1,337 results on '"Symplectic vector space"'
Search Results
2. Decomposition of (Co)isotropic Relations
- Author
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Lorand, Jonathan and Weinstein, Alan
- Subjects
linear relation ,duality ,symplectic vector space ,coisotropic relation ,Mathematical Sciences ,Physical Sciences ,Mathematical Physics - Abstract
We identify 13 isomorphism classes of indecomposable coisotropic relations between Poisson vector spaces and show that every coisotropic relation between finite-dimensional Poisson vector spaces may be decomposed as a direct sum of multiples of these indecomposables. We also find a list of 13 invariants, each of which is the dimension of a space constructed from the relation, such that the 13-vector of multiplicities and the 13-vector of invariants are related by an invertible matrix over Z. It turns out to be simpler to do the analysis above for isotropic relations between presymplectic vector spaces. The coisotropic/Poisson case then follows by a simple duality argument.
- Published
- 2016
3. Efficient quantum gate teleportation in higher dimensions.
- Author
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Silva, Nadish de
- Subjects
- *
QUANTUM teleportation , *QUANTUM gates , *QUANTUM computing , *ALGORITHMS , *TELEPORTATION - Abstract
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 higherdimensional qudit setting. Our contribution is to leverage the discrete Stone–von Neumann theorem and the symplectic formalism of qudit stabilizer theory towards extending the results of Zeng et al. (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 recognizing and diagonalizing semi-Clifford gates, and a concise proof of the classification of the diagonal Clifford hierarchy gates due to Cui et al. (2016) for the single-qudit case. We generalize 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 supports the conjecture that higher-level gates can be implemented efficiently. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
4. (Co)isotropic triples and poset representations.
- Author
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HERRMANN, CH., LORAND, J., and WEINSTEIN, A.
- Subjects
PARTIALLY ordered sets ,SYMPLECTIC spaces ,VECTOR spaces ,VECTOR fields ,REPRESENTATION theory - Abstract
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 assume only 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 C 2 C 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. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
5. Selective categories and linear canonical relations
- Author
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Li-Bland, D and Weinstein, A
- Subjects
symplectic vector space ,canonical relation ,rigid monoidal category ,highly selective category ,quantization ,Applied Mathematics ,Mathematical Physics ,Pure Mathematics - Abstract
A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are "good". We then apply this notion to the category SLREL of linear canonical relations and the result WW(SLREL) of our version of the WW construction, identifying the morphisms in the latter with pairs (L; k) consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in SLREL itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts.
- Published
- 2014
6. Selective Categories and Linear Canonical Relations
- Author
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Li-Bland, David
- Subjects
Applied Mathematics ,Pure Mathematics ,Mathematical Sciences ,symplectic vector space ,canonical relation ,rigid monoidal category ,highly selective category ,quantization ,Mathematical Physics ,Applied mathematics ,Pure mathematics - Abstract
A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are "good". We then apply this notion to the category SLREL of linear canonical relations and the result WW(SLREL) of our version of the WW construction, identifying the morphisms in the latter with pairs (L; k) consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in SLREL itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts.
- Published
- 2014
7. The Maslov cycle as a Legendre singularity and projection of a wavefront set
- Author
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Weinstein, Alan
- Subjects
symplectic vector space ,lagrangian grassmannian ,Fourier integral distribution ,Maslov cycle ,Pure Mathematics ,General Mathematics - Abstract
A Maslov cycle is a singular variety in the lagrangian grassmannian Λ(V) of a symplectic vector space V consisting of all lagrangian subspaces having nonzero intersection with a fixed one. Givental has shown that a Maslov cycle is a Legendre singularity, i.e. the projection of a smooth conic lagrangian submanifold S in the cotangent bundle of Λ(V). We show here that S is the wavefront set of a Fourier integral distributionwhich is "evaluation at 0 of the quantizations". © 2013 Sociedade Brasileira de Matemática.
