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The aim of this paper is to show that the canonical quantization of the moment maps on symplectic vector spaces naturally gives rise to the oscillator representations. More precisely, let $(W,\omega)$ denote a real symplectic vector space, on which a Lie group $G$ acts symplectically on the left, where $G$ denotes a real reductive Lie group $\mathrm{Sp}(n,\mathbb R), \mathrm U(p,q)$ or $\mathrm O^*(2n)$ in this paper. Then we quantize the moment map $\mu: W \to \mathfrak g_0^*$, where $\mathfrak g_0^*$ denotes the dual space of the Lie algebra $\mathfrak g_0$ of $G$. Namely, after taking a complex Lagrangian subspace $V$ of the complexification of $W$, we assign an element of the Weyl algebra for $V$ to $< \mu, X >$, which we denote by $< \hat{\mu}, X >$, for each $X \in \mathfrak g_0$. It is shown that the map $X \mapsto \mathrm i $ gives a representation of $\mathfrak g_0$ which extends to the one of $\mathfrak g$, the complexification of $\mathfrak g_0$, by linearity. With a suitable choice of the complex Lagrangian subspace $V$ in each case, the representation coincides with the oscillator representation of $\mathfrak g$., Comment: 24pages, no figure; corrected some typos (v2); 27pages, added Section 5 describing a relation between the image of complex Lagrangian subspaces by the moment map and the associated variety of the corresponding irreducible (g,K)-modules (v3); some references are added and replaced (v4)

Let K be an arbitrary field, let n , k , l be nonnegative integers satisfying n ⩾ 1 , 1 ⩽ k ⩽ 2 n , 0 ⩽ l ⩽ min ( n , k ) , and let V be a 2 n -dimensional vector space over K equipped with a nondegenerate alternating bilinear form f . Let W k , l denote the subspace of ⋀ k V generated by all vectors v ¯ 1 ∧ ⋯ ∧ v ¯ k , where v ¯ 1 , … , v ¯ k are k linearly independent vectors of V such that 〈 v ¯ 1 , … , v ¯ l 〉 is totally isotropic with respect to f . We prove that dim ( W k , l ) = 2 n k - 2 n 2 l - k - 2 . We give a recursive method for constructing a basis of W k , l and give a decomposition of W k , l relative to a given hyperbolic basis of V . We also study two linear mappings, one between the spaces W k , l and W k - 2 , l - 1 and another one between W k , l and W 2 n - k , n + l - k .

Let V be a 6-dimensional vector space over a field F , let f be a nondegenerate alternating bilinear form on V and let Sp ( V , f ) ≅ Sp 6 ( F ) denote the symplectic group associated with ( V , f ) . The group GL ( V ) has a natural action on the third exterior power ⋀ 3 V of V and this action defines five families of nonzero trivectors of V. Four of these families are orbits for any choice of the field F . The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of F that are contained in a fixed algebraic closure F ¯ of F . In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of Sp ( V , f ) ⊆ GL ( V ) on ⋀ 3 V .

Let (E, a) be a real 2n-dimensional symplectic vector space with symplectic form a, i.e., a is a nondegenerate skew-symmetric bilinear form on E. Then an n-dimensional subspace λ of E will be called a Lagrangean subspace if alλ = 0 holds. The set Λ(E) of all Lagrangean subspaces of (E, a) has a structure of n(n + l)-dimensional compact connected regular algebraic variety. If we put A\λ): = [μ 6 Λ(E) I dim (λ Π μ) = k} for λ £ Λ(E), then Λ°U) is a cell (i.e., diίϊeomorphic to i r ( n + 1 ) / 2 ) for any λ e Λ(E). Moreover Σ tf): = |J*2>i Λ*W) is an algebraic subvariety of Λ(E), and defines an oriented cycle of codimension one, whose Poincare dual is a generator of H(Λ(E), Z) = Z and defines the Maslov-Arnold index [1], [3], [4]. This index plays an important role in the proof of Morse index theorem in the calculus of variations [4]. In the present note, we shall give a differential geometric characterization of 2 U)> i ^ ? by introducing an appropriate riemannian metric on Λ{E) we shall show that 2 (λ) is the cut locus of some μ e Λ(E) and Λ°(X) is the interior set of μ. In fact, take a basis {eu fά] (1 , and ao((p, q), (p\ q')): = (q, p'y —

