# Lie algebra of a compact Lie group and derivations of the Hopf algebra of representative functions

Let $$\mathcal{G}$$ be a compact (real) Lie group. We know that the Lie algebra $$\mathfrak{g}$$ of $$\mathcal{G}$$ is, by definition, the space of all left-invariant (smooth) vector fields over $$\mathcal{G}$$ with bracket given by the commutator. We also know that it is isomorphic, as a Lie algebra, with $$T_e\mathcal{G}$$ (the tangent space to $$\mathcal{G}$$ at the neutral element $$e$$).

Consider the Hopf algebra of representative functions $$H:=\mathcal{R}_{\mathbb{R}}(\mathcal{G})$$ associated to $$\mathcal{G}$$ and recall that $$H\subseteq \mathcal{C}^\infty(\mathcal{G})$$ is a dense subspace with respect to the supremum norm (Proposition I.3.12 of Brocker, Dieck, Representations of Compact Lie Groups together with Peter-Weyl Throem). The Lie algebra $$\mathcal{P}(H^\circ)$$ of primitive elements of its finite dual is isomorphic the Lie algebra of left-invariant derivations $${^{H}\mathsf{Der}_{\mathbb{R}}(H,H)}$$ (i.e., derivations $$\delta:H\to H$$ such that $$\Delta\delta=(H\otimes \delta)\Delta$$).

I would expect to have an isomorphism $$\mathfrak{g}\cong {^{H}\mathsf{Der}_{\mathbb{R}}(H,H)}$$.

Q1. Is this true?

I didn't find it anywhere in the literature, whence I am trying to provide one by myself.

Let $$X$$ be a left-invariant vector field and let $$\varphi$$ be a smooth function on $$\mathcal{G}$$. Then $$\boldsymbol{X}(\varphi):\mathcal{G}\to \mathbb{R}$$ given by $$\boldsymbol{X}(\varphi)(g) = X_g(\varphi)$$ ($$X_g\in T_g\mathcal{G}$$) is a smooth function and hence we have an assignment $$\boldsymbol{X}:\mathcal{C}^\infty(\mathcal{G})\to \mathcal{C}^\infty(\mathcal{G})$$. One can verify that this induces a Lie algebra map $$\mathfrak{g}\to {^{H}\mathsf{Der}_{\mathbb{R}}(H,H)}: X\mapsto \boldsymbol{X}.$$ To provide a candidate inverse to this morphism, I consider a left-invariant derivation $$\delta$$ and for every $$g$$ in $$\mathcal{G}$$ the function $$\delta_g:\mathcal{R}_{\mathbb{R}}(\mathcal{G})\to \mathbb{R}, \varphi\mapsto\delta(\varphi)(g)$$. Now, I would expect to be able to extend such a function to the whole $$\mathcal{C}^\infty(\mathcal{G})$$, maybe resorting to the Continuous Linear Extension Theorem (see Theorem 5.7.6 in Foundations of Applied Mathematics, Volume I: Mathematical Analysis by Humpherys, Jarvis, Evans), but I didn't manage to.

Q2. Could somebody suggest a way to do this?

Q3. Is there some reference in which this is treated in some detail?

OT: I already asked it on MSE, but maybe it could be that this question is more suitable for MO.

• As far as I know, $\mathcal{G}$ is used for Lie groupoids.. you can use normal $G$?? – Praphulla Koushik Sep 11 '18 at 10:10
• Of course I could, but it does not make too much sense in my opinion. It's just a matter of notation and I am used to use normal $G$ for discrete groups. If you really think that using $G$ instead of $\mathcal{G}$ improves the readability of the question, you have my approval in editing it (presently it's a bit difficult for me as I don't have my laptop) – Ender Wiggins Sep 11 '18 at 11:25
• When you define these spaces of derivations $H\to H$ are you imposing any a priori assumption of continuity? (I have faint alarm bells ringing in my head in view of the various answers and comments at mathoverflow.net/questions/6074/… but this could easily be a false alarm on my part) – Yemon Choi Jul 8 '19 at 2:38
• @YemonChoi No, actually I am not imposing any additional assumption apart from those explicitly mentioned. In particular, I am using Abe's book on Hopf algebras as main reference for these notions. So, derivations $\delta:H\to H$ are simply $\mathbb{R}$-linear maps such that $\delta(ab)=\delta(a)b+a\delta(b)$ (or, if you prefer, derivations from $H$ into the $H$-bimodule $H$ itself). – Ender Wiggins Jul 8 '19 at 13:24

If you have a Hopf algebra $$H$$, then the counit gives you a map $$\varepsilon_* :\mathrm{Hom}(H,H)\to \mathrm{Hom}(H,k)$$. This map has a nice and natural splitting $$H^*\to \mathrm{End}(H)$$ $$f\mapsto F$$ where $$F(h)=h_1f(h_2)$$. This splitting actually lift to a splitting of the maps $$\varepsilon_*:\mathrm{Hom}(H^{\otimes n},H)\to \mathrm{Hom}(H^{\otimes n},k)$$ for all $$n$$, in a way that it is compatible with Hochschild coboundary, and stable under Gerstenhaber bracket (see the proof of Theorem 1.5 in the reference above).
In particular, $$\mathrm{Der}(H,k)$$ injects into $$\mathrm{Der}(H,H)$$. For $$H=\mathcal O(G)$$, $$\mathrm{Der}(H,k)$$ (with $$k$$ viewed as $$H$$-module via $$\varepsilon$$) is -almost by definition- $${\frak g}=T_e(G)$$, and $$\mathrm{Der}(H,H)$$ is the algebraic version of $${\frak X}(G)$$. The compatibility with Gerstenhaber structure says that these maps are Lie algebra maps. So you get a "Hopf"-way of defining "the unique left invariant vector field" associated to an element in the tangent space of the identity, and for $$H=\mathcal O(G)$$ you get a precise map $$\mathrm{Der}(\mathcal O(G),\mathcal O(G)) \cong \mathcal O(G)\otimes \mathrm{Der}(\mathcal O(G),k) =\mathcal O(G)\otimes\frak g.$$
Regarding the $$C^\infty$$-problems, I think that if you write the formula " $$F(h)=h_1f(h_2)$$" in a diagrammatic way, you should be able to do the $$C^\infty$$-case.
• Thank you very much, I will have a look at your reference and see if it helps. However, my main concern is that in the algebraic setting that "identification" between $T_e\mathcal{G}$, $^H\mathsf{Der}_{\Bbbk}(H,H)$ and $\mathcal{P}(H^\circ)$ is more or less well established and widely used (see Humphreys, Linear Algebraic Groups, §9.1; Hochschild, Basic Theory of Algebraic Groups and Lie Algebras, §III.3; Abe, Hopf Algebras, §4.3.1), but it seems that in the differential setting things are much more complicated (Hochschild and Mostow have a whole bunch of papers on the topic). – Ender Wiggins Jun 8 '19 at 9:07
• For these reasons I am really interested in the $\mathcal{C}^{\infty}$-setting – Ender Wiggins Jun 8 '19 at 9:08