- Published
- 2013
8. An elementary proof of the symplectic spectral theorem.
- Author
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Malagón, Camilo Sanabria
- Subjects
ORTHOGRAPHIC projection ,MATHEMATICS theorems ,VECTOR spaces ,LAGRANGE equations ,DARBOUX transformations - Abstract
The classical spectral theorem completely describes self-adjoint operators on finite-dimensional inner product vector spaces as linear combinations of orthogonal projections onto pairwise orthogonal subspaces. We prove a similar theorem for self-adjoint operators on finite-dimensional symplectic vector spaces over perfect fields. We show that these operators decompose according to a polarization, i.e. as the product of an operator on a Lagrangian subspace and its dual on a complementary Lagrangian. Moreover, if all the eigenvalues of the operator are in the base field, then there exists a Darboux basis such that the matrix representation of the operator is 2 × 2 blocks block-diagonal, where the first block is in Jordan normal form and the second block is the transpose of the first one. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
9. (Co)isotropic triples and poset representations
- Author
-
Alan Weinstein, Jonathan Lorand, and Christian Herrmann
- Subjects
Pure mathematics ,Algebra and Number Theory ,Symmetric bilinear form ,Duality (order theory) ,Symplectic vector space ,15A21, 53D99, 16G20 ,Mathematics - Symplectic Geometry ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,Geometry and Topology ,Representation Theory (math.RT) ,Partially ordered set ,Indecomposable module ,Mathematics - Representation Theory ,Mathematical Physics ,Analysis ,Symplectic geometry ,Vector space ,Mathematics ,Dual pair - Abstract
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
- Published
- 2021
10. Automorphisms of Finite Orthoalgebras, Exceptional Root Systems and Quantum Mechanics
- Author
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Ruuge, Artur E., Van Oystaeyen, Fred, Silvestrov, Sergei, editor, Paal, Eugen, editor, Abramov, Viktor, editor, and Stolin, Alexander, editor
- Published
- 2009
- Full Text
- View/download PDF
11. Some Facts About the Wick Calculus
- Author
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Lerner, N., Morel, J. -M., editor, Takens, F., editor, Teissier, B., editor, Feichtinger, Hans G., Helffer, Bernard, Lamoureux, Michael P., Lerner, Nicolas, Toft, Joachim, Rodino, Luigi, editor, and Wong, M. W., editor
- Published
- 2008
- Full Text
- View/download PDF
12. From momentum maps and dual pairs to symplectic and Poisson groupoids
- Author
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Marle, Charles-Michel, Bass, Hyman, editor, Oesterlé, Joseph, editor, Weinstein, Alan, editor, Marsden, Jerrold E., editor, and Ratiu, Tudor S., editor
- Published
- 2005
- Full Text
- View/download PDF
13. Symplectic, Product and Complex Structures on 3‐Lie Algebras
- Author
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Rong Tang and Yunhe Sheng
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Structure (category theory) ,FOS: Physical sciences ,01 natural sciences ,Symplectic vector space ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,Linear complex structure ,0101 mathematics ,Symplectomorphism ,Moment map ,Mathematical Physics ,Mathematics ,Algebra and Number Theory ,Symplectic group ,010308 nuclear & particles physics ,Direct sum ,010102 general mathematics ,Mathematical Physics (math-ph) ,Mathematics - Rings and Algebras ,Symplectic representation ,Differential Geometry (math.DG) ,Rings and Algebras (math.RA) ,If and only if ,Product (mathematics) ,Phase space ,Vector space ,Symplectic geometry - Abstract
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
- Published
- 2021
14. Proof of Theorem I.38
- Author
-
Audin, Michèle, da Silva, Ana Cannas, Lerman, Eugene, Castellet, Manuel, editor, Audin, Michèle, da Silva, Ana Cannas, and Lerman, Eugene
- Published
- 2003
- Full Text
- View/download PDF
15. Root Systems In Finite Symplectic Vector Spaces.
- Author
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Lentner, Simon
- Subjects
INVARIANT subspaces ,VECTOR analysis ,MATHEMATICAL analysis ,VECTOR spaces ,FROBENIUS algebras ,ASSOCIATIVE algebras ,FROBENIUS groups ,GROUP theory - Abstract
We study realizations of root systems in possibly degenerate symplectic vector spaces over finite fields, up to symplectic isomorphisms. The main result of this article is the classification of such realizations for the field 𝔽2. Thereby, each root system requires a specific degree of degeneracy of the symplectic vector space. Our main motivation for this article is that for each such realization of a root system we can construct a Nichols algebra over a nonabelian group. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
16. Global Description of Action-Angle Duality for a Poisson–Lie Deformation of the Trigonometric $$\varvec{\mathrm {BC}_n}$$ BC n Sutherland System
- Author
-
Ian Marshall and László Fehér
- Subjects
Physics ,Nuclear and High Energy Physics ,Pure mathematics ,Integrable system ,Statistical and Nonlinear Physics ,Torus ,Fixed point ,Conserved quantity ,symbols.namesake ,Symplectic vector space ,Phase space ,symbols ,Hamiltonian (quantum mechanics) ,Mathematical Physics ,Poisson algebra - Abstract
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.
