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A field acting on an abelian group is called a vector space. Is there a name for a field acting on a non-abelian group?

What I mean is a group $G$ a field $F$, and an operation $F \times G \to G$, $(a, x) \mapsto x^a$ such that:

  • $x^0 = e$
  • $x^1 = x$
  • $x^{a+b} = x^ax^b$
  • $x^{ab} = (x^a)^b$

Is there a name for this kind of structure? Why aren't they studied?

Aleks
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  • Do you also want $(xy)^{a} = x^{a}y^{a}$? If you do, then $G$ must be abelian. – Daniel Fischer Aug 16 '20 at 21:34
  • No, $(xy)^a \neq x^a y^a$ in general since $x$ and $y$ might not commute – Aleks Aug 16 '20 at 21:35
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    Do you have an example of such a thing? – Randall Aug 16 '20 at 21:37
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    If $F$ has characteristic $2$ then $1+1=0$, so $e=x^0=x^{1+1}=x\cdot x$. Thus $G$ is elementary abelian. – David A. Craven Aug 16 '20 at 21:40
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    @Randall More generally, if $F$ has characteristic $p$ then $G$ is a $p$-group of exponent $p$. If $F=\mathbb{F}_p$ then every $p$-group of exponent $p$ is an example of such a thing. – David A. Craven Aug 16 '20 at 21:56
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    Do you want $e^a=e$ for all $a\in F$? This is normally taken care of via $(xy)^a=x^ay^a$, but this axiom has been dropped. – David A. Craven Aug 16 '20 at 22:05
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    Typically when one speaks of a structure $A$ acting on a structure $B$, the set of endomorphisms $\textrm{End} (B)$ has the same kind of structure as $A$ and one has a homomorphism from $A \to \textrm{End} (B)$. If $G$ is an abelian group then $\textrm{End} (G)$ is a ring (not necessarily commutative), and that is how we end up with the notion of module; but if $G$ is a general abelian group then $\textrm{End} (G)$ is not even a ring. In my view the right question to ask is, what algebraic structure does $\mathrm{End} (G)$ actually have, other than the inherent composition of endomorphisms? – Zhen Lin Aug 17 '20 at 01:16
  • In general $g^a,$ where $g$ is an element in a group, is only defined for integer $a$ so $F$ must be a subset of $\mathbb Z,$ which I guess requires that $F = \mathbb Z_p$ for some prime $p.$ But for a simply connected Lie group over $F$ we can define $g^a$ by $g^a = \exp (a \log g),$ where $\log g$ is an element in the corresponding Lie algebra such that $\exp(\log g)=g.$ – md2perpe Aug 17 '20 at 14:37

1 Answers1

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In the case when $F$ has characteristic zero, the restriction $\mathbb{Z} \times G\rightarrow G$ of the external law is just the operation of iterates. We see that the restriction of $(x,a)\mapsto x^a$ to $\mathbb{Q} \times G$ must be an operation of fractional iterates.

The group $G$ can be seen as a group of bijective endomorphisms under composition (let it act by translations on itself). And the existence of a non-ambiguous notion of fractional iterates for such endomorphisms is a rather rare event.

In general, given a fixed $x \in G$, the set $x^F$ is included in the centralizer $\mathcal{C}(x)$ of $x$. So the centralizer of $x$ contains a quotient of the underlying group of $F$. This is also a rather rare event.

So I would say those structures are not much studied because they don't come up often. One other obstacle is that it seems hard to imagine one would be able to say much on the class of those non-commutative vector spaces without specifying some properties of $(x y)^a$ for $x,y \in G$ and $a\in F$, when $x$ and $y$ don't commute.

However I know of similar examples in the case when $F=\mathbb{R}$.

One example is in the group $P_n$ of positive definite $n \times n$ matrices over $\mathbb{R}$. Since their spectrum is contained in $\mathbb{R}^{>0}$, one has a calculus of real-analytic functions on $P_n$. In particular, there is a well-defined operation of real powers $(M,a)\longmapsto M^a$ which satisfies your conditions.

One other example I know is the case of transseries of exponentiality $0$, or even formal Laurent series under composition. In those cases, a positive infinite series has "real iterates" and in fact the tools to define them are formal versions of the functional calculus of the previous example. See G. A. Edgar's article Fractional Iteration of Series and Transseries. This also generalizes to larger fields of transseries and in certain cases the set $x^{\mathbb{R}}$ coincides with $\mathcal{C}(x)$.

Note that my examples are all linked to an order on $G$, and this may be a feature of the characteristic zero case. In any case, there are obstructions to generalizing those two examples to the field of complex numbers.

nombre
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