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Coalgebras of distributions

Mathematicians are very fond of thinking about algebras. In particular, it’s common to think of commutative algebras as consisting of functions of some sort on spaces of some sort.

Less commonly, mathematicians sometimes think about coalgebras . In general it seems that mathematicians find these harder to think about, although it’s sometimes unavoidable, e.g. when discussing Hopf algebras. The goal of this post is to describe how to begin thinking about cocommutative coalgebras as consisting of  distributions of some sort on spaces of some sort.

Functions vs. distributions

Distributions are typically defined as being duals (spaces of continuous linear functionals) to topological vector spaces of functions. Loosely speaking, a distribution is something you can integrate a class of functions against; it’s a kind of generalized measure.

For example, the dual of the space Coalgebras of distributions of continuous functions on a compact Hausdorff space Coalgebras of distributions (with the sup norm topology) is a space of (signed)  Radon measures on Coalgebras of distributions . A class of examples closer to the examples we’ll be considering, although it involves more technicalities than we’ll need, is the dual of the space Coalgebras of distributions of smooth functions on a smooth manifold Coalgebras of distributions (with the  Fréchet topology ), which can be thought of as “distributions with compact support” on Coalgebras of distributions .

The simplest examples of distributions are the Dirac delta distributions, definable in great generality: as linear functionals on spaces of functions they are precisely the evaluation functionals

Coalgebras of distributions .

When we take duals to spaces of smooth functions, as opposed to continuous functions, we get more interesting distributions “supported at a point” given by taking derivatives. For example, on Coalgebras of distributions , at every point Coalgebras of distributions there are linear functionals on Coalgebras of distributions given by

Coalgebras of distributions .

These distributions are named using derivative notation because they are the distributional derivatives of Coalgebras of distributions .

The two most important things to keep in mind about the difference between functions and distributions is the following.

  1. Functions pull back, while distributions push forward.
  2. Functions form commutative algebras, while distributions form cocommutative coalgebras.

These points are closely related: the multiplication on functions resp. the comultiplication on functions, can be described using pullback resp. pushforward along the diagonal map

Coalgebras of distributions .

Namely, because we can multiply functions on Coalgebras of distributions by functions on Coalgebras of distributions to get functions on Coalgebras of distributions , for any reasonable notion of functions Coalgebras of distributions we get a dual map

Coalgebras of distributions

giving the multiplication on functions.

The situation for distributions is similar but less straightforward: if Coalgebras of distributions is any reasonable notion of distributions we get a map

Coalgebras of distributions

To get a comultiplication from this we’d like for there to be an isomorphism, or at least a map, from Coalgebras of distributions to Coalgebras of distributions . Unfortunately, the map that exists usually goes in the other direction, and usually will not be an isomorphism unless Coalgebras of distributions is some kind of completed tensor product.

Nevertheless, in some examples, and/or with the right modified notion of tensor product, the required maps do exist and we do get a comultiplication on distributions.

In addition to comultiplication, coalgebras also need a counit. In the case of distributions on spaces this counit comes from pushing forward along the unique map Coalgebras of distributions , getting a map

Coalgebras of distributions

which, if we think of distributions as generalized measures, computes the “total measure” of a measure.

The diagonal

The appearance of the diagonal map above can be put into a more abstract context.Recallthat in any category Coalgebras of distributions with finite products, every object is canonically a cocommutative comonoid in a unique way, via the diagonal map

Coalgebras of distributions .

A typical example for us will be Coalgebras of distributions , and in general we’ll want to think of Coalgebras of distributions as a category of “spaces.” We can get both commutative monoids and cocommutative comonoids out of diagonal maps as follows.

If Coalgebras of distributions is a contravariant functor out of Coalgebras of distributions (describing a notion of “functions”) to a symmetric monoidal category Coalgebras of distributions (typically something like Coalgebras of distributions ) which is lax symmetric monoidal in the sense that it is equipped with natural transformations

Coalgebras of distributions

compatible with symmetries (plus some unit stuff), then pulling back along the diagonal endows each Coalgebras of distributions with the structure of a commutative monoid in Coalgebras of distributions .

Example. If Coalgebras of distributions , then we can take Coalgebras of distributions to consist of all functions Coalgebras of distributions , where Coalgebras of distributions is the underlying field. If Coalgebras of distributions , then Coalgebras of distributions is even symmetric monoidal in the sense that the natural transformations above are isomorphisms.

