Showing papers on "Symplectic vector space published in 2021"

Split Canonical Relations

Abstract: A Lagrangian subspace $L$ of a weak symplectic vector space is called \emph{split Lagrangian} if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for $L$ to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace $C$ of a weak symplectic space $V$ which imply that the induced canonical relation $L_C$ from $V$ to $C/C^{\omega}$ is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations.

Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes

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 <= t. The results hold in odd charachteristic and the "stable range" t <= 1/2 dim V. Our work has implications for the representation theory of the Clifford group. It can be thought of as a generalization of the known characterization of the invariants of the Clifford group in terms of self-dual codes.

Einstein tori and crooked surfaces

Abstract: In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. The classification contributes to a complete disjointness criterion for crooked surfaces in the 3-dimensional Einstein universe.

Co)isotropic triples and poset representations

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.

Optimal unit triangular factorization of symplectic matrices

Abstract: We prove that any symplectic matrix can be factored into no more than 5 unit triangular symplectic matrices, moreover, 5 is the optimal number. This result improves the existing triangular factorization of symplectic matrices which gives proof of 9 blocks. We also show the corresponding improved conclusions for structured subsets of symplectic matrices.

Understanding Quaternions from Modern Algebra and Theoretical Physics

Abstract: Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius' theorem, which says that "the only finite-dimensional real division algebra are the real field ${\bf R}$, the complex field ${\bf C}$ and the algebra ${\bf H}$ of quaternions" was derived. They appear also through Hamilton formulation of mechanics, as elements of rotation groups in the symplectic vector spaces. Quaternions were used in the solution of 4-dimensional Dirac equation in QED, and also in solutions of Yang-Mills equation in QCD as elements of noncommutative geometry. We present how quaternions are formulated in Clifford Algebra, how it is used in explaining rotation group in symplectic vector space and parallel transformation in holonomic dynamics. When a dynamical system has hysteresis, pre-symplectic manifolds and nonholonomic dynamics appear. Quaternions represent rotation of 3-dimensional sphere ${\bf S}^3$. Artin's generalized quaternions and Rohlin-Pontryagin's embedding of quaternions on 4-dimensional manifolds, and Kodaira's embedding of quaternions on ${\bf S}^1\times {\bf S}^3$ manifolds are also discussed.

Semisymplectic group and Hamiltonian vector fields

Abstract: The purpose of this paper is presenting a theoretical basis for the study of $\omega$-Hamiltonian vector fields in a more general approach than the classical one. We introduce the concepts of $\omega$-symplectic group and $\omega$-semisymplectic group, and describe some of their properties. We show that the Lie algebra of such groups is a useful tool in the recognition of an $\omega$-Hamiltonian vector field defined on a symplectic vector space $(V,\omega)$ with respect to coordinates that are not necessarily symplectic.

On Parabolic Subgroups of Symplectic Reflection Groups

Abstract: Using Cohen's classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group are themselves symplectic reflection groups. This is the symplectic analogue of Steinberg's Theorem for complex reflection groups. Using computational results required in the proof, we show the non-existence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

The moment map on the space of symplectic 3D Monge-Amp\`ere equations

Abstract: For any second-order scalar PDE $\mathcal{E}$ in one unknown function, that we interpret as a hypersurface of a second-order jet space $J^2$, we construct, by means of the characteristics of $\mathcal{E}$, a sub-bundle of the contact distribution of the underlying contact manifold $J^1$, consisting of conic varieties. We call it the contact cone structure associated with $\mathcal{E}$. We then focus on symplectic Monge-Ampere equations in 3 independent variables, that are naturally parametrized by a 13-dimensional real projective space. If we pass to the field of complex numbers $\mathbb{C}$, this projective space turns out to be the projectivization of the 14-dimensional irreducible representation of the simple Lie group $\mathsf{Sp}(6,\mathbb{C})$: the associated moment map allows to define a rational map $\varpi$ from the space of symplectic 3D Monge-Ampere equations to the projectivization of the space of quadratic forms on a $6$-dimensional symplectic vector space. We study in details the relationship between the zero locus of the image of $\varpi$, herewith called the cocharacteristic variety, and the contact cone structure of a 3D Monge-Ampere equation $\mathcal{E}$: under the hypothesis of non-degenerate symbol, we prove that these two constructions coincide. A key tool in achieving such a result will be a complete list of mutually non-equivalent quadratic forms on a $6$-dimensional symplectic space, which has an interest on its own.

Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations

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 $Q$.

Dirac and normal states on Weyl–von Neumann algebras

Abstract: We study particular classes of states on the Weyl algebra $$\mathcal {W}$$ associated with a symplectic vector space S and on the von Neumann algebras generated in representations of $$\mathcal {W}$$ . 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 on S, and it turns out that those corresponding to non-trivial Dirac states are typically discontinuous. We discuss general aspects of this interplay between functions on S and 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 $$S = L^2(\mathbb {R}^n)$$ or test functions on $$\mathbb {R}^n$$ and relate properties of states on $$\mathcal {W}$$ with those of generalized functions on $$\mathbb {R}^n$$ or with harmonic analysis aspects of corresponding Borel measures on Schwartz functions and on temperate distributions.