Lie groups – Francis Bischoff

In my last post, I told you how to encode the zero locus of a polynomial function $f: V \to E$ in terms of an $L_{\infty}$ -algebra structure on $L = V[-1] \oplus E[-2]$ , where $V$ lies in degree $+1$ and $E$ lies in degree $+2$ . Namely, we simply defined the $n-$ ary bracket on $L$ to be the $n^{th}$ Taylor coefficient of $f.$ This gave us one of the simplest examples of a derived manifold: the derived vanishing locus of $f .$ It also illustrated a simple ‘principle’ of derived algebraic geometry: if the equations defining a space are not independent, then don’t impose them. Instead treat the equations as geometric spaces in their own right. This is useful in part because it allows us to avoid dealing directly with the space defined by the equations, which can often be quite pathological.

In this post, I want to discuss the other side of the story: quotients. What if we are trying to define a space by imposing an equivalence relation that fails to be independent in some way? Again the result can be quite pathological and the solution to this problem is, once again, to do nothing: just don’t take the quotient and treat the equivalence relation as a geometric space in its own right. This will lead us to Lie groupoids and then to stacks, which are Lie groupoids up to Morita equivalence. But what I’m really aiming towards is the infinitesimal counterpart to a Lie groupoid, which is called a Lie algebroid. After reviewing the definition I will explain that, by taking the Taylor coefficients of the structure maps, a Lie algebroid can locally be encoded by an $L_{\infty}$ -algebra, this time concentrated in degrees $0$ and $+1$ . This $L_{\infty}$ -algebra, considered up to quasi-isomorphisms, encodes the formal quotient stack associated to the Lie algebroid. This post will be a little longer and a bit less concrete than the last one, but I still hope that it will help demystify both stacks and $L_{\infty}$ -algebras.

Continue reading →

Consider the exponential map for matrices ${\exp : \text{End}(\mathbb{C}^{m}) \rightarrow \text{Gl}(m, \mathbb{C})}$ , which can be defined by the formula

$\displaystyle e^{x} = \sum_{k = 0}^{\infty} \frac{(2 \pi i)^k}{k!} x^{k},$

where ${x}$ is a square matrix. A matrix logarithm is defined to be a right-inverse to the exponential; that is to say, it is a function ${\log: \text{Gl}(m, \mathbb{C}) \rightarrow \text{End}(\mathbb{C}^{m})}$ such that ${\exp \circ \log = \text{id}_{ \text{Gl}(m, \mathbb{C})}}$ . It is a standard result of Lie theory that the exponential is a diffeomorphism in a neighbourhood of ${0}$ in ${\text{End}(\mathbb{C}^{m})}$ , and hence there is a well-defined choice of logarithm in a neighbourhood of the identity in ${\text{Gl}(m, \mathbb{C})}$ . However, it is not at all clear that a logarithm can be globally defined, or that there is a unique way of doing so. This problem also shows up in the more simple case of the exponential for complex numbers. Indeed, there are several choices for the complex logarithm, and none of them are analytic (or even continuous) on all of ${\mathbb{C}^{\times}}$ . However, it turns out that choosing a complex logarithm is enough to determine a unique choice of a matrix logarithm in any dimension. More precisely, we have the following:

Theorem. Consider the exponential sequence for complex numbers

$\displaystyle 0 \rightarrow \mathbb{Z} \rightarrow \mathbb{C} \mathop{\rightarrow}^{\exp} \mathbb{C}^{\times} \rightarrow 0,$

and let ${\tau : \mathbb{C}^{\times} \rightarrow \mathbb{C}}$ be a set-theoretical splitting of this sequence, in other words, a (not-necessarily continuous) branch of the complex logarithm. Then for any positive integer ${m}$ , there is a uniquely defined matrix logarithm