We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we only assume to be perfect and not of characteristic 2. Our work uses the theory of representations of partially ordered sets with (order reversing) involution; for (co)isotropic triples, the relevant poset is "$2 + 2 + 2$" consisting of three independent ordered pairs, with the involution exchanging the members of each pair. A key feature of the classification is that any indecomposable (co)isotropic triple is either "split" or "non-split". The latter is the case when the poset representation underlying an indecomposable (co)isotropic triple is itself indecomposable. Otherwise, in the "split" case, the underlying representation is decomposable and necessarily the direct sum of a dual pair of indecomposable poset representations; the (co)isotropic triple is a "symplectification". In the course of the paper we develop the framework of "symplectic poset representations", which can be applied to a range of problems of symplectic linear algebra. The classification of linear Hamiltonian vector fields, up to conjugation, is an example; we briefly explain the connection between these and (co)isotropic triples. The framework lends itself equally well to studying poset representations on spaces carrying a non-degenerate symmetric bilinear form; we mainly keep our focus, however, on the symplectic side., Comment: 91 pages

In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. A 3-Lie algebra enjoys a product structure if and only if it is the direct sum (as vector spaces) of two subalgebras. We find that there are four types special integrability conditions, and each of them gives rise to a special decomposition of the original 3-Lie algebra. They are also related to $\huaO$-operators, Rota-Baxter operators and matched pairs of 3-Lie algebras. Parallelly, we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special integrability conditions. Finally, we add compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, to introduce the notions of a complex product structure, a para-K\"{a}hler structure and a pseudo-K\"{a}hler structure on a 3-Lie algebra. We use 3-pre-Lie algebras to construct these structures. Furthermore, a Levi-Civita product is introduced associated to a pseudo-Riemannian 3-Lie algebra and deeply studied., Comment: 37 pages

Integrable many-body systems of Ruijsenaars–Schneider–van Diejen type displaying action-angle duality are derived by Hamiltonian reduction of the Heisenberg double of the Poisson–Lie group $$\mathrm{SU}(2n)$$ . New global models of the reduced phase space are described, revealing non-trivial features of the two systems in duality with one another. For example, after establishing that the symplectic vector space $$\mathbb {C}^n\simeq \mathbb {R}^{2n}$$ underlies both global models, it is seen that for both systems the action variables generate the standard torus action on $$\mathbb {C}^n$$ , and the fixed point of this action corresponds to the unique equilibrium positions of the pertinent systems. The systems in duality are found to be non-degenerate in the sense that the functional dimension of the Poisson algebra of their conserved quantities is equal to half the dimension of the phase space. The dual of the deformed Sutherland system is shown to be a limiting case of a van Diejen system.

We consider moduli spaces of cyclic configurations of $N$ lines in a $2n$-dimensional symplectic vector space, such that every set of $n$ consecutive lines generates a Lagrangian subspace. We study geometric and combinatorial problems related to these moduli spaces, and prove that they are isomorphic to quotients of spaces of symmetric linear difference operators with monodromy $-1$. The symplectic cross-ratio is an invariant of two pairs of $1$-dimensional subspaces of a symplectic vector space. For $N = 2n+2$, the moduli space of Lagrangian configurations is parametrized by $n+1$ symplectic cross-ratios. These cross-ratios satisfy a single remarkable relation, related to tridiagonal determinants and continuants, given by the Pfaffian of a Gram matrix., Comment: 29 pages, minor revisions and corrections

