Is there a theory of lie "rings"?

Question

A Lie group is a group that is a differentiable manifold and addition and inversion are differentiable maps. Is there a theory for rings that are differential manifolds and have differentiable addition, inversion and multiplication?

If so, how are these rings called? (Obviously, "Lie ring" is already in use for a different structure.)

Answer 1

Let $R$ be a connected Lie ring. As you-sir-33433 mentions, connected abelian Lie groups are products of $\mathbb{R}$ and $S^1$, so we can write the underlying abelian Lie group of $R$ as $V \oplus T$ where $V$ is a finite-dimensional real vector space and $T$ is a torus. Left multiplication gives a continuous embedding $R \to \text{End}(R)$, and we can compute $\text{End}(R)$ as follows:

$$\text{Hom}(V \oplus T, V \oplus T) \cong \text{Hom}(V, V) \oplus \text{Hom}(V, T) \oplus \text{Hom}(T, V) \oplus \text{Hom}(T, T).$$

We know what all four of these look like:

• $\text{Hom}(V, V) \cong M_n(\mathbb{R})$ where $V \cong \mathbb{R}^n$.
• $\text{Hom}(T, T) \cong \mathbb{Z}^m$ where $T \cong (S^1)^m$, and since we assumed that $R$ is connected the image of $R$ necessarily lies in the identity component of $\text{Hom}(T, T)$, so we can ignore it.
• $\text{Hom}(T, V)$ is trivial.
• $\text{Hom}(V, T) \cong (V^{\ast})^m$.

In particular, $\text{End}(R)$ is torsion-free. This implies that no $S^1$ factors can occur, so $T = 0$ and the underlying abelian Lie group of $R$ is $V$. Furthermore, the map $V \cong R \to \text{End}(R) \cong \text{End}(V)$ must be $\mathbb{R}$-linear by continuity. I've left out a few details, but the conclusion should be that

$R$ must be a finite-dimensional $\mathbb{R}$-algebra.

So in some sense there is no Lie theory to do. Everything is completely algebraic from the start.

Answer 2

There are certainly examples, like the set of $n\times n$ matrices. But any connected abelian Lie group is homeomorphic to a product of copies of $\mathbb{R}$ and $S^1$, so a "Lie ring" (which is, in particular, an abelian Lie group) is geometrically quite boring.