On Raising Your Hand

To cap off this post with a mathematical gemstone, below is the full computation of the automorphism group of the graded noncommutative ring mentioned on the previous page.

Any automorphism of the noncommutative $\mathbb{C}$-algebra $$R_q:=\mathbb{C}\langle x,y\rangle/(xy-qyx)$$ that preserves the grading by degree can be thought of as an element of $\mathrm{GL}_2(\mathbb{C})$, since the degree-one generators $x$ and $y$ must map to linear combinations of $x$ and $y$, and these images determine the map. In other words, we can represent the automorphism that sends $x$ to $ax+by$ and $y$ to $cx+dy$ by the $2\times 2$ matrix $$\left(\begin{array}{cc} a & b \\ c & d\end{array}\right).$$

For a map of this form to extend to an automorphism, it is necessary and sufficient that the ideal generator $xy-qyx$ maps to a scalar multiple of itself. In this case we have
$$\begin{align*}
xy-qyx &\mapsto (ax+by)(cx+dy)-q(cx+dy)(ax+by) \\
&= (1-q)acx^2+(ad-qbc)xy+(bc-qad)yx+(1-q)bdy^2
\end{align*}$$

If $q=1$, this simplifies to $(ad-bc)(xy-yx)$, which lies in the desired ideal. Thus the group of degree-preserving automorphisms is all of $\mathrm{GL}_2(\mathbb{C})$ in this case.

If $q\neq 1$, since the $x^2$ and $y^2$ terms must vanish, we have either $a=0$ or $c=0$, and either $b=0$ or $d=0$. If $a=0$ and $b=0$ simultaneously then the $xy$ coefficient would be $0$, and similarly for $c$ and $d$. Thus we must have either $a=d=0$ or $b=c=0$.

In the case that $a=d=0$, the above simplifies to $bc(-qxy+yx)$, and in the case that $b=c=0$, the it simplifies to $ad(xy-qyx)$. The latter is clearly a scalar multiple of $xy-qyx$, but the former is so only if $q=-1$. Hence if $q\neq \pm 1$, the automorphism group is isomorphic to $\mathbb{C}^\times \times \mathbb{C}^\times$, consisting of the matrices of the form $$\left(\begin{array}{cc} a & 0 \\ 0 & d\end{array}\right)$$ with $a,d\neq 0$. And if $q=-1$, we have a second copy of $\mathbb{C}^\times \times \mathbb{C}^\times$, from the matrices of the form $$\left(\begin{array}{cc} 0 & b \\ c & 0\end{array}\right)$$ with $b,c\neq 0$.

The upshot is that because this ring is generated in degree $1$ with a single quadratic homogeneous relation, the higher degree $q$-numbers do not appear in the computation, and only $q=\pm 1$ are special values. A nice little gemstone!

4 thoughts on “On Raising Your Hand

  1. In my experience with teaching, different classes can have vastly different question-asking habits. The standard is usually set the first lecture – if no questions are asked that lecture, it will be the same every lecture thereafter, and it is really hard to change the mentality. As a lecturer, one really need to encourage questions on the very first lecture. I feel like the same might be true for talks at conferences, where the first talk sort of sets the atmosphere for the entire event.

    Perhaps it would be an interesting experiment, to ask a few people in advance, to prepare questions for the first talk (to ask both during and after) to set a good example.

    Finally, a women-only venue is maybe a bit more prone to question-shyness, as in my personal experience, and some studies (see reference below) indicate that women are better at presenting ideas in a clearer fashion. Hence, there might not be any questions needed during a talk if it is easy to follow.

    Diane F. Halpern, Camilla P. Benbow, David C. Geary,
    Ruben C. Gur, Janet Shibley Hyde, and Morton Ann
    Gernsbacher. The science of sex differences in science and
    mathematics. Psychological Science in the Public Interest,
    8(1):1–51, aug 2007

    • Interesting, thanks for sharing! You raise a good point about women also giving more clear presentations. In my experience the clearer presentations actually get more questions though, so I’d think the correlation would go the other way? Unless it’s the combination of a shy audience and clear presentations that make for a silent questions time.

  2. I considered PMing you, but I suppose that inclination proves your point, so I will instead ‘raise my hand’ in a public comment. I am glad for your encouragement in the last paragraph for women to take part more actively in mathematical discussion, including asking questions at talks, as it is certainly needed. It could be read, though, as a call to women to change their behavior without an attempt to understand what motivates it. I’m very glad your experience in academia has not (yet) been clouded by the struggles typically discussed at AWM panels. The experience of those struggles contributes to women’s reluctance to speak up, so there’s more to the issue than simply asking women to change. The culture of an event (even, apparently, an all-women meeting!) needs to adapt as well to be more welcoming of the contributions/comments of women. I also agree with Per’s comment, that the first talk (or first day of class) is key to communicating that participation by all in the audience (classroom) is welcomed and encouraged.

    • Thanks for your reply, Jessica. I agree that my post is lacking in addressing the motivating factors behind women being less likely to speak up, and I’m glad you raised that point. I don’t feel I have a good sense of what those motivations may be – do you have specific ways in mind that a culture or environment may not be welcoming to women speaking up?

      I like the idea of establishing the interactive environment in the first talk for a class, though I’d think at a research conference it’s assumed by default that questions are welcome, particularly at the end of a talk. I remember Chelsea, who was the first plenary speaker at the AWM meeting, going out of her way to encourage questions at the start, but it didn’t seem to make too much of a difference.

      I’m not sure my encouragement is going to do much either, of course. I guess my hope is more that there is some young woman out there who is just getting started in math, who reads my post and realizes that working past her own shyness might help her progress in a competitive field. It may make the difference for a few shy people here and there to just give some encouragement. I totally agree that it’s not going to solve any big systemic issue though.

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