- Published
- 2019
17. Dirac and normal states on Weyl–von Neumann algebras
- Author
-
Hörmann, Günther
- Subjects
Physics ,Class (set theory) ,Pure mathematics ,Weyl algebra ,Generalized function ,Quantization with constraints ,Generalized functions ,Dirac (software) ,FOS: Physical sciences ,81R10 (Primary) 46F99 (Secondary) ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Gauge (firearms) ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Symplectic vector space ,Nonlinear system ,symbols.namesake ,FOS: Mathematics ,symbols ,Measures on non-locally compact groups ,Mathematical Physics ,Von Neumann architecture - Abstract
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.
- Published
- 2021
18. Generalized complex manifolds
- Author
-
Akkan, Eray Emre, Şahin, Bayram, and Ege Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Ana Bilim Dalı
- Subjects
Simplektik Vektör Uzayı ,Genelleştirilmiş Demet ,Calibrated Complex Structures ,Generalized Tangent Bundle ,Generalized Complex Structure ,Symplectic Vector Space ,Genelleştirilmiş Kompleks Yapı ,Dirac Yapı ,Kalibre Kompleks Yapı ,Distribüsyon ,Kompleks Manifold ,Riemann Manifold ,Complex Manifold ,Kompleks Yapı ,Complex Structure ,Dirac Structure - Abstract
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.
- Published
- 2021
19. Irreducible self-adjoint representations of quantum Teichm\'uller space and the phase constants
- Author
-
Hyun Kyu Kim
- Subjects
20G05, 43A25, 47A67, 81R60, 22D10, 16G99 ,Pure mathematics ,010102 general mathematics ,Hilbert space ,General Physics and Astronomy ,01 natural sciences ,Linear subspace ,Fock space ,Quantization (physics) ,Symplectic vector space ,symbols.namesake ,0103 physical sciences ,symbols ,Quantum gravity ,010307 mathematical physics ,Geometry and Topology ,Unitary operator ,0101 mathematics ,Self-adjoint operator ,Mathematics - Representation Theory ,Mathematical Physics ,Mathematics - Abstract
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.
- Published
- 2020
20. Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations
- Author
-
Yuki Hirano
- Subjects
Pure mathematics ,Derived category ,Pfaffian ,matrix factorization ,Noncommutative geometry ,Representation theory ,equivariant tilting module ,Mathematics - Algebraic Geometry ,Symplectic vector space ,Mathematics::K-Theory and Homology ,Pfaffian variety ,14F08, 18G80, 16E35 ,FOS: Mathematics ,Equivariant map ,Geometry and Topology ,Variety (universal algebra) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics::Symplectic Geometry ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory ,Mathematical Physics ,Analysis ,Symplectic geometry ,Mathematics - Abstract
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 $Q$.
- Published
- 2020
- Full Text
- View/download PDF
21. Efficient quantum gate teleportation in higher dimensions
- Author
-
Nadish de Silva
- Subjects
General Mathematics ,Diagonal ,General Physics and Astronomy ,FOS: Physical sciences ,01 natural sciences ,Teleportation ,010305 fluids & plasmas ,Symplectic vector space ,Computer Science::Hardware Architecture ,Quantum gate ,Computer Science::Emerging Technologies ,0103 physical sciences ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Representation Theory (math.RT) ,010306 general physics ,Mathematical Physics ,Quantum computer ,Physics ,Sequence ,Quantum Physics ,Hierarchy (mathematics) ,General Engineering ,Mathematical Physics (math-ph) ,Algebra ,Quantum Physics (quant-ph) ,Mathematics - Representation Theory - Abstract
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
- Published
- 2020
- Full Text
- View/download PDF
22. Gaiotto’s Lagrangian Subvarieties via Derived Symplectic Geometry
- Author
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Victor Ginzburg and Nick Rozenblyum
- Subjects
Pure mathematics ,Symplectic group ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Symplectic representation ,01 natural sciences ,Symplectic matrix ,Algebra ,Symplectic vector space ,0103 physical sciences ,0101 mathematics ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic geometry ,Mathematics ,Symplectic manifold - Abstract
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.
