Symplectic Group Actions Research Articles

Equivariant categories of symplectic surfaces and fixed loci of Bridgeland moduli spaces

Given an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(\mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(\mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed loci of symplectic group actions on moduli spaces of stable objects on symplectic surfaces.

Equivariant Lagrangian Floer cohomology via semi-global Kuranishi structures

Using a simplified version of Kuranishi perturbation theory that we call semi-global Kuranishi structures, we give a definition of the equivariant Lagrangian Floer cohomology of a pair of Lagrangian submanifolds that are fixed under a finite symplectic group action and satisfy certain simplifying assumptions.

Differential Invariants of Linear Symplectic Actions

We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem.

Homologically trivial group actions on elliptic surfaces

In this paper, we study the minimal symplectic elliptic surfaces [Formula: see text] with homologically trivial symplectic finite group actions, and get a rigidity theorem under some restriction.

The Hochschild cohomology ring of a global quotient orbifold

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields TXpoly on X, and is generated as a TXpoly-algebra by the sum of the determinants det⁡(NXg) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne–Mumford stacks in general. We discuss, in the case of a symplectic group action on a symplectic variety X, relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, we employ compatible trivializations of the determinants det⁡(NXg), which requires (for the cup product) a nontrivial normalization missing in previous literature.

Hamiltonian and symplectic symmetries: An introduction

Classical mechanical systems are modeled by a symplectic manifold $(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $\omega$. These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact abelian Lie groups that are, in addition, of type, i.e. they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, finite dimensional integrable Hamiltonian systems, more general symplectic actions, and topology, have flourished. In this paper we review classical and recent results on Hamiltonian and non Hamiltonian symplectic group actions roughly starting from the results of these authors. The paper also serves as a quick introduction to the basics of symplectic geometry.

Symplectic group actions on homotopy elliptic surfaces

In this paper, the authors study the homologically trivial symplectic group actions on homotopy elliptic surfaces E(n) and get some rigidity results.

Fundaments of Quaternionic Clifford Analysis I: Quaternionic Structure

Introducing a quaternionic structure on Euclidean space, the fundaments for quaternionic and symplectic Clifford analysis are studied in detail from the viewpoint of invariance for the symplectic group action.

Symplectic group actions and covering spaces

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied.

The permutation action of finite symplectic groups of odd characteristic on their standard modules

We study the space of functions on a finite-dimensional vector space over a field of odd order as a module for a symplectic group. We construct a basis of this module with the following special properties. Each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule. From these properties, the dimension and composition factors of the submodule generated by any subset of the basis can be determined. These results apply to incidence geometry of the symplectic polar space, yielding the symplectic analogue of Hamada's additive formula for the p-ranks of the incidence matrices between points and flats. A special case leads to a closed formula for the p-rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle over a field of odd order. Together with earlier results on the 2-ranks, this result completes the determination of the p-ranks for these quadrangles.

The reduced spaces of a symplectic Lie group action

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux et al. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.

Classification of Lagrangian stars and their symplectic reductions

The Lagrangian star is a germ of the system of Lagrangian submanifolds in the symplectic manifold . We investigate the symplectic group action on Lagrangian stars and construct the basic invariants of such action. The Kashiwara signature for 3-Lagrangian linear stars is generalized to the nonlinear case and the generalized contact classes for Lagrangian stars are constructed. Finally, we obtain the generic classification of simple normal forms of reduced Lagrangian stars with respect to a hypersurface.

Persistence and stability of relative equilibria

We consider relative equilibria in symmetric Hamiltonian systems, and their persistence or bifurcation as the momentum is varied. In particular, we extend a classical result about persistence of relative equilibria from values of the momentum map that are regular for the coadjoint action, to arbitrary values, provided that either (i) the relative equilibrium is at a local extremum of the reduced Hamiltonian or (ii) the action on the phase space is (locally) free. The first case uses just point-set topology, while in the second we rely on the local normal form for (free) symplectic group actions, and then apply the splitting lemma. We also consider the Lyapunov stability of extremal relative equilibria. The group of symmetries is assumed to be compact.

