Fix a pair of relatively prime integers $n>kge 1$, and a point $(eta , | , tau) in mathbb{C} times mathbb{H}$, where $mathbb{H}$ denotes the upper-half complex plane, and let ${{a ; ,b} choose {c , ; d}} in mathrm{SL}(2,mathbb{Z})$. We show that Feigin and Odesskiis elliptic algebras $Q_{n,k}(eta , | , tau)$ have the property $Q_{n,k} big( frac{eta}{ctau+d} ,bigvert , frac{atau+b}{ctau+d} big) cong Q_{n,k}(eta , | , tau)$. As a consequence, given a pair $(E,xi)$ consisting of a complex elliptic curve $E$ and a point $xi in E$, one may unambiguously define $Q_{n,k}(E,xi):=Q_{n,k}(eta , | , tau)$ where $tau in mathbb{H}$ is any point such that $mathbb{C}/mathbb{Z}+mathbb{Z}tau cong E$ and $eta in mathbb{C}$ is any point whose image in $E$ is $xi$. This justifies Feigin and Odesskiis notation $Q_{n,k}(E,xi)$ for their algebras.
تحميل البحث