Category: Algebraic geometry

How does a Lie algebra encode a space? (Part 2)

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.

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How does a Lie algebra encode a space? (Part 1)

A fundamental principle of derived deformation theory is that a formal “space” (i.e. formal moduli problem) can be encoded by a differential graded Lie algebra, or more generally, an L_{\infty} -algebra. This is usually attributed to a number of famous mathematicians, such as Quillen, Deligne, Drinfeld, Goldman, Millson, Feigin, Manetti, and Kontsevich. More recently, a formalization of this principle was independently obtained by Lurie and by Pridham.

In this post, I want to discuss a simple example of this: how to encode the ‘derived’ vanishing-locus of a function using an L_{\infty} -algebra. My goal is to show that, far from being a high-brow construction, this essentially boils down to the familiar Taylor series expansion. After explaining this fact, I will go through a number of concepts and constructions in the world of L_{\infty} -algebras and show how they can be translated into the setting of a function and its zero locus. Hopefully, this will help to demystify the notion of derived manifolds for myself and maybe for others. In a future post, I will discuss the ‘stacky’ version of this story: how to describe a Lie algebra action, or a Lie algebroid, in terms of an L_{\infty} -algebra. As usual, there might be some (hopefully minor) errors in what follows, so proceed with caution!

Continue reading “How does a Lie algebra encode a space? (Part 1)” →