- Published
- 2018
23. Symplectic integrators for second-order linear non-autonomous equations
- Author
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Sergio Blanes, Enrique Ponsoda, Nikita Kopylov, Fernando Casas, and Philipp Bader
- Subjects
magnus expansion ,second-order linear differential equations ,010103 numerical & computational mathematics ,Symplectic integrators ,non-autonomous ,01 natural sciences ,G.1.0 ,Second-order linear differential equations ,Symplectic vector space ,Magnus expansion ,FOS: Mathematics ,65L07, 65L05, 65Z05 ,Mathematics - Numerical Analysis ,Symplectic integrator ,0101 mathematics ,Variational integrator ,Moment map ,Matrix Hill s equation ,Mathematics ,Symplectic manifold ,Non-autonomous ,symplectic integrators ,Applied Mathematics ,Mathematical analysis ,Numerical Analysis (math.NA) ,Symplectic representation ,Symplectic matrix ,matrix Hill’s equation ,010101 applied mathematics ,Computational Mathematics ,MATEMATICA APLICADA - Abstract
[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.
- Published
- 2018
24. Berezin–Toeplitz Quantization for Eigenstates of the Bochner Laplacian on Symplectic Manifolds
- Author
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Louis Ioos, Wen Lu, Xiaonan Ma, and George Marinescu
- Subjects
010102 general mathematics ,Mathematics::Spectral Theory ,Symplectic representation ,01 natural sciences ,Symplectic matrix ,Quantization (physics) ,Symplectic vector space ,0103 physical sciences ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Mathematics ,Mathematical physics ,Symplectic manifold ,Symplectic geometry - Abstract
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.
- Published
- 2018
25. The quantization of gravity
- Author
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Claus Gerhardt
- Subjects
Physics ,Hamiltonian mechanics ,General Mathematics ,General Physics and Astronomy ,Cosmological constant ,General Relativity and Quantum Cosmology ,Symplectic vector space ,Quantization (physics) ,symbols.namesake ,Bundle ,Quantum mechanics ,symbols ,Spectral resolution ,Special case ,Unified field theory ,Mathematical physics - Abstract
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.
- Published
- 2018
26. On a noncontractible family of representations of the canonical commutation relations
- Author
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Hyunmoon Kim
- Subjects
Pure mathematics ,Complexification ,General Physics and Astronomy ,Space (mathematics) ,Linear subspace ,Symplectic vector space ,symbols.namesake ,Irreducible representation ,symbols ,Geometry and Topology ,Mathematics::Representation Theory ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Lagrangian ,Schrödinger's cat ,Mathematics - Abstract
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.
- Published
- 2021
27. Injectivity of Generalized Wronski Maps
- Author
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Frank Sottile, Igor Zelenko, and Yanhe Huang
- Subjects
0209 industrial biotechnology ,Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Linear system ,02 engineering and technology ,Differential operator ,01 natural sciences ,Linear subspace ,Linear map ,Mathematics - Algebraic Geometry ,Symplectic vector space ,020901 industrial engineering & automation ,Cover (topology) ,Dimension (vector space) ,Grassmannian ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
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
- Published
- 2017
28. Canonical symplectic structure and structure-preserving geometric algorithms for Schrödinger–Maxwell systems
- Author
-
Hong Qin, Jian Liu, Jianyuan Xiao, Yang He, Yulei Wang, Ruili Zhang, and Qiang Chen
- Subjects
Quantum Physics ,Numerical Analysis ,Symplectic group ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Symplectic representation ,01 natural sciences ,Physics - Plasma Physics ,Symplectic matrix ,Physics - Atomic Physics ,010305 fluids & plasmas ,Computer Science Applications ,Computational Mathematics ,Symplectic vector space ,Mathematics - Symplectic Geometry ,Modeling and Simulation ,0103 physical sciences ,Symplectic integrator ,010306 general physics ,Symplectomorphism ,Algorithm ,Moment map ,Mathematics ,Symplectic manifold - Abstract
An infinite dimensional canonical symplectic structure and structure-preserving geometric algorithms are developed for the photon-matter interactions described by the Schr\"odinger-Maxwell equations. The algorithms preserve the symplectic structure of the system and the unitary nature of the wavefunctions, and bound the energy error of the simulation for all time-steps. This new numerical capability enables us to carry out first-principle based simulation study of important photon-matter interactions, such as the high harmonic generation and stabilization of ionization, with long-term accuracy and fidelity., Comment: 17 pages, 7 figures
