Characterizing subadjoint varieties among Legendrian varieties

In this paper we construct a symplectic Banach space (X, Ü) which does not split as a direct sum of closed isotropic subspaces.Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form Y X Y* is settled in the negative.The proof also shows that £(A") admits a nontrivial continuous homomorphism into £(//) where H is a Hilbert space.1. Introduction.Given a Banach space E, a linear symplectic form on F is a continuous bilinear map ß: E X E -> R which is alternating and nondegenerate in the (strong) sense that the induced map ß: E -E* given by Û(e)(f) = ü(e, f) is an isomorphism of E onto E*.A Banach space with such a form is called a symplectic Banach space.It can be shown, by essentially the argument of Lemma 2 below, that any symplectic Banach space can be renormed so that ß is an isometry.Any symplectic Banach space is reflexive.Standard examples of symplectic Banach spaces all arise in the following way.Let F be a reflexive Banach space and set E -Y © Y*.Define the linear symplectic form fiyby Qy[(^.y% (z>z*)] = z*(y) ~y*(z)-We define two symplectic spaces (£,, ß,) and (E2, ß2) to be equivalent if there is an isomorphism A : Ex -» E2 such that Q2(Ax, Ay) = ß,(x, y).A. Weinstein [10] has asked the question whether every symplectic Banach space is equivalent to one of the form (Y © Y*, üy).(See also [8].)A closed subspace F of a symplectic space E is isotropic if ß(x, y) = 0 for every x, v G F. F is Lagrangian if it is isotropic and possesses an isotropic complement.As Weinstein notes [10] the question is thus whether every symplectic space has a Lagrangian subspace.An obvious approach to this question is to construct a maximal isotropic subspace of any given symplectic space.However a maximal isotropic subspace of a symplectic space need not be Lagrangian.Indeed, let F be a reflexive Banach space and

Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

We consider the Weyl quantization on a flat non-standard symplectic vector space. We focus mainly on the properties of the Wigner functions defined therein. In particular we show that the sets of Wigner functions on distinct symplectic spaces are different but have non-empty intersections. This extends previous results to arbitrary dimension and arbitrary (constant) symplectic structure. As a by-product we introduce and prove several concepts and results on non-standard symplectic spaces which generalize those on the standard symplectic space, namely, the symplectic spectrum, Williamson’s theorem, and Narcowich-Wigner spectra. We also show how Wigner functions on non-standard symplectic spaces behave under the action of an arbitrary linear coordinate transformation.

It is shown how Darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear Schrödinger equations are realized as isospectral Hamiltonian flows. The parametrization follows from a moment map embedding of the symplectic vector space, reduced by suitable group actions, into the dual ■g̃+*̃ of the algebra ■g̃+ of positive frequency loops in a Lie algebra 𝔤 . The resulting phase space is identified with a Poisson subspace of ■g̃+* consisting of elements that are rational in the loop parameter. Reduced coordinates associated to the various Hermitian symmetric Lie algebras (𝔤 ,𝔨) corresponding to the classical Lie algebras are obtained.

This chapter is divided into three parts. The first and most substantial of these is devoted to symplectic vector spaces. Using methods of exterior algebra which are due to Elie Cartan, we study the rank of an exterior 2-form on a real vector space as well as that of the 2-form induced on a vector subspace. By means of associated canonical bases we show that there is only one model of a symplectic form and that vector subspaces may be classified by the rank of the induced form. In mechanics one uses mainly isotropic and coisotropic subspaces, particularly Lagrangian subspaces which will be discussed here in some detail. In particular we study the reduction of a symplectic vector space.

Introduction Part I. Symmetries and Integrals: 1. Distributions 2. Ordinary differential equations 3. Model differential equations and Lie superposition principle Part II. Symplectic Algebra: 4. Linear algebra of symplectic vector spaces 5. Exterior algebra on symplectic vector spaces 6. A Symplectic classification of exterior 2-forms in dimension 4 7. Symplectic classification of exterior 2-forms 8. Classification of exterior 3-forms on a 6-dimensional symplectic space Part III. Monge-Ampere Equations: 9. Symplectic manifolds 10. Contact manifolds 11. Monge-Ampere equations 12. Symmetries and contact transformations of Monge-Ampere equations 13. Conservation laws 14. Monge-Ampere equations on 2-dimensional manifolds and geometric structures 15. Systems of first order partial differential equations on 2-dimensional manifolds Part IV. Applications: 16. Non-linear acoustics 17. Non-linear thermal conductivity 18. Meteorology applications Part V. Classification of Monge-Ampere Equations: 19. Classification of symplectic MAEs on 2-dimensional manifolds 20. Classification of symplectic MAEs on 2-dimensional manifolds 21. Contact classification of MAEs on 2-dimensional manifolds 22. Symplectic classification of MAEs on 3-dimensional manifolds.

