Seminar of Algebra

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Deformations of rings of differential operators in one variable and their combinatorics

Samuel A. Lopes (University of Porto and CMUP)
Departamento de Álgebra
Mon, 20 nov 2017 11:30
Samuel Lopes is a professor at the Mathematics Department of the Faculty of Sciences of the University of Porto, and vice-director of the research center CMUP. His research interests lie in representation theory and ring theory. He likes to flirt with algebraic combinatorics, always with mutual consent

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.