- Published
- 2017
29. Spreads of nonsingular pairs in symplectic vector spaces.
- Author
-
Weintraub, Steven H.
- Subjects
VECTOR spaces ,LINEAR algebra ,VECTOR analysis ,FUNCTIONAL analysis ,CALCULUS of variations - Abstract
Let V be a vector space of dimension 2 n, n even, over a field F, equipped with a nonsingular symplectic form. We define a new algebraic/combinatorial structure, a spread of nonsingular pairs, or nsp-spread, on V and show that nsp-spreads exist in considerable generality. We further examine in detail some particular cases. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
30. Lorentzian affine hypersurfaces with an almost symplectic form
- Author
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Michal Szancer
- Subjects
Pure mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Symplectic representation ,01 natural sciences ,Symplectic matrix ,010101 applied mathematics ,Symplectic vector space ,Hypersurface ,Mathematics::Differential Geometry ,Geometry and Topology ,0101 mathematics ,Symplectomorphism ,Moment map ,Mathematical Physics ,Symplectic geometry ,Mathematics ,Symplectic manifold - Abstract
In this paper, we study affine hypersurfaces with a Lorentzian second fundamental form additionally equipped with an almost symplectic structure ω . We prove that the rank of the shape operator is at most one if the hypersurface is of dimension at least 6 and R k ⋅ ω = 0 or ∇ k ω = 0 for some positive integer k .
- Published
- 2017
31. Second-order evaluations of orthogonal and symplectic Yangians
- Author
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R. Kirschner and D. R. Karakhanyan
- Subjects
Pure mathematics ,Symplectic group ,010308 nuclear & particles physics ,Statistical and Nonlinear Physics ,Symplectic representation ,01 natural sciences ,Algebra ,Symplectic vector space ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Mathematics::Quantum Algebra ,0103 physical sciences ,Order (group theory) ,Symmetry (geometry) ,010306 general physics ,Symplectomorphism ,Moment map ,Mathematical Physics ,Symplectic geometry ,Mathematics - Abstract
Orthogonal or symplectic Yangians are defined by the Yang–Baxter RLL relation involving the fundamental R-matrix with the corresponding so(n) or sp(2m) symmetry. We investigate the second-order solution conditions, where the expansion of L(u) in u −1 is truncated at the second power, and we derive the relations for the two nontrivial terms in L(u).
- Published
- 2017
32. Spreads of nonsingular pairs in symplectic vector spaces.
- Author
-
Weintraub, Steven
- Subjects
SYMPLECTIC geometry ,DIFFERENTIAL geometry ,ORDERED algebraic structures ,ALGEBRA ,VECTOR spaces - Abstract
Let V be a vector space of dimension 2 n, n even, over a field F, equipped with a nonsingular symplectic form. We define a new algebraic/combinatorial structure, a spread of nonsingular pairs, or nsp-spread, on V and show that nsp-spreads exist in considerable generality. We further examine in detail some particular cases. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
33. Modified symplectic schemes with nearly-analytic discrete operators for acoustic wave simulations
- Author
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Wenshuai Wang, Zhide Pan, Dinghui Yang, Shaolin Liu, and Chao Lang
- Subjects
010504 meteorology & atmospheric sciences ,Discretization ,Semi-implicit Euler method ,Mathematical analysis ,General Physics and Astronomy ,010502 geochemistry & geophysics ,01 natural sciences ,Hamiltonian system ,Symplectic vector space ,Hardware and Architecture ,Acoustic wave equation ,Symplectic integrator ,Temporal discretization ,Mathematics::Symplectic Geometry ,0105 earth and related environmental sciences ,Mathematics ,Symplectic geometry - Abstract
Using a structure-preserving algorithm significantly increases the computational efficiency of solving wave equations. However, only a few explicit symplectic schemes are available in the literature, and the capabilities of these symplectic schemes have not been sufficiently exploited. Here, we propose a modified strategy to construct explicit symplectic schemes for time advance. The acoustic wave equation is transformed into a Hamiltonian system. The classical symplectic partitioned Runge–Kutta (PRK) method is used for the temporal discretization. Additional spatial differential terms are added to the PRK schemes to form the modified symplectic methods and then two modified time-advancing symplectic methods with all of positive symplectic coefficients are then constructed. The spatial differential operators are approximated by nearly-analytic discrete (NAD) operators, and we call the fully discretized scheme modified symplectic nearly analytic discrete (MSNAD) method. Theoretical analyses show that the MSNAD methods exhibit less numerical dispersion and higher stability limits than conventional methods. Three numerical experiments are conducted to verify the advantages of the MSNAD methods, such as their numerical accuracy, computational cost, stability, and long-term calculation capability.
