Last update: 22 October 2012
The basic material on completions given in §1 can be found in many books, in particular, [AMa1969] Chapt 10. The book [SSt1993] has a comprehensive treatment of deformation theory. Theorem (2.6) is stated and proved in [SSt1993] Prop. 11.3.1.
We will be working with algebras over the ring of formal power series in a variable with coefficients in A typical element of which is not in is the element
The ring is just extended a little bit so that some nice elements that we want to write down, like are in
An algebra over is a vector space over i.e. a free which has a multiplication and an identity which satisfy the conditions in I (1.1) If is an algebra over then we can extend coefficients and get a new algebra which is over But sometimes this new algebra is not quite big enough so we need to extend it a little bit and work with the completion which contains all the nice elements that we want to write down.
Continuing in thie vein we will want to consider the tensor product Again, this algebra is not quite big enough and we extend it to get a slightly bigger object so that all the elements we are are available.
If is an algebra over then set
of formal power series with coefficients in is the completion of the the in the topology. The extension of the multiplication in gives the structure of a The ring is, in general larger than For each element the element
is a well defined element of
Let be a field and let be an indeterminate. The ring is a local ring with unique maximal ideal Let be a The sets
form a basis for a topology on called the topology. Define a map by
where is a real number and is the largest nonnegative integer such that Then is a metric on which generates the topology.
Let be a The completion of the metric space is a metric space which contains in a natural way and which has a natural structure. The completion of is defined in the usual way, as a set of equivalence classes of Cauchy sequences of elements of Let us review this construction.
A sequence of elements in is a Cauchy sequence in the topology if for every positive integer there exists a positive integer such that
i.e. is "divisible" by for all Two Cauchy sequences and are equivalent if the sequence converges to 0, i.e.
The set of all equivalence classes of Cauchy sequences in is the completion of
The completion is a where the operations are determined by
where and are Cauchy sequences with elements in and Define a map
i.e. is the equivalence class of the sequence such that for all This map is injective and thus we can view as a submodule of
We are going to make the quantum group by deforming the enveloping algebra of a complex simple Lie algebra as a Hopf algebra. This last condition is important because the enveloping algebra does not have any deformations as an algebra.
Assume that is a Hopf algebra over Let denote the completion of in the topology. A deformation of as a Hopf algebra is a tuple where
are maps which are continuous in the topology, satisfy axioms (1) - (7) in the definition of a Hopf algebra, and can be written in the form
where, for each positive integer
are maps which are extended first and then to the completion. We shall abuse language (only slightly) and call a Hopf algebra over
Two Hopf algebra deformations and of a Hopf algebra are equivalent if there is an isomorphism
of complete Hopf algebras over which can be written in the form
such that, for each positive integer is a map which is extended to and then to the completion
Let be a Hopf algebra. The trivial deformation of as a Hopf algebra is the Hopf algebra over such that and (extended to
Assume that is an algebra over Let denote the completion of in the topology. A deformation of as an algebra is a tuple where
are maps which are continuous in the topology, satisfy the axioms the definition of an algeebra (see I (1.1)) and can be written in the form
where, for each positive integer
are maps which are extended first and then to the completion. We shall abuse language (only slightly) and call an algebra over
This definition is exactly like the definition of a deformation as a Hopf algebra in (2.2) above except that we only need to start with an algebra and we only require the result to be an algebra. We can define equivalence of deformations as algebras in exactly the same way that we defined them for deformations as Hopf algebras except that we only require the isomorphism to be an algebra isomorphism instead of a Hopf algebra isomorphism.
Let be an algebra. The trivial deformation of as an algebra is the algebra over such that and (extended to The deformation of the quantum group given in V (1.3) is even more incredible if one keeps the following theorem in mind.
Let be a finite dimensional complex simple Lie algebra and let be the enveloping algebra of Then has no deformations as an algebra (up to equivalence of deformations).
In other words, all deformations of as an algebra are equivalent to the trivial deformation of
This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.