Quantization of the Teichm\"uller space of a non-compact Riemann surface has emerged in 1980's as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate functions on the edges form a coordinate system for the Teichm\"uller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change of triangulations, one must construct a unitary operator between the Hilbert spaces intertwining the quantum coordinate operators and satisfying the composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a family of reducible representations, and the phase constants are all trivial. In the present paper, we employ the harmonic-analytic theory of the Shale-Weil intertwiners for the Schr\"odinger representations, as well as Faddeev-Kashaev's quantum dilogarithm function, to construct a family of irreducible representations of the quantum shear coordinate functions and the corresponding intertwiners for the changes of triangulations. The phase constants are explicitly computed and described by the Maslov indices of the Lagrangian subspaces of a symplectic vector space, and by the pentagon relation of the flips of triangulations. The present work may generalize to the cluster $\mathscr{X}$-varieties.

We show that equivariant tilting modules over equivariant algebras induce equivalences of derived factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived factorization category of a noncommutative gauged Landau-Ginzburg model $(\Lambda,\chi, w)^{\mathbb{G}_m}$, where $\Lambda$ is a noncommutative resolution of the quotient singularity $W/\operatorname{GSp}(Q)$ arising from a certain representation $W$ of the symplectic similitude group $\operatorname{GSp}(Q)$ of a symplectic vector space $Q$.

In a former paper we proposed a model for the quantization of gravity by working in a bundle $E$ where we realized the Hamilton constraint as the Wheeler-DeWitt equation. However, the corresponding operator only acts in the fibers and not in the base space. Therefore, we now discard the Wheeler-DeWitt equation and express the Hamilton constraint differently, either with the help of the Hamilton equations or by employing a geometric evolution equation. There are two modifications possible which both are equivalent to the Hamilton constraint and which lead to two new models. In the first model we obtain a hyperbolic operator that acts in the fibers as well as in the base space and we can construct a symplectic vector space and a Weyl system. \nd In the second model the resulting equation is a wave equation in $\so \times (0,\infty)$ valid in points $(x,t,\xi)$ in $E$ and we look for solutions for each fixed $\xi$. This set of equations contains as a special case the equation of a quantized cosmological Friedmann universe without matter but with a cosmological constant, when we look for solutions which only depend on $t$. Moreover, in case $\so$ is compact we prove a spectral resolution of the equation.

We construct an explicit family of irreducible representations of the canonical commutation relations parametrized by the space of pairs of transverse Lagrangian subspaces in the complexification of a symplectic vector space. We identify different restrictions of this family with previously studied families of Lion-Vergne, Satake, Grossmann, and Mumford. The classical Schrodinger and Fock-Segal-Bargmann representations are also derived.

We study linear projections on Pluecker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of m-dimensional linear subspaces in a symplectic vector space of dimension 2m, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples., 14 pages

We study particular classes of states on the Weyl algebra$$\mathcal {W}$$Wassociated with a symplectic vector spaceSand on the von Neumann algebras generated in representations of$$\mathcal {W}$$W. Applications in quantum physics require an implementation of constraint equations, e.g., due to gauge conditions, and can be based on the so-called Dirac states. The states can be characterized by nonlinear functions onS, and it turns out that those corresponding to non-trivial Dirac states are typically discontinuous. We discuss general aspects of this interplay between functions onSand states, but also develop an analysis for a particular example class of non-trivial Dirac states. In the last part, we focus on the specific situation with$$S = L^2(\mathbb {R}^n)$$S=L2(Rn)or test functions on$$\mathbb {R}^n$$Rnand relate properties of states on$$\mathcal {W}$$Wwith those of generalized functions on$$\mathbb {R}^n$$Rnor with harmonic analysis aspects of corresponding Borel measures on Schwartz functions and on temperate distributions.