A Group Algebra for Inductive Limit Groups. Continuity Problems of the Canonical Commutation Relations

Given an inductive limit group $$G = \underrightarrow {\lim }G_\beta ,\beta \in \Gamma$$ where each $$G_\beta$$ is locally compact, and a continuous two-cocycle $$\rho \in Z^2 (G,T)$$ , we construct a C*-algebra $$ L $$ group algebra $$ C_\rho ^* (G_d )$$ is imbedded in its multiplier algebra $$ M(L) $$ , and the representations of $$ L $$ are identified with the strong operator continuous $$ \rho - {\text{representation}} $$ of G. If any of these representations are faithful, the above imbedding is faithful. When G is locally compact, $$ L $$ is precisely $$ C_\rho ^* (G) $$ , the twisted group algebra of G, and for these reasons we regard $$ L $$ in the general case as a twisted group algebra for G. Applying this construction to the CCR-algebra over an infinite dimensional symplectic space (S,\,B),we realise the regular representations as the representation space of the C*-algebra $$ L $$ , and show that pointwise continuous symplectic group actions on (S,\, B) produce pointwise continuous actions on $$ L $$ , though not on the CCR-algebra. We also develop the theory to accommodate and classify 'partially regular' representations, i.e. representations which are strong operator continuous on some subgroup H of G (of suitable type) but not necessarily on G, given that such representations occur in constrained quantum systems.

On adjacencies of orbits of a symplectic group action on the manifold of complete flags in a symplectic space

On adjacencies of orbits of a symplectic group action on the manifold of complete flags in a symplectic space

Critical Point Theory for Indefinite Functionals with Symmetries

LetXbe a Hilbert space andφ∈C1(X, R) be strongly indefinite. Assume in addition that a compact Lie groupGacts orthogonally onXand thatφis invariant. In order to find critical points ofφwe develop the limit relative category of Fournieret al. in the equivariant context. We use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem. In the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension. Also other conditions of the classical symmetric mountain pass theorem forG=Z/2 (due to Ambrosetti and Rabinowitz) can be weakened considerably. For example, we are able to deal with infinite-dimensional fixed point spaces. The proofs consist of a direct reduction to a relative Borsuk–Ulam type theorem. This provides a new proof even for the classical mountain pass theorem. The abstract critical point theorems are applied to an elliptic system with Dirichlet boundary conditions. We only need a weak version of the usual superquadraticity condition. The linking theorem can be applied to asymptotically linear Hamiltonian systems which are symmetric with respect to a (generalized) symplectic group action.

Dynamical algebras in classical mechanics

For a spectrum-generating algebra of classical observables, it is proven that the phase space dynamics simplifies to a Hamiltonian system on submanifolds of the algebra's dual. These submanifolds are coadjoint orbits if the algebra arises from a symplectic group action. If the Hamiltonian splits into the sum of a function of the algebra generators plus a commuting part, then the dynamics transfers to the dual space and an explicit formula is given for the flow vector field on the coadjoint orbits. A unique feature of the presentation is that all constructions are at the Lie algebra level.

Quantum mechanics of non-linear systems

A generalisation of Weyl's correspondence rule and the associated Moyal bracket for systems with non-linear phase spaces that admit symplectic transitive group actions is presented. This provides a simple account with explicit constructions of some of the general results of Bayen et al. (1978).

All local classical symmetries in Hamiltonian mechanics

Using the formalism of symplectic group actions and coadjoint orbits, we give a complete list of all classical simple Lie algebras which are local symmetries for a given Hamiltonian vector field.

Completely integrable systems and symplectic actions

We study results on a class of completely integrable systems, for instance, with Hamiltonian H (x,y) = (1/2) 𝒥ni=1y2i +𝒥i<j(xi−yj)−2 +α𝒥ni=1x2i, using quotient manifolds induced by symplectic group actions, which enables us to integrate the systems and understand their complete integrability. In addition, we give a natural interpretation for the scattering maps associated with these systems.