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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).

### 3 Responses

1. on July 18, 2020 at 4:55 pm | Reply Anonymous

What do you mean infinitesimals went out of fashion? Artin’s Algebra textbook gives a totally rigorous construction of the Lie algebra of a linear algebraic group by looking at the groups G(k), G(k[e]/e^2), and \$G(k[e]/e^3). I imagine it’s pretty close to what Lie originally did. (Artin plays a shell game about the difference between a Lie group and a linear algebraic group.)

Not to ruin the joke or belabor the obvious, but if you take ind-finite cocommutative coalgebras and take the linear dual, you get pro-finite commutative algebras (with the completed tensor product) and their spectra are called formal schemes and group objects in them are called formal groups and are a pretty popular way of making sense of “infinitesimal groups.” (You might want to link between your posts on distributions, but I don’t think you get to the end of the story.)

Not super relevant, but while I’m on the topic: the universal enveloping algebra is the ring of left-invariant differential operators on the group and its dual, the ring of functions on the formal group, is the formal completion at the identity of functions on the group. The pairing is to apply the operator to the function to get a new function and evaluate at the identity.

2. on January 14, 2016 at 10:03 am | Reply Christos

Nice post! Could you say more about more the phrase: ”… which in turn can be interpreted as a category of infinitesimal spaces” by the end of the post?

• on January 14, 2016 at 12:15 pm | Reply Qiaochu Yuan

Over a field, the category of cocommutative coalgebras is the ind-completion of the category of finite-dimensional cocommutative coalgebras, which is equivalent to the opposite of the category of finite-dimensional commutative algebras.