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complete Hopf algebra

Definition

A complete Hopf algebra is a complete augmented algebra AA equipped with a diagonal map Δ:AAA\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 AAAA\otimes A'\to A and AAAA\otimes A'\to A' induced by the augmentation maps of AA' respectively AA.

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 AA to the group

G(A)={x1+A¯Δx=x^x}G(A)=\{x\in 1+\bar A\mid \Delta x=x\hat\otimes x\}

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

In the case k=Qk=\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 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 AA to the Lie algebra

P(A)={xA¯Δx=x^1+1^x}P(A)=\{x\in \bar A\mid \Delta x=x\hat\otimes 1+1\hat\otimes x\}

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

In the case kk 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 Quillen.)

Properties in the case of zero characteristic

Assume that the field kk has characteristic 0.

Then the exponential map

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

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

The exponential map is an isomorphism of sets, both of which have natural filtrations induced from AA, 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)G(A), the Lie bracket on its associated graded abelian group is given by the commutator. The associated graded abelian group of G(A)G(A) is also equipped with a structure of a kk-module transferred from the associated graded kk-module of P(A)P(A) via the exponential isomorphism.

References

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

Last revised on March 28, 2021 at 02:51:35. See the history of this page for a list of all contributions to it.