Let $L^2$ be the vector space of bivariate, square-normable functions, i.e., $L^2 := \{ f(x,y) | \iint_{[0,1]^2}dxdy|f(x)|^2 < \infty \}$. And let the Poisson bracket be defined on $L^2$ as $\{f, g\} := \frac{\partial f}{\partial x} \frac{\partial g}{\partial y} - \frac{\partial f}{\partial y} \frac{\partial g}{\partial x}$.
Since the Poisson bracket satisfies bilinearity, alternativity ($\{f, f\}=0$), and the Jacobi identity, it is clear that $L^2$ equipped with $\{\cdot,\cdot\}$ forms a Lie algebra. My question is, can one identify a Lie group whose Lie algebra is isomorphic to $L^2$?
Since $L^2$ is clearly infinite-dimensional, it is clear that Lie's third theorem does not apply. Not being familiar with the Lie theory that much, I'm not sure what kind of conditions I should look for in this case.
Thank you.