Dually, if Coalgebras of distributions is a covariant functor out of Coalgebras of distributions (describing a notion of “distributions”) to a symmetric monoidal category Coalgebras of distributions which is oplax symmetric monoidal in the sense that it is equipped with natural transformations

Coalgebras of distributions

compatible with symmetries (plus unit stuff as above), then pushing forward along the diagonal endows each Coalgebras of distributions with the structure of a cocommutative comonoid in Coalgebras of distributions .

Example. If Coalgebras of distributions , then we can take Coalgebras of distributions to consist of the free Coalgebras of distributions -vector space Coalgebras of distributions on Coalgebras of distributions , where Coalgebras of distributions is the underlying field. Without any finiteness hypotheses, this is even symmetric monoidal.

Sets as coalgebras

Let’s slightly generalize the construction above. Let Coalgebras of distributions be a commutative ring (in fact we could take a commutative semiring here). Then we have a free Coalgebras of distributions -module functor from sets to Coalgebras of distributions -modules. The above construction shows that this functor can be regarded as taking values in cocommutative coalgebras over Coalgebras of distributions , so in fact we have a functor

Coalgebras of distributions .

At this point it will be convenient to introduce the following definition.

Definition:An element Coalgebras of distributions of a coalgebra Coalgebras of distributions (where Coalgebras of distributions is the comultiplication and Coalgebras of distributions is the counit) is  setlike if Coalgebras of distributions and Coalgebras of distributions . If Coalgebras of distributions is a coalgebra, we’ll write Coalgebras of distributions for its set of setlike elements.

(The more common term is grouplike , but that term is really only appropriate to the case of Hopf algebras, since in that case the setlike elements form a group. Here the setlike elements only form a set.)

Now we can describe Coalgebras of distributions , as a coalgebra, as being freely generated by setlike elements. Thinking in terms of distributions, setlike elements correspond to Dirac distributions, and so it’s reasonable to think of them as the “points” of a coalgebra, or more precisely of a hypothetical space on which the coalgebra is distributions.

Proposition:The functor Coalgebras of distributions from sets to coalgebras above has a right adjoint sending a coalgebra Coalgebras of distributions to its set Coalgebras of distributions of setlike elements.

Proof. We want to show that if Coalgebras of distributions is a set and Coalgebras of distributions is a coalgebra, we have a natural bijection

Coalgebras of distributions .

But this is clear from the observation that Coalgebras of distributions is a free Coalgebras of distributions -module on setlike elements, from which it follows that a coalgebra homomorphism Coalgebras of distributions is uniquely and freely determined by what it does to each element Coalgebras of distributions . These elements must be sent to some setlike element of Coalgebras of distributions and can be sent to any such element. Coalgebras of distributions

In praticular, the functor Coalgebras of distributions is represented by the coalgebra Coalgebras of distributions (of “distributions on a point”).

Lemma:Suppose Coalgebras of distributions has no nontrivial idempotents (that is, it is a connected ring ). Then the setlike elements of Coalgebras of distributions are precisely the elements Coalgebras of distributions : that is, the unit Coalgebras of distributions of the above adjunction is an isomorphism.

Proof. Suppose Coalgebras of distributions is a setlike element. Then

Coalgebras of distributions

must be equal to

Coalgebras of distributions

which happens if and only if Coalgebras of distributions if Coalgebras of distributions and Coalgebras of distributions otherwise. The counit condition is

Coalgebras of distributions .

Altogether, the condition that Coalgebras of distributions is primitive is precisely the condition that the elements Coalgebras of distributions are a complete set of orthogonal idempotents in Coalgebras of distributions . Since Coalgebras of distributions has no nontrivial idempotents by assumption, each Coalgebras of distributions is equal to either Coalgebras of distributions or Coalgebras of distributions . Since they are orthogonal (meaning Coalgebras of distributions if Coalgebras of distributions ), at most one of them is equal to Coalgebras of distributions . And since they sum to Coalgebras of distributions , exactly one of them is equal to Coalgebras of distributions . Hence our setlike element is some Coalgebras of distributions . Coalgebras of distributions

The correct statement without the hypothesis that Coalgebras of distributions is connected, which is not hard to extract from the above argument, is that the setlike elements of Coalgebras of distributions in general correspond to functions from the set of connected components of Coalgebras of distributions to Coalgebras of distributions with finite image, or equivalently to continuous functions from the Pierce spectrum Coalgebras of distributions to Coalgebras of distributions .

