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Separable algebras

Let k be a commutative ring and let A be a k-algebra. In this post we’ll investigate a condition on A which generalizes the condition that A is a finite separable field extension (in the case that k is a field). It can be stated in many equivalent ways, as follows. Below, “bimodule” always means “bimodule over k.”

Definition-Theorem: The following conditions on A are all equivalent, and all define what it means for A to be a separable k-algebra:

  1. A is projective as an (A, A)-bimodule (equivalently, as a left A \otimes_k A^{op}-module).
  2. The multiplication map A \otimes_k A^{op} \ni (a, b) \xrightarrow{m} ab \in A has a section as an (A, A)-bimodule map.
  3. A admits a separability idempotent: an element p \in A \otimes_k A^{op} such that m(p) = 1 and ap = pa for all a \in A (which implies that p^2 = p).

(Edit, 3/27/16: Previously this definition included a condition involving Hochschild cohomology, but it’s debatable whether what I had in mind is the correct definition of Hochschild cohomology unless k is a field or A is projective over k. It’s been removed since it plays no role in the post anyway.)

When k is a field, this condition turns out to be a natural strengthening of the condition that A is semisimple. In general, loosely speaking, a separable k-algebra is like a “bundle of semisimple algebras” over \text{Spec } k.

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Previously we suggested that if we think of commutative algebras as secretly being functions on some sort of spaces, we should correspondingly think of cocommutative coalgebras as secretly being distributions on some sort of spaces. In this post we’ll describe what these spaces are in the language of algebraic geometry.

Let D be a cocommutative coalgebra over a commutative ring k. If we want to make sense of D as defining an algebro-geometric object, it needs to have a functor of points on commutative k-algebras. Here it is:

\displaystyle D(-) : \text{CAlg}(k) \ni R \mapsto |D \otimes_k R| \in \text{Set}.

In words, the functor of points of a cocommutative coalgebra D sends a commutative k-algebra R to the set |D \otimes_k R| of setlike elements of D \otimes_k R. In the rest of this post we’ll work through some examples.

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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.

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The principle of maximum entropy asserts that when trying to determine an unknown probability distribution (for example, the distribution of possible results that occur when you toss a possibly unfair die), you should pick the distribution with maximum entropy consistent with your knowledge.

The goal of this post is to derive the principle of maximum entropy in the special case of probability distributions over finite sets from

We’ll do this by deriving an arguably more fundamental principle of maximum relative entropy using only Bayes’ theorem.

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Monads are idempotents

It’s common to think of monads as generalized algebraic theories; the most familiar examples, such as the monads on \text{Set} encoding groups, rings, and so forth, have this flavor. However, this intuition is really only appropriate for certain monads (e.g. finitary monads on \text{Set}, which are the same thing as Lawvere theories).

It’s also common to think of monads as generalized monoids; previously we discussed why this was a reasonable thing to do.

Today we’ll discuss a different intuition: monads are (loosely) categorifications of idempotents.

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Lie algebras are groups

Once upon a time I imagine people were very happy to think of Lie algebras as “infinitesimal groups,” but presumably when infinitesimals fell out of favor this interpretation did too. In this post I’ll record an observation that can justify thinking of Lie algebras as groups in a strong sense: they are group objects in a certain category which can be interpreted as a category of “infinitesimal spaces.”

Below we work throughout over a field of characteristic zero.

For starters, the universal enveloping algebra functor \mathfrak{g} \mapsto U(\mathfrak{g}), which a priori takes values in algebras (it’s left adjoint to the forgetful functor from algebras to Lie algebras), in fact takes values in Hopf algebras. This upgraded functor continues to be a left adjoint, although the forgetful functor is less obvious. Given a Hopf algebra H, its primitive elements are those elements x \in H satisfying

\Delta x = x \otimes 1 + 1 \otimes x

where \Delta is the comultiplication. The primitive elements of a Hopf algebra form a Lie algebra, and this gives a forgetful functor from Hopf algebras to Lie algebras whose left adjoint is the upgraded universal enveloping algebra functor.

The key observation is that this upgraded functor \mathfrak{g} \to U(\mathfrak{g}) is fully faithful; that is, there is a natural bijection between Lie algebra homomorphisms \mathfrak{g} \to \mathfrak{h} and Hopf algebra homomorphisms U(\mathfrak{g}) \to U(\mathfrak{h}). This is more or less equivalent to the claim that the natural inclusion \mathfrak{g} \to U(\mathfrak{g}) induces an isomorphism from \mathfrak{g} to the Lie algebra of primitive elements of U(\mathfrak{g}), which can be proven using the PBW theorem.

Hence Lie algebras embed as a full subcategory of Hopf algebras; that is, they can be thought of as Hopf algebras satisfying certain properties, rather than having extra structure (in the nLab sense). What are these properties? For starters, they are all cocommutative. This is important because cocommutative Hopf algebras are group objects in the category of cocommutative coalgebras (this is not true with “cocommutative” dropped!), which in turn can be interpreted as a category of infinitesimal spaces. (For example, this category is cartesian closed, and in particular distributive.)

Hence Lie algebras are group objects in cocommutative coalgebras satisfying some property (for example, “conilpotence”; see Theorem 3.8.1 here).

Previously we learned how to count the finite index subgroups of the modular group \Gamma = PSL_2(\mathbb{Z}). The worst thing about that post was that it didn’t include any pictures of these subgroups. Today we’ll fix that.

The pictures in this post can be interpreted in at least two ways. On the one hand, they are graphs of groups in the sense of Bass-Serre theory, and on the other hand, they are also dessin d’enfants (for the rest of this post abbreviated to “dessins”) in the sense of Grothendieck. But you don’t need to know that to draw and appreciate them.

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