Üç bölümden oluşan bu çalışmanın ilk bölümü tezdeki araştırma konusunun kısa tarihçesine ve konunun önemine ayrılmıştır. İkinci bölüm ise tezin diğer bölümlerinde kullanılacak temel kavramlara ayrılmıştır. Bu bölümde önce temel cebirsel kavramlar verilmiş ve sonrasında simplektik vektör uzayları tanıtılmıştır. Ayrıca kompleks manifoldlar ve üzerindeki temel yapılar kısaca sunulmuştur. Bu bölümde son olarak manifoldlar, manifoldlar üzerindeki temel kavramlar ve bazı manifold çeşitleri verilmiştir. Son olarak, üçüncü bölümde ise genelleştirilmiş manifoldlar kavramı genelleştirilmiş demet kavramı yardımıyla verilmiştir. Burada öncelikle genelleştirilmiş geometride önemli olan Courant braketi ve özellikleri detaylı olarak sunulmuştur. Ayrıca örnekler verilmiş ve genelleştirilmiş kompleks manifoldlar kavramı tanıtılmıştır. Bu bölümde son olarak özel bir genelleştirilmiş kompleks manifold olan kalibre kompleks manifoldları incelenmiş ve kalibre kompleks manifoldların sağladığı özellikler sırasıyla sunulmuştur., This thesis consists of three chapters. In chapter 1, the brief history of the topic of the thesis is given and the motivation of the research problem is presented. In chapter 2, we first review basic algebraic notions which will be used throughout thesis. In section 2, we give certain definitions and theorems on symplectic vector spaces. We also present brief notes on complex vector spaces in section 3. In the last section, we consider manifold theory and give many notions related to the manifold theory and discuss some special manifolds. In the last chapter, we study the generalized geometry which is the main topic of this thesis. We first present generalized bundle, generalized metric, Dirac structure and Courant bracket. Then we introduce generalized complex structures supported by examples and investigate the main properties. In the last section, we consider calibrated complex structures as a special generalized complex structure and investigate their main properties.

The hyperplanes of the symplectic dual polar space DW(5; F) that arise from the Grassmann embedding have been classied in [B.N. Cooperstein and B. De Bruyn. Points and hyperplanes of the universal embedding space of the dual polar space DW(5; q), q odd. Michigan Math. J., 58:195{212, 2009.] in case F is a finite field of odd characteristic, and in [B. De Bruyn. Hyperplanes of DW(5;K) with K a perfect eld of characteristic 2. J. Algebraic Combin., 30:567{584, 2009.] in case F is a perfect eld of characteristic 2. In the present paper, these classifications are extended to arbitrary fields. In the case of characteristic 2 however, it was not possible to provide a complete classification. The main tool in the proof is the classification of the quasi-Sp(V; f)-equivalence classes of trivectors of a 6-dimensional symplectic vector space (V; f) obtained in [B. De Bruyn and M. Kwiatkowski. A 14-dimensional module for the symplectic group: orbits on vectors. Comm. Algebra,43:4553{4569, 2015.

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $${{\mathbb{R}}^4}$$ , the projective plane $${{\mathbb{C}}P^2}$$ , and the monotone $${S^2 \times S^2}$$ . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for $${T^*{\mathbb{T}}^2}$$ , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

We introduce a novel commutative C*-algebra C R ( X ) of functions on a symplectic vector space ( X , σ ) admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of C R ( X ) to the resolvent algebra introduced by Buchholz and Grundling [2] . The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [1] . We also define a Berezin-type quantization map on all of C R ( X ) , which continuously and bijectively maps it onto the resolvent algebra. The commutative resolvent algebra C R ( X ) , generally defined on a real inner product space X, intimately depends on the finite dimensional subspaces of X. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum.

This chapter is a brief introduction to symplectic manifolds. We will start this chapter by defining a symplectic vector space (Sect. 1.1). After briefly reviewing the notion of an almost complex structure on a vector space, we will see how the compatibility condition between the symplectic form and an almost complex structure gives rise to an inner product. In Sect. 1.3, we will discuss the definition of symplectic manifolds, describe some of their basic properties and will finally see some examples in Sect. 1.4. Section 1.2 contains a review of results from differential topology which are essential material for what follows.

