The relation $(X+Y)^n = X^n + Y^n$, and miraculous cancellations in quantum $SL_2$

The relation $(X+Y)^n = X^n + Y^n$, and miraculous cancellations in quantum $SL_2$

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Francis Bonahon, USC
Fine Hall 314

The convenient formula $(X+Y)^n = X^n + Y^n$ is (unfortunately) frequently used by our calculus students. Our more advanced students know that this relation does hold in some special cases, for instance in prime characteristic $n$ or when $YX=qXY$ with $q$  a primitive $n$-root of unity.  I will discuss similar ``miraculous cancellations’’ for 2-by-2 matrices, in the context of the quantum group $U_q(sl_2)$.