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Let $X$ be an algebraic variety, $at(E) \in Ext^1(E, E \otimes T)$ the Atiyah class of a complex $E \in D(Coh X)$ (see Markaryan, $\S$1.1-1.2).

Then $at(\Omega[1])$ gives a "shifted Lie algebra" on $T_X[-1]$, that is a morphism $T_X[-1] \otimes T_X[-1] \to T_X[-1]$ in the derived category $D(Coh X)$, satisfying certain properties.

How can one think about such an object? For example, can one explicitly (but, maybe up to quasi-isomorphism or something?) describe it for $X=\mathbb P^1$?

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    $\begingroup$ The Atiyah class of a line bundle coincides with the first Chern class. So, for $X=\mathbb{P}^1$ you get an element of $H^1(X, \mathrm{Sym}^2(\Omega^1_X)\otimes T_X)\cong H^1(X, \Omega^1_X)\cong \mathbb{C}$ which is $\pm 2$. $\endgroup$
    – Pavel Safronov
    Commented Sep 8, 2017 at 15:22

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