Semigroups in Lie groups, semialgebras in Lie algebras

Abstract:

Consider a subsemigroup of a Lie group containing the identity and being ruled by one-parameter semigroups near the identity. We associate with it the set $W$ of its tangent vectors at the identity and obtain a subset of the Lie algebra $L$ of the group. The set $W$ has the following properties: (i) $W + W = W$, (ii) ${{\mathbf {R}}^ + } \cdot \;W \subset W$, (iii) ${W^ - } = W$, and, the crucial property, (iv) for all sufficiently small elements $x$ and $y$ in $W$ one has $x \ast y = x + y + \frac {1} {2}[x,y] + \cdots$ (Campbell-Hausdorff!) $\in W$. We call a subset $W$ of a finite-dimensional real Lie algebra $L$ a Lie semialgebra if it satisfies these conditions, and develop a theory of Lie semialgebras. In particular, we show that a subset $W$ satisfying (i)-(iii) is a Lie semialgebra if and only if, for each point $x$ of $W$ and the (appropriately defined) tangent space ${T_x}$ to $W$ in $x$, one has $[x,{T_x}] \subset {T_x}$. (The Lie semialgebra $W$ of a subgroup is always a vector space, and for vector spaces $W$ we have ${T_x} = W$ for all $x$ in $W$, and thus the condition reduces to the old property that $W$ is a Lie algebra.) In the introduction we fully discuss all Lie semialgebras of dimension not exceeding three. Our methods include a full duality theory for closed convex wedges, basic Lie group theory, and certain aspects of ordinary differential equations.