Following Weyl’s account in The Classical Groups we develop an analogue of the (first and second) Fundamental Theorems of Invariant Theory for rings of differential operators: when V is a k-dimensional complex vector space with the standard SL kC action, we give a presentation of the ring of invariant differential operators D(C[V n])SLkC and a description of the ring of differential operators on the G.I.T. quotient, D(C[V n]SLkC), which is the ring of differential operators on the (affine cone over the) Grassmann variety of k-planes in NumberingDepth=0-dimensional space. We also compute the Hilbert series of the associated graded rings GrD(C[V n])SLk and Gr(D(C[V n]SLkC)). This computation shows that earlier claims that the kernel of themap from D(C[V n])SLkC to D(C[V n]SLkC) is generated by the Casimir operator are incorrect. Something can be gleaned from these earlier incorrect computations though: the kernel meets the universal enveloping algebra of slkC precisely in the central elements of U(slkC).

This paper analyzes the action {\delta} of a Lie algebra X by derivations on a C*-algebra A. This action satisfies an "almost inner" property which ensures affiliation of the generators of the derivations {\delta} with A, and is expressed in terms of corresponding pseudo-resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*-algebra A, it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo- resolvents. If the Lie action {\delta} is ergodic, i.e. the only elements of A on which all the derivations in {\delta}_x vanish are multiples of the identity, then this extension is given by a (non-degenerate) symplectic form {\sigma} on X. Moreover, the algebra generated by the pseudo-resolvents coincides with the resolvent algebra based on the symplectic space (X, {\sigma}). Thus the resolvent algebra of the canonical commutation relations, which was recently introduced in physically motivated analyses of quantum systems, appears also naturally in the representation theory of Lie algebras of derivations acting on C*-algebras.

Let $$G \subset GL(V)$$ be a reductive algebraic subgroup acting on the symplectic vector space $$W=(V \oplus V^*)^{\oplus m}$$ , and let $$\mu :\ W \rightarrow Lie(G)^*$$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $$\mu ^{-1}(0)/\!/G$$ for classes of examples where $$G=GL(V)$$ , $$O(V)$$ , or $$Sp(V)$$ . For these classes of examples, $$\mu ^{-1}(0)/\!/G$$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert–Chow morphism with the (well-known) symplectic desingularizations of $$\mu ^{-1}(0)/\!/G$$ .

We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its "purification". As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein–Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein–Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.

In the preceding chapters we analyzed some properties of a vector space E\(_{\textit{n}}\). In this chapter we introduce into E\(_{\textit{n}}\) two other operations: the scalar product and the antiscalar product. A vector space equipped with the first operation is called a Euclidean vector space, whereas when it is equipped with the second operation, it is said to be a symplectic vector space. These operations allow us to introduce into E\(_{\textit{n}}\) many other geometric and algebraic concepts such as length of a vector, orthogonality between two vectors, etc. Further, eigenvalues and eigenvectors of a linear map are analyzed together with orthogonal transformations of \(E_n\). Finally, symplectic vector spaces are introduced and some their properties studied.

Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.

We briefly survey the general scheme of deformation quantization on symplectic vector spaces and analyze its functional analytic aspects. We treat different star products in a unified way by systematically using an appropriate space of analytic test functions for which the series expansions of the star products in powers of the deformation parameter converge absolutely. The star products are extendable by continuity to larger functional classes. The uniqueness of the extension is guaranteed by suitable density theorems. We show that the maximal star product algebra with the absolute convergence property, consisting of entire functions of an order at most 2 and minimal type, is nuclear. We obtain an integral representation for the star product corresponding to the Cahill-Glauber s-ordering, which connects the normal, symmetric, and antinormal orderings continuously as s varies from 1 to −1. We exactly characterize those extensions of the Wick and anti-Wick correspondences that are in line with the known extension of the Weyl correspondence to tempered distributions.

Let (V, Ω) be a symplectic vector space and let $${\phi : M \rightarrow V}$$ be a symplectic immersion. We show that $${\phi(M) \subset V}$$ is locally an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of Cahen et al. (J Geom Phys 59(4):409f́b-425, 2009) if and only if the second fundamental form of $${\phi}$$ is parallel. Furthermore, we show that any symmetric space, which admits an immersion as an e.s.s.s., also admits a full such immersion, i.e., such that $${\phi(M)}$$ is not contained in a proper affine subspace of V, and this immersion is unique up to affine equivalence. Moreover, we show that any extrinsic symplectic immersion of M factors through to the full one by a symplectic reduction of the ambient space. In particular, this shows that the full immersion is characterized by having an ambient space V of minimal dimension.

In this paper we construct a symplectic Banach space ( X , Ω ) (X,\Omega ) which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form Y × Y ∗ Y \times {Y^ \ast } is settled in the negative. The proof also shows that L ( X ) \mathfrak {L}(X) admits a nontrivial continuous homomorphism into L ( H ) \mathfrak {L}(H) where H H is a Hilbert space.