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.