instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
Deformations of rings of differential operators in one variable and their combinatorics
Seminario de Álgebra

Consider an arbitrary derivation $\delta=h\frac{d}{dx}$, where $h$ is a smooth function of $x$. What is the general rule for computing  the iterations $\delta^k(f)$, formally? How is this related to representation theory? For example: $$\begin{eqnarray} \delta(f) &=& f^{(1)}h \\\\ \delta^2(f) &=&  f^{(2)}h^2 + f^{(1)} h^{(1)} h \\\\ \delta^3(f) &=& f^{(3)} h^3 + 3 f^{(2)} h^{(1)} h^2 + f^{(1)} h^{(2)} h^2 + f^{(1)}( h^{(1)})^2 h \\\\ \delta^3(f) & & \mbox{has $7$ summands with coefficients $1, 6, 4, 7, 1, 4, 1$}. \end{eqnarray}$$ Together with the multiplication by $x$ operator, $\delta$ generates a noncommutative algebra $A_h$ whose elements can be written as differential operators with coefficients in $\mathbb{F}[x]$, where $\mathbb{F}$ is a field of arbitrary characteristic. I will talk about some features of this algebra related to Hochschild cohomology and representation theory, also addressing the combinatorial problem described above.

Parts of this talk are based on joint work with G. Benkart and M. Ondrus, and work in progress with A. Solotar.