Related papers: Introduction to the symplectic group Sp(2)
We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with…
We continue the analysis, started by Abreu, McDuff and Anjos, of the topology of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the areas of the two spheres lies in the interval (1,2]. We express the group, up to…
In this paper we will study both the finite and infinite-dimensional representations of the symplectic Lie algebra $\mathfrak{sp}(2n)$ and develop a polynomial model for these representations. This means that we will associate a certain…
After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We…
It is shown that any symplectic $2n\times 2n$-matrix, whose entries are complex holomorphic functions on a reduced Stein space, can be decomposed into a finite product of elementary symplectic matrices if and only if it is null-homotopic.…
We construct and study the holomorphic discrete series representation and the principal series representation of the symplectic group $\mathrm{Sp}(2n,F)$ over a $p$-adic field $F$ as well as a duality between some sub-representations of…
We present a new derivation of Hamilton's equations that shows that they have a symmetry group Sp(2n) *s H(n). Sp(2n) is the symplectic group and H(n) is mathematically a Weyl-Heisenberg group that is parameterized by velocity, force and…
We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular…
We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety $X$, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification…
We introduce the symplectic group $\mathrm{Sp}_2(G, \sigma)$ associated to a Lie subgroup $G$ of a (possibly noncommutative) associative algebra $A$ equipped with an anti-involution $\sigma$. Our construction recovers several classical Lie…
The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times…
It is noted that two-by-two S-matrices in multilayer optics can be represented by the Sp(2) group whose algebraic property is the same as the group of Lorentz transformations applicable to two space-like and one time-like dimensions. It is…
Suppose $\ell$ is a prime number, $\ell >3$, $K$ is a field that is an unramified finite extension of the field $\Q_\ell$ of $\ell$-adic numbers, and $G$ is a finite group that is a semi-direct product of a normal $\ell'$-subgroup $H$ and a…
Classical mechanical systems are modeled by a symplectic manifold $(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $\omega$. These actions, which are called "symplectic",…
We solve a classical problem of centrality of symplectic $\mathrm K_2$, namely we show that for an arbitrary commutative ring $R$, $l\geq3$ the symplectic Steinberg group $\mathrm{StSp}(2l,\,R)$ as an extension of the elementary symplectic…
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete…
By elementary matrix algebra we show that every real $2n \times 2n$ matrix admits a dilation to an element of the real symplectic group $Sp (2(n+m))$ for some nonnegative integer $m.$ Our methods do not yield the minimum value of $m,$ for…
The internal low-energy symmetry of the massless Lorentz-invariant Dirac Hamiltonian in $2+1$ dimensions is known to be $O(2N)$, where $N$ is the number of two-component Dirac fermions. Here we point out that there exists an analogous…
Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to…
In this paper, we show the relation between $sp(4,\mathbb{R})$, the Lie algebra of the symplectic group, and the elements of $Sp(4,\mathbb{R})$. We use this result to obtain some special cases of symplectic matrices relevant to the study of…