Corollary:Let Coalgebras of distributions have no nontrivial idempotents. Then the functor Coalgebras of distributions is an equivalence of categories from sets to cocommutative coalgebras over Coalgebras of distributions which are free on setlike elements.

In other words, as a slogan, sets are coalgebras of Dirac deltas.

Proof. We showed that the unit of the adjunction between sets and coalgebras is an isomorphism on sets. In general, an adjunction restricts to an equivalence of categories between the subcategories on which the unit resp. the counit of the adjunction are isomorphisms. So it remains to determine for which coalgebras the counit of the adjunction is an isomorphism. Explicitly, the counit is the natural map

Coalgebras of distributions

from the free Coalgebras of distributions -module on the setlike elements of a coalgebra Coalgebras of distributions to Coalgebras of distributions . If this is an isomorphism, then Coalgebras of distributions must in particular be free on some setlike elements. Conversely, if Coalgebras of distributions is free on setlike elements, then the lemma above shows that Coalgebras of distributions naturally, so that Coalgebras of distributions is an isomorphism. Coalgebras of distributions

This equivalence induces an equivalence between groups and cocommutative Hopf algebras over Coalgebras of distributions which are free (as modules) on setlike (here “grouplike”) elements.

Beyond Dirac deltas

We’ve said a lot about setlike elements of coalgebras, or equivalently about Dirac delta distributions. But coalgebras have lots of other kinds of elements in general. For example, if Coalgebras of distributions is a Lie algebra, its universal enveloping algebra Coalgebras of distributions has a natural comultiplication given by extending

Coalgebras of distributions

where Coalgebras of distributions ; that is, each Coalgebras of distributions is primitive. In a geometric story about distributions, where do the primitives?

The first observation is that in an arbitrary coalgebra there isn’t an element called Coalgebras of distributions , so coalgebras don’t have a notion of primitive element. What makes the element Coalgebras of distributions special is that it is in fact the unique setlike element: it satisfies Coalgebras of distributions and is the only element of Coalgebras of distributions with this property. So whatever primitivity means, geometrically it has something to do with a fixed setlike element, or in distributional terms with a fixed Dirac delta.

Definition:Let Coalgebras of distributions be a setlike element of a coalgebra Coalgebras of distributions . An element Coalgebras of distributions is primitive with respect to Coalgebras of distributions if

Coalgebras of distributions

and Coalgebras of distributions .

We can get a big hint about what this definition means by going back to the example of distributions coming from taking the dual of the space of smooth functions Coalgebras of distributions . Consider the distribution

Coalgebras of distributions .

How does comultiplication act on this distribution? To answer that question we need to see what this distribution does to a product Coalgebras of distributions of functions (since this describes the action of the distribution on at least a dense subspace of the pullback of functions along the diagonal map Coalgebras of distributions ). The answer, using the product rule, is that

Coalgebras of distributions .

This gives that

Coalgebras of distributions

and tells us that primitivity is a reflection of the Leibniz rule for derivations: saying that an element Coalgebras of distributions is primitive with respect to a setlike element Coalgebras of distributions means that if Coalgebras of distributions is a “point,” or more precisely a Dirac delta at a point, then Coalgebras of distributions is a “directional derivative” in a tangent direction at that point. Similarly, computing the pushforward to a point means differentiating constant functions (which are the functions pulled back from a point), which gives zero.

More formally, we can say the following.

Theorem:Let Coalgebras of distributions be a setlike element of a cocommutative coalgebra Coalgebras of distributions over Coalgebras of distributions , and let Coalgebras of distributions be an arbitrary element. Then Coalgebras of distributions is primitive with respect to Coalgebras of distributions iff Coalgebras of distributions is a setlike element of Coalgebras of distributions .

Proof. Computation. Coalgebras of distributions

Intuitively, Coalgebras of distributions is primitive with respect to Coalgebras of distributions iff both Coalgebras of distributions and Coalgebras of distributions are “points,” where the Coalgebras of distributions indicates that they are “infinitesimally close” points.

The fact that Coalgebras of distributions , as a Hopf algebra, is generated by primitive elements can be interpreted geometrically as saying that it corresponds to distributions “supported at a point.” In fact it is possible to describe Coalgebras of distributions as distributions supported at the identity on a Lie group Coalgebras of distributions with Lie algebra Coalgebras of distributions .

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