Research

To quantize the reduction of a coisotropic submanifold of a Poisson manifold, Cattaneo and Felder constructed a homotopy Poisson algebra together with a deformation quantization. Its degree-zero cohomology is the reduced Poisson algebra, while the full homotopy structure retains information about the formal embedding even when the reduced space is singular or trivial. On the integration side, Cattaneo showed that a coisotropic submanifold integrates to a Lagrangian subgroupoid of any symplectic groupoid integrating the ambient Poisson manifold. It is now recognized that both constructions fit into shifted symplectic geometry: the Lagrangian subgroupoid presents a 1-shifted Lagrangian morphism of differentiable stacks, and the homotopy Poisson algebra describes the associated 0-shifted Poisson structure. However, such morphisms form only a special class of 1-shifted Lagrangians. We extend this picture to 1-shifted Lagrangians and, more generally, 1-shifted coisotropics in 1-shifted symplectic differentiable stacks. We establish an integration theorem for their infinitesimal description in terms of twisted Dirac structures and construct a natural Poisson bracket on invariant functions. Under regularity hypotheses, we equip the associated Lie algebroid complex with a flat $P_\infty$-structure and a curved $A_\infty$-quantization, independent of the auxiliary choices up to isomorphism. The induced bracket in degree zero recovers the reduced Poisson bracket, and suitable cohomological vanishing conditions yield a deformation quantization of invariant functions. This extends the Cattaneo–Felder picture to more general reduction procedures, with the same infinitesimal data governing integration and quantization.

We study Ore extensions $R[t;\sigma,\delta]$ of rings graded by an arbitrary group $\Gamma$. The grading of $R$ extends to a grading with $t$ homogeneous of degree $\gamma$ if and only if $\sigma$ and $\delta$ are compatible with conjugation and translation by $\gamma$, and this grading is then unique. For nonabelian $\Gamma$ the degree of $t$ need not centralize the support of $R$; this twist can be removed by a change of variable when $R_\gamma$ contains a unit, but not in general. The graded forms of the classical necessary conditions for simplicity are not sufficient, as the quantized Weyl algebra shows. If $R$ is graded-simple and $\sigma$ is an automorphism, then $R[t;\sigma,\delta]$ is graded-simple if and only if it contains no homogeneous normal element that is monic of positive $t$-degree. For Ore extensions of group algebras, an affine action of the group on the line describes all graded ideals; in characteristic zero, graded simplicity is equivalent to the nonvanishing of a class in $H^1(G,k_\chi)$, and simplicity forces the group to be metabelian.

We study differential polynomial rings $R[t;\delta]$ over $\Gamma$-graded rings, where $\Gamma$ is an arbitrary group. We show that $R[t;\delta]$ admits a $\Gamma$-grading compatible with that of $R$ if and only if $\delta$ is a $\gamma$-derivation for some $\gamma\in C_\Gamma(\Gamma_R)$, and that this grading is unique once $\deg(t)=\gamma$ is fixed; if $\delta\neq0$, then $\gamma$ is itself uniquely determined by $\delta$. We characterize the resulting graded ring by a universal property. We prove a characteristic-free center criterion for gr-simplicity whenever $Z(R[t;\delta])$ is a graded subring; in characteristic zero, gr-simplicity is equivalent to $\delta$-gr-simplicity of $R$ and $\gamma$-outerness of $\delta$, extending Jordan's simplicity criterion to the graded setting. We further show that $R[t;\delta]$ is gr-prime if and only if $R$ is $\delta$-gr-prime, and that gr-Noetherianity of $R$ passes to $R[t;\delta]$, recovering a graded Hilbert basis theorem as a special case. When $\Gamma$ is abelian, gr-simplicity and gr-primality are shown to be invariants of homogeneous graded Morita equivalence, and every ring homogeneously graded equivalent to $R[t;\delta]$ via a compatible idempotent is again a graded differential polynomial ring.

We introduce a notion of generalized homogeneous derivations on graded rings, extending the homogeneous derivations of Kanunnikov, and single out the gr-generalized derivations among them, those preserving the degree of every homogeneous component. Several results originally established for prime rings are extended to gr-prime rings. We also study the algebraic and module-theoretic structure of these maps, establish their functorial properties, and construct categories that record their derivation structure in both ring and module contexts.

We establish results related to homogeneous derivations, a concept introduced by Kanunnikov (2018). First, we prove the existence of a non-trivial family of derivations that are not homogeneous on graded rings. Furthermore, based on homogeneous derivations, we extend certain existing significant results in the context of prime (resp. semiprime) rings to gr-prime (resp. gr-semiprime) rings, such as Posner's and Herstein's theorems.