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I am reading a book on algebraic geometry and I google some keywords, eventually come up with this post in Terry Tao's blog:

http://terrytao.wordpress.com/2009/10/19/grothendiecks-definition-of-a-group/

I think I got good intuition on the different thoughts on a group, but not this one:

(6) Dynamic: A group represents the passage of time (or of some other variable(s) of motion or action) on a (reversible) dynamical system.

Can anyone explain to me how a group represents the passage of time? I can only think of Noether's theorem on conservation law when combining the concept of time and algebra.

Alp Uzman
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wh0
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    I'm not sure if that's what he meant, but (the action of) the semigroup of positive reals is often used to denote some kind of passage of time, like in the case of the heat semigroups. You could have the negative reals too if you allowed reversal of time, though that wouldn't be quite as well defined as in the case of usual heat semigroup. ;) – tomasz Jan 04 '14 at 16:17
  • Googling "dynamic group" brings up lots of relevant stuff. "Dynamic" is the operative modifier here for the connection with time. – rschwieb Jan 04 '14 at 16:28

2 Answers2

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It's hard to say what Terry meant exactly unless you ask him. Nevertheless, it seems quite probable that he is talking about flows on manifolds (or more general kind of spaces but let's stick with this).

A reversible flow on a manifold $M$ is a family of diffeomorphisms $\phi_t:M \to M, t \in {\bf R}$ that obey the rule of composition $\phi_t \circ \phi_s = \phi_{s+t}$. In other words, flow is an action of the group $({\bf R}, +)$ on the manifold. That this describes dynamics is immediate: for any $x \in M$, $\phi_t(x)$ is a curve that shows how $x$ moves under the flow.

Marek
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  • OK. We think the action of a group is to make the time go forward or backward. I think get this point. If we think of a group (as passage of time) like this, then the time defined by a group may not be a straight line. What I mean it that the time could be a recurrence of time $\mathbb{Z}/n\mathbb{Z}$ (going forward for some time and eventually I will get back to the origin) or I have different way to go forward (or backward) in time like $\mathbb{D}_4$ – wh0 Jan 05 '14 at 06:49
  • I can also define a time passage look like $\mathbb{R}/\mathbb{Z}$ so that when I go forward with time, I will never meet the points I visited. – wh0 Jan 05 '14 at 07:03
  • @wonghang: precisely. Another often occurring group is integers that occurs in discrete dynamics. You obtain it by simply taking any map $f$ on a given space and iterating it and its inverse. Although to be fair, most dynamics are not reversible, maps not invertible, so one uses natural numbers instead and moves from groups to monoids or semigroups. But dynamics coming from physics usually are reversible (with the notable exception of heat equation). – Marek Jan 05 '14 at 11:19
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It seems Tao is alluding to the following:

The discipline called dynamical systems study the time evolution of a closed system. The most general ($\ast$) mathematical model of this is a triple $(X,T,\alpha)$, where $X$ and $T$ are two spaces and $\alpha:T\times X\to X$ is a map. Currying $\alpha$ gives

$$\alpha_\bullet: T\to[X\to X],$$

so that we may think of $\alpha$ as a family of self-maps of $X$ parameterized by $T$. Here $X$ models the "closed system", $T$ models the "time" and $\alpha$ models the "evolution". As Marek mentioned often the components of the triple $(X,T,\alpha)$ have compatible structures (measurable, topological, differentiable, algebraic,...). One natural structure is the algebraic structure of the space $X\to X$; it's a monoid under composition. So it is natural (from the syntactic point of view) to restrict attention to the case when $T$ has a monoid structure also and $\alpha_\bullet$ respects this monoid structure. Similarly one can restrict the codomain of $\alpha_\bullet$ to the invertible elements $\operatorname{Aut}(X)$ of $\operatorname{End}(X)=[X\to X]$, which naturally has a group structure, and in this case it would be natural to endow $T$ with a group structure and demand that $\alpha_\bullet$ respect the group structure. To distinguish this algebraic case from the general case one can (and does, typically) call the former a nonautonomous dynamical system and the latter an autonomous dynamical system.

Observe that Marek's answer alludes to this: classical dynamical systems are flows of autonomous vector fields (and their discrete analogs), where the group property is emergent from the Existence and Uniqueness theorem for ODE's (thinking of a flow on $M$ as a group homomorphism $\mathbb{R}\to\operatorname{Diff}^r(M)$ (or $\mathbb{R}\to\operatorname{Aut}(X)$) is a relatively modern perspective).

To compare, for instance one can think of a representation in this framework, however in representation theory interpreting the group $T$ that is being represented as time is rare. Further, even when one uses representation theory in dynamics ("Koopmanisms" in the language of B. Simon; Mathematical framework of the Koopman operator, Ergodicity of surjective continuous endomorphism of compact abelian group (confused about a step)) the acting group still preserves the time connotation, from the dynamical point of view.

See e.g. Question regarding integration and diffeomorphisms on $\mathbb{R}$, Random elements and random processes for contexts in which the nonautonomous perspective is relevant. See What are some interesting examples of non-classical dynamical systems? (Group action other than $\mathbb{Z}$ or $\mathbb{R}$) examples of dynamical systems with acting groups more complicated than $\mathbb{Z}$ or $\mathbb{R}$.

($\ast$) "most general" for the purposes of this answer; e.g. some consider a foliation (or more generally equivalence relations) as a dynamical system with no specified time parameterization.

Alp Uzman
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