A quick comparison of Lie algebras and formal group laws

This post assumes that you are familiar with the definition of Lie group/algebra, and that you are comfortable with the Lazard ring. Note: This is less of an expository post and more of an unfinished question.

Why care about formal group laws? Well, we want to study smooth algebraic groups, but Lie algebras fail us in characteristic p (for example, $\frac{d}{dx}(x^p) = 0$), so, rather than a tangent bundle, we take something closer to a jet bundle.

Lie algebras are to smooth groups over $\mathbb{R}$ or $\mathbb{C}$ as formal group laws are to smooth algebraic groups over any ring $R$.

I want to apply this analogy! I want this deeply. I’m trying to puzzle out how to see if this analogy is deep or superficial. How deep does the rabbit hole go? Let’s look at an example.

Given the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, we might think of this as a deformation of the Symmetric algebra:

$$U_{\epsilon}(\mathfrak{g}) := T(\mathfrak{g})/(x \otimes y – y \otimes x – \epsilon[x, y])$$

$$\mathbb{C}[\mathfrak{g}^*] \simeq Symm[\mathfrak{g}] := T(\mathfrak{g})(x \otimes y – y \otimes x)$$

Action on all of this is the adjoint action, that is, the action which takes an element $g$ of a Lie group $G$ sends $X \mapsto gXg^{-1}$. Orbits of this action stratify the dual Lie algebra, and there is a symplectic form that lives on each orbit.

I want to think of the adjoint action as directly analogous to the compositional conjugation action on the spectrum of the Lazard ring (over a ring $R$). Continue reading A quick comparison of Lie algebras and formal group laws

Complex Analysis: Poles, Residues, and Child’s Drawings

Thanks to Laurens Gunnarsen for his superb pedagogy and for this amazing explanation on the incredible depth of connections springing from the Sperner lemma. All errors are mine not his. This started with a chain of events, sitting in on number theory seminars and encountering Abel’s differentials of the first and second kind, interest in the dessin, and led up to asking Laurens:

How do I understand poles and residues?

By understanding Riemann-Roch. But, first of all, you should know the zeroes and the poles of an analytic function, together with the residues of the function at each, pretty much tell you all you need to know about the function.

This is a little bit like saying that if you know the zeroes of a rational function — which is to say, the zeros of the polynomial that is its numerator — and the poles of a rational function — which is to say, the zeroes of the polynomial that is its denominator — then you basically know the rational function.  Continue reading Complex Analysis: Poles, Residues, and Child’s Drawings

Bordism with singularities construction of elliptic homology

This post assumes familiarity with the Landweber exact functor theorem, elliptic genera, and bordism theories.

An ongoing desire of mine is to geometrically approach elliptic spectra.

Note that I’m not talking about geometric cocycles for tmf. I’m talking about the vague goal of understanding elliptic spectra in their own right using “geometric” techniques (properties of the object that are invariant under a chosen collection of transformations of that object).

There is a presentation of an elliptic homology theory as a bordism theory with singularities outlined by Landweber in Elliptic Cohomology and Modular Forms (based on this paper which presents $H_*(-;\mathbb{Z})$ as a bordism theory with singularities).

This seemed like the beginning of the answer to my desire, but I now think that this construction is basically a less intuitive version of $MSO_*(-) \otimes_{MSO_*} R$, thinking about “tensoring out” classes in $MSO$ as coning them off. I’ll explain what I mean by “coning them off” in a bit, first let me outline the construction:

Outline:

  1. Start with an elliptic genus $\pi_*(MSO) \xrightarrow{\phi} R$
  2. Mod out the ring spectrum $MSO$ by $ker(\phi)$ to get a spectrum “$MSO/ker(\phi)$” whose homotopy groups are $R$
    i.e., construct $F: \pi_*(MSO/ker(\phi)) \to R$ s.t. $ker(F) = 0$
  3. Check that the spectrum $MSO/ker(\phi)$ is a ring spectrum

Let’s go through it! Continue reading Bordism with singularities construction of elliptic homology

$Pic(X)$ vs. $CP^\infty$

Thank you to Edward Frenkel for kindly explaining the difference between $CP^\infty$ and $Pic(X)$ (both classifying spaces of line bundles), and to Qiaochu Yuan for explaining why on earth $CP^\infty$ is the moduli space of line bundles over a point. Any errors are mine, not theirs.

As we saw in a Precursor to Characteristic Classes, $CP^\infty$ is the classifying space of complex line bundles over $X$.

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$CP^\infty$ is, in some sense, the moduli space of line bundles over a point. There’s only one isomorphism class of line bundles over a point — but then this one line bundle has automorphism group $C^\times$ (which is homotopy equivalent to $U(1)$).

Allow me to introduce you to something that looks a LOT like $CP^\infty$. Continue reading $Pic(X)$ vs. $CP^\infty$

A Precursor to Characteristic Classes

I’ll assume that you know what a line bundle is, and are comfortable with the following equivalences; if you aren’t familiar with the notation in these equivalences, John Baez might help. Note that integral cohomology := cohomology with coefficients in $\mathbb{Z}$.

$U(1) \simeq S^1 \simeq K(\mathbb{Z}, 1)$

$BU(1) \simeq CP^\infty \simeq K(\mathbb{Z}, 2)$

The aim of this post is to give you a taste of the beautiful world of characteristic classes and their intimate relationship to line bundles via the concrete example of how the second integral cohomology group of a space is actually the isomorphism classes of line bundles over that space.

That’s right! $H^2(X; Z) \simeq$ the isomorphism classes of (complex) line bundles over X.  It is in fact, a group homomorphism — the group operations being tensor product of line bundles and the usual addition on cohomology. This isn’t something that I understood at first glance. I mean, hot damn, it’s unexpectedly rich. Continue reading A Precursor to Characteristic Classes