I'm trying to understand the definition of "Group Affine" systems (from Theorem 1 in this paper). I'll restate it here:
Let $\frac{d}{dt}X_t = f_{u_t}(X_t)$ be a vector field describing the system dynamics on a Lie Group $G$.
A system is group affine if the dynamics, $f_{u_t}(\cdot)$, satisfies:
$f_{u_t}(X_1X_2) = f_{u_t}(X_1)X_2 + X_1 f_{u_t}(X_2) - X_1 f_{u_t}(I_d)X_2$
for all $t>0$ and $X_1, X_2 \in G$.
First I want to make sure that I understand what each of these objects are, please correct me if I get anything wrong, because I'm not sure where my understanding is breaking down.
- $X \in G$ is a group element, that one's easy.
- $\frac{d}{dt}X \in T_XG$ is a vector in the tangent space of $G$ at the element $X$ representing the "flow" of $X$ through the group.
- $f(X) : G \to T_XG$ is a vector field on $G$ that maps each group element, $X$, to a vector in the tangent space $T_XG$.
If that's all correct, then I'm not sure how to interpret the operations $f_{u_t}(X_1)X_2$ and $X_1f_{u_t}(X_2)$. What are these binary operations? What does it mean to operate on $X_2$ (an element of $G$) with $f_{u_t}(X_1)$ which is an element of $T_{X_1}G$? What space does the result live in? From the lhs it seems like it should live in $T_{X_1X_2}G$.
This notation is specific to matrix Lie groups, but is easily extended to general Lie groups, as explained below.
Suppose first that $G$ is a matrix Lie group, so $G \subset M$ is a manifold inside a vector space $M$ of square matrices of a given size.
Now the following holds.
If $X \in G$, and $u \in T_XG$, in the sense that $u$ is a vector in $M$ tangent to $G$ at $X$ (in $M$), then, for any $Y\in G$: $$ Y u \in T_{YX}G \qquad u Y \in T_{XY}G $$ where $Y u$ and $uY$ are matrix multiplications in $M$.
It looks specific to matrix Lie groups, but is in fact quite general. For a general Lie group now, pick $Y\in G$ and define the functions $L_Y \colon G \to G$ by $L_Y X := YX$, and $R_Y \colon G \to G$ by $R_Y Xh := XY$. Their tangent maps at $X\in G$ are so that $TL_Y \colon T_X G \to T_{YX}G$ and $TR_Y \colon T_X G \to T_{XY}G$.
Now, if $G$ is a matrix Lie group, and if $u \in T_X G$, then the tangent vector $Yu$ above is exactly $TL_Y u \in T_{YX}G$ (when embedded in $M$), and $uY$ is exactly $TR_Y u \in T_{XY}G$ (when embedded in $M$).