- Published
- 2017
34. Each 2n-by-2n complex symplectic matrix is a product of n+ 1 commutators of J-symmetries
- Author
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Ralph John de la Cruz and Kennett L. Dela Rosa
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Symplectic group ,Mathematical analysis ,0211 other engineering and technologies ,021107 urban & regional planning ,010103 numerical & computational mathematics ,02 engineering and technology ,Symplectic representation ,01 natural sciences ,Symplectic matrix ,Combinatorics ,Symplectic vector space ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,0101 mathematics ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic geometry ,Mathematics ,Symplectic manifold - Abstract
A 2 n × 2 n complex matrix A is symplectic if A ⊤ [ 0 I − I 0 ] A = [ 0 I − I 0 ] . If A is symplectic and rank ( A − I ) = 1 , then it is called a J-symmetry. For each n, we prove that every 2 n × 2 n symplectic matrix M is a product of n + 1 commutators of J-symmetries and this number cannot be smaller for some M.
- Published
- 2017
35. Every 2n-by-2n complex matrix is a sum of three symplectic matrices
- Author
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Agnes T. Paras, Dennis I. Merino, and Ralph John de la Cruz
- Subjects
Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,Symplectic group ,0211 other engineering and technologies ,021107 urban & regional planning ,010103 numerical & computational mathematics ,02 engineering and technology ,Symplectic representation ,01 natural sciences ,Symplectic matrix ,Combinatorics ,Symplectic vector space ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,0101 mathematics ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic geometry ,Mathematics ,Symplectic manifold - Abstract
Let J 2 n = [ 0 I n − I n 0 ] . An A ∈ M 2 n ( C ) is called symplectic if A T J 2 n A = J 2 n . If n = 1 , then we show that every matrix in M 2 n ( C ) is a sum of two symplectic matrices. If n > 1 , then we show that every matrix in M 2 n ( C ) is a sum of three symplectic matrices; moreover, we show that some matrices cannot be written with less than three symplectic matrices. We also show that for every A ∈ M 2 n ( C ) , there exist symplectic P, Q ∈ M 2 n ( C ) and B, C, D ∈ M n ( C ) such that P A Q = [ B C 0 D ] . If A is skew Hamiltonian ( J 2 n − 1 A T J 2 n = A ), then we show that A is a sum of two symplectic matrices.
- Published
- 2017
36. The hyperplanes of DW(5,F) arising from the Grassmann embedding
- Author
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Bart De Bruyn and Mariusz Kwiatkowski
- Subjects
Combinatorics ,Symplectic vector space ,Algebra and Number Theory ,Grassmann number ,Symplectic group ,Hyperplane ,Embedding ,Polar space ,Symplectic representation ,Symplectic geometry ,Mathematics - Abstract
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.
- Published
- 2017
37. Yet more elementary proofs that the determinant of a symplectic matrix is 1
- Author
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Florian Bünger and Siegfried M. Rump
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Symplectic group ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,Symplectic representation ,01 natural sciences ,Symplectic matrix ,Algebra ,Symplectic vector space ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,0101 mathematics ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic manifold ,Symplectic geometry ,Mathematics - Abstract
It seems to be of recurring interest in the literature to give alternative proofs for the fact that the determinant of a symplectic matrix is one. We state four short and elementary proofs for symplectic matrices over general fields. Two of them seem to be new.