Let V be a symplectic vector space and let $\mu$ be the oscillator representation of Sp(V). It is natural to ask how the tensor power representation $\mu^{\otimes t}$ decomposes. If V is a real vector space, then Howe-Kashiwara-Vergne (HKV) duality asserts that there is a one-one correspondence between the irreducible subrepresentations of Sp(V) and the irreps of an orthogonal group O(t). It is well-known that this duality fails over finite fields. Addressing this situation, Gurevich and Howe have recently assigned a notion of rank to each Sp(V) representation. They show that a variant of HKV duality continues to hold over finite fields, if one restricts attention to subrepresentations of maximal rank. The nature of the rank-deficient components was left open. Here, we show that all rank-deficient Sp(V)-subrepresentations arise from embeddings of lower-order tensor products of $\mu$ and $\bar\mu$ into $\mu^{\otimes t}$. The embeddings live on spaces that have been studied in quantum information theory as tensor powers of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes. We then find that the irreducible Sp(V) subrepresentations of $\mu^{\otimes t}$ are labelled by the irreps of orthogonal groups O(r) acting on certain r-dimensional spaces for r, Comment: 22 pages, 3 figures. v2: presentation updated, title changed

The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and 'nearly diagonal' semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states. Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting. Our contribution is to leverage the discrete Stone-von Neumann theorem and the symplectic formalism of qudit stabiliser mechanics towards extending results of Zeng-Cheng-Chuang (2008) and Beigi-Shor (2010) to higher dimensions in a uniform manner. We further give a simple algorithm for recursively enumerating all gates of the Clifford hierarchy, a simple algorithm for recognising and diagonalising semi-Clifford gates, and a concise proof of the classification of the diagonal Clifford hierarchy gates due to Cui-Gottesman-Krishna (2016) for the single-qudit case. We generalise the efficient gate teleportation protocols of semi-Clifford gates to the qudit setting and prove that every third level gate of one qudit (of any prime dimension) and of two qutrits can be implemented efficiently. Numerical evidence gathered via the aforementioned algorithms support the conjecture that higher-level gates can be implemented efficiently., Comment: Draft version. Comments, questions, & corrections welcome

The main aim of this paper is to show that one can construct [Formula: see text]-modules of [Formula: see text] associated with the finite-dimensional representation of [Formula: see text] by quantizing the moment map on the symplectic vector space [Formula: see text] and using the fact that [Formula: see text] is a dual pair. Then one obtains the [Formula: see text]-type formula, the Gelfand–Kirillov dimension and the Bernstein degree of them for all non-negative integers [Formula: see text] satisfying [Formula: see text] when [Formula: see text] and [Formula: see text] is even. In fact, one finds that the Gelfand–Kirillov dimension is equal to [Formula: see text] and the Bernstein degree is equal to [Formula: see text].

Let BunG be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, Gaiotto (2016) associated to any symplectic representation of G a Lagrangian subvariety of T∗BunG. We give a simple interpretation of (a generalization of) Gaiotto’s construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.

[EN] Two families of symplectic methods specially designed for second-order time-dependent linear systems are presented. Both are obtained from the Magnus expansion of the corresponding first-order equation, but otherwise they differ in significant aspects. The first family is addressed to problems with low to moderate dimension, whereas the second is more appropriate when the dimension is large, in particular when the system corresponds to a linear wave equation previously discretised in space. Several numerical experiments illustrate the main features of the new schemes. (C) 2017 Elsevier B.V. All rights reserved., Bader, Blanes, Casas and Kopylov acknowledge the Ministerio de Economia y Competitividad (Spain) for financial support through the coordinated project MTM2013-46553-C3-3-P. Additionally, Kopylov has been partly supported by fellowship GRISOLIA/2015/A/137 from the Generalitat Valenciana.

We study the Berezin–Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product.