebra is a complete augmented algebra $A$ equipped with a diagonal map $\Delta\colon A\to A\otimes A$ that is a morphism of complete augmented algebras, is coassociative and cocommutative and has the augmentation map as a counit map. (See the article Hopf algebra for a definition of these terms.)

Examples

The completion of a Hopf algebra as an augmented algebra (see complete augmented algebra for a construction) is a complete Hopf algebra.

In particular, the completion of the group algebra of a group and the universal enveloping algebra of a Lie algebra is a complete Hopf algebra.

Properties

The monoidal product of the underlying complete augmented algebras of complete Hopf algebras can be equipped in the obvious way with a structure of a complete Hopf algebra. The resulting monoidal structure $\hat\otimes$ on complete Hopf algebras is the cartesian monoidal structure, with the projection maps $A\otimes A'\to A$ and $A\otimes A'\to A'$ induced by the augmentation maps of $A'$ respectively $A$.

The associated graded functor sends complete Hopf algebras to graded Hopf algebras.

Constructions

The forgetful functor from the category of complete Hopf algebras to the category of groups sends a complete Hopf algebra $A$ to the group

$$G(A)=\{x\in 1+\bar A\mid \Delta x=x\hat\otimes x\}$$

of group-like elements in $A$. Its left adjoint functor sends a group to the completion of its group algebra. (See Proposition A.2.5 in Quillen.)

In the case $k=\mathbf{Q}$, the forgetful functor is fully faithful and its essential image comprises precisely Malcev groups. Thus, the category of rational complete Hopf algebras is equivalent to the category of Malcev groups. (Theorem A.3.3 in Quillen.)

The monad induced by the adjunction between groups and complete Hopf algebras is known as the Malcev completion of groups.

The forgetful functor from the category of complete Hopf algebras to the category of Lie algebras sends a complete Hopf algebra $A$ to the Lie algebra

$$P(A)=\{x\in \bar A\mid \Delta x=x\hat\otimes 1+1\hat\otimes x\}$$

of primitive elements in $A$. Its left adjoint functor sends a Lie algebra to the completion of its universal enveloping algebra. (See Proposition A.2.5 in Quillen.)

In the case $k$ has characteristic 0, the forgetful functor is fully faithful and its essential image comprises precisely Malcev Lie algebras. Thus, the category of complete Hopf algebras over a field of characteristic 0 is equivalent to the category of Malcev Lie algebras over the same field. (Theorem A.3.3 in Quillen.)

Properties in the case of zero characteristic

Assume that the field $k$ has characteristic 0.

Then the exponential map

$$\exp\colon P(A)\to G(A)$$

can be defined using formal power series for any complete Hopf algebra $A$.

The exponential map is an isomorphism of sets, both of which have natural filtrations induced from $A$, which turn them into a complete filtered Lie algebra and a complete filtered group respectively.

The exponential map induces an isomorphism of the associated graded Lie algebra over integers. For the filtered group $G(A)$, the Lie bracket on its associated graded abelian group is given by the commutator. The associated graded abelian group of $G(A)$ is also equipped with a structure of a $k$-module transferred from the associated graded $k$-module of $P(A)$ via the exponential isomorphism.

Related concepts

• augmented algebra

• complete augmented algebra

• Malcev group

• Malcev Lie algebra

• Malcev completion

References

• Daniel Quillen, Rational homotopy theory, Annals of Mathematics 90:2 (1969), 205. doi.