- Published
- 2017
38. Quantum quaternion spheres
- Author
-
Bipul Saurabh
- Subjects
Discrete mathematics ,Symplectic group ,Quantum group ,General Mathematics ,010102 general mathematics ,Symplectic representation ,Quotient space (linear algebra) ,01 natural sciences ,Symplectic vector space ,Universal C*-algebra ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Finite set ,Moment map ,Mathematics - Abstract
For the quantum symplectic group SP q (2n), we describe the C ∗-algebra of continuous functions on the quotient space S P q (2n)/S P q (2n−2) as an universal C ∗-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the K-groups of this C ∗-algebra in terms of generators of the C ∗-algebra.
- Published
- 2017
39. SYMPLECTIC RUNGE-KUTTA METHODS OF HIGH ORDER BASED ON W-TRANSFORMATION
- Author
-
Yuhao Cong, Geng Sun, and Kaifeng Xia
- Subjects
General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Symplectic vector space ,Runge–Kutta methods ,Transformation (function) ,Symplectic integrator ,0101 mathematics ,High order ,Symplectic geometry ,Symplectic manifold ,Mathematics - Published
- 2017
40. Vanishing of local symplectic periods for cuspidal representations of the unitary group
- Author
-
Omer Offen and Arnab Mitra
- Subjects
Pure mathematics ,Symplectic group ,Mathematics::Number Theory ,Cuspidal representation ,010102 general mathematics ,General Medicine ,Symplectic representation ,01 natural sciences ,Symplectic matrix ,Algebra ,Symplectic vector space ,Metaplectic group ,Unitary group ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Moment map ,Mathematics - Abstract
We show that, for a quadratic extension of p-adic fields, no cuspidal representation of the quasi-split unitary group admits a non-trivial linear form invariant by the symplectic subgroup. Our proof is purely local.
- Published
- 2017
41. ABOUT GENERAL CONFORMAL ALMOST SYMPLECTIC N-LINEAR CONNECTIONS ON K-COTANGENT BUNDLE
- Author
-
Monica Purcaru
- Subjects
Symplectic vector space ,Pure mathematics ,Normal bundle ,Connection (vector bundle) ,Cotangent bundle ,Geometry ,General Medicine ,Symplectomorphism ,Frame bundle ,Moment map ,Symplectic manifold ,Mathematics - Published
- 2016
42. Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems
- Author
-
Jialin Hong, Peng Wang, and Dongsheng Xu
- Subjects
Physics and Astronomy (miscellaneous) ,Mathematical analysis ,010103 numerical & computational mathematics ,Symplectic representation ,01 natural sciences ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Symplectic vector space ,Applied mathematics ,Symplectic integrator ,0101 mathematics ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,First class constraint ,Symplectic geometry ,Symplectic manifold ,Mathematics - Abstract
We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.
- Published
- 2016
43. Symplectic S 5 action on symplectic homotopy K3 surfaces
- Author
-
Hongxia Li
- Subjects
Pure mathematics ,Symplectic vector space ,Symplectic group ,General Mathematics ,Homotopy ,Mathematical analysis ,Symplectomorphism ,Symplectic representation ,Mathematics::Symplectic Geometry ,Moment map ,Mathematics ,Symplectic geometry ,Symplectic manifold - Abstract
Let X be a symplectic homotopy K3 surface and G = S 5 act on X symplectically. In this paper, we give a weak classification of the G action on X by discussing the fixed-point set structure. Besides, we analyse the exoticness of smooth structures of X under the action of G.
- Published
- 2016
44. Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2
- Author
-
Elizabeth Goodman, Alexander Ivrii, and Georgios Dimitroglou Rizell
- Subjects
Pure mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Torus ,02 engineering and technology ,Submanifold ,01 natural sciences ,Symplectic vector space ,Monotone polygon ,Isotopy ,Cotangent bundle ,Geometry and Topology ,Projective plane ,0101 mathematics ,Mathematics::Symplectic Geometry ,Analysis ,021101 geological & geomatics engineering ,Mathematics ,Symplectic geometry - Abstract
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.
