L∞ - derivations and the argument shift method for deformation quantization algebras

The argument shift method is a well-known method for generating commutative families of functions in Poisson algebras from central elements and a vector field, verifying a special condition with respect to the Poisson bracket. In this notice we give an analogous construction, which gives one a way to create commutative subalgebras of a deformed algebra from its center (which is as it is well known describable in the terms of the center of the Poisson algebra) and an L∞-differentiation of the algebra of Hochschild cochains, verifying some additional conditions with respect to the Poisson structure.

Poisson-Commutative Subalgebras of 𝒮(𝔤) associated with Involutions

The symmetric algebra ${\mathcal{S}}({{\mathfrak{g}}})$ of a reductive Lie algebra ${{\mathfrak{g}}}$ is equipped with the standard Poisson structure, that is, the Lie–Poisson bracket. Poisson-commutative subalgebras of ${\mathcal{S}}({{\mathfrak{g}}})$ attract a great deal of attention because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset{\mathcal{S}}({{\mathfrak{g}}})$ is bounded by the “magic number” ${\boldsymbol{b}}({{\mathfrak{g}}})$ of ${{\mathfrak{g}}}$. There are two classical constructions of $\mathcal C$ with ${\textrm{tr.deg}}\,{\mathcal C}={\boldsymbol{b}}({{\mathfrak{g}}})$. The 1st one is applicable to $\mathfrak{gl}_n$ and $\mathfrak{so}_n$ and uses the Gelfand–Tsetlin chains of subalgebras. The 2nd one is known as the “argument shift method” of Mishchenko–Fomenko. We generalise the Gelfand–Tsetlin approach to chains of almost arbitrary symmetric subalgebras. Our method works for all types. Starting from a symmetric decompositions ${{\mathfrak{g}}}={{\mathfrak{g}}}_0\oplus{{\mathfrak{g}}}_1$, Poisson-commutative subalgebras ${{\mathcal{Z}}},\tilde{{\mathcal{Z}}}\subset{\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ of the maximal possible transcendence degree are constructed. If the ${{\mathbb{Z}}}_2$-contraction ${{\mathfrak{g}}}_0\ltimes{{\mathfrak{g}}}_1^{\textsf{ab}}$ has a polynomial ring of symmetric invariants, then $\tilde{{\mathcal{Z}}}$ is a polynomial maximal Poisson-commutative subalgebra of ${\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ and its free generators are explicitly described.