- Published
- 2016
45. Quantization and the resolvent algebra
- Author
-
Teun D.H. van Nuland
- Subjects
Quantization (signal processing) ,010102 general mathematics ,Spectrum (functional analysis) ,Subalgebra ,01 natural sciences ,Linear subspace ,Functional Analysis (math.FA) ,Algebra ,Mathematics - Functional Analysis ,Symplectic vector space ,Inner product space ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Commutative property ,Mathematics ,Analysis ,Resolvent - Abstract
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.
- Published
- 2019
46. Symplectic Vector Spaces
- Author
-
Shubham Dwivedi, Lisa C. Jeffrey, Theo van den Hurk, and Jonathan Herman
- Subjects
Pure mathematics ,Symplectic vector space ,Differential topology ,Mathematics::Symplectic Geometry ,Vector space ,Mathematics ,Symplectic geometry - Abstract
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.
- Published
- 2019
47. Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes
- Author
-
David Gross and Felipe Montealegre-Mora
- Subjects
Quantum Physics ,Rank (linear algebra) ,Oscillator representation ,Group (mathematics) ,010102 general mathematics ,Duality (optimization) ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,01 natural sciences ,Representation theory ,Combinatorics ,Symplectic vector space ,Mathematics (miscellaneous) ,Tensor product ,0103 physical sciences ,FOS: Mathematics ,Orthogonal group ,010307 mathematical physics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Quantum Physics (quant-ph) ,Mathematics - Representation Theory ,Mathematical Physics ,Mathematics - Abstract
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
- Published
- 2019
- Full Text
- View/download PDF
48. Lagrangian configurations and symplectic cross-ratios
- Author
-
Charles H. Conley, Valentin Ovsienko, and Centre National de la Recherche Scientifique (CNRS)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Tridiagonal matrix ,General Mathematics ,010102 general mathematics ,Pfaffian ,Dynamical Systems (math.DS) ,01 natural sciences ,Linear subspace ,Moduli space ,Symplectic vector space ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,[MATH]Mathematics [math] ,Mathematics::Symplectic Geometry ,ComputingMilieux_MISCELLANEOUS ,Symplectic geometry ,Gramian matrix ,Mathematics - Abstract
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
- Published
- 2019
49. On the conditioning of factors in the SR decomposition
- Author
-
Heike Faßbender and Miroslav Rozložník
- Subjects
Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,Hamiltonian matrix ,Symplectic group ,0211 other engineering and technologies ,Block matrix ,021107 urban & regional planning ,010103 numerical & computational mathematics ,02 engineering and technology ,Symplectic representation ,01 natural sciences ,Symplectic matrix ,Matrix decomposition ,Combinatorics ,Symplectic vector space ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,0101 mathematics ,Mathematics ,Symplectic manifold - Abstract
Almost every nonsingular matrix A∈R2m,2m can be decomposed into the product of a symplectic matrix S and an upper J-triangular matrix R. This decomposition is not unique. In this paper we analyze the freedom of choice in the symplectic and the upper J-triangular factors and review several existing suggestions on how to choose the free parameters in the SR decomposition. In particular we consider two choices leading to the minimization of the condition number of the diagonal blocks in the upper J-triangular factor and to the minimization of the conditioning of the corresponding blocks in the symplectic factor. We develop bounds for the extremal singular values of the whole upper J-triangular factor and the whole symplectic factor in terms of the spectral properties of even-dimensioned principal submatrices of the skew-symmetric matrix associated with the SR decomposition. The theoretical results are illustrated on two small examples.
- Published
- 2016
50. Automorphisms of Symplectic and Contact Structures
- Author
-
V. I. Panzhensky and N. A. Tyapin
- Subjects
Statistics and Probability ,Pure mathematics ,Symplectic group ,Automorphisms of the symmetric and alternating groups ,Applied Mathematics ,General Mathematics ,Symplectic representation ,Algebra ,Symplectic vector space ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Mathematics ,Symplectic manifold ,Symplectic geometry - Abstract
The survey contains main results of the theory of automorphisms of symplectic (almost symplectic) and contact (almost contact) structures and the original results of the authors of estimates of the maximal dimension of Lie groups of automorphisms of symplectic and contact structures that preserve an associated linear connection.
- Published
- 2016
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