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Alice Barbora Tumpach

Publications and source records attributed to Alice Barbora Tumpach.

11 recordsLinked to original sources
arXiv2026

The Restricted Schatten-class Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$ as affine coadjoint orbit

In this paper, we consider the restricted $p$-Schatten class Grassmannian $\mathrm {Gr}_{{\rm res}, p}(\mathcal{H})$ consisting of infinite-dimensional and infinite codimensional subspaces $W$ of a polarized complex separable Hilbert space $\mathcal{H} = \mathcal{H}_+\oplus \mathcal{H}_-$ such that the orthogonal projection from $W$ onto $\mathcal{H}_+$ is Fredholm and the orthogonal projection from $W$ onto $\mathcal{H}_-$ is in the Schatten ideal $L_p$, $p\geq 1$. The aim of this paper is to show that, for $1\leq p\leq 2$, the restricted $p$-Schatten class Grassmannian $\mathrm {Gr}_{{\rm res}, p}(\mathcal{H})$ is an affine (co-)adjoint orbit of an infinite-dimensional restricted unitary group $\operatorname{U}_{{\rm res}, p}(\mathcal{H})$, and that it admits natural weak symplectic structures. These results follow from the fact that the Lie algebra of the restricted $p$-Schatten class unitary group $\operatorname{U}_{{\rm res}, p}(\mathcal{H})$ admits a non-trivial $2$-cocycle.

math.FA↗
arXiv2025

Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions

In this paper, we investigate the theory of $R$-brackets, Baxter brackets and Nijenhuis brackets in the Banach setting, in particular in relation with Banach Poisson-Lie groups. The notion of Banach Lie-Poisson space with respect to an arbitrary duality pairing is crucial for the equations of motion to make sense. In the presence of a non-degenerate invariant pairing on a Banach Lie algebra, these equations of motion assume a Lax form. We prove a version of the Adler-Kostant-Symes theorem adapted to $R$-matrices on infinite-dimensional Banach algebras. Applications to the resolution of Lax equations associated to some Banach Manin triples are given. The semi-infinite Toda lattice is also presented as an example of this approach.

nlin.SI↗
arXiv2025

Special vector fields on Riemannian manifolds of constant negative sectional curvature and conservation laws

We show that any $n$-dimensional Riemannian manifold with constant negative sectional curvature admits local orthonormal vector fields such that one of them $v_1$ is tangent to geodesics and the other $n-1$ vector fields are tangent to horocycles. We prove that the $1$-form dual to $v_1$ is a closed form. We show how the closed form can be used to obtain conservation laws for PDEs whose generic solutions define metrics on open subsets with constant negative sectional curvature. These results extend to higher dimensions the $2$-dimensional case proved in the 1980s. We prove that there exist local coordinates on the manifold such that the coordinate curves are tangent to the orthonormal vector fields. We apply the theory to obtain conservation laws for the Camassa-Holm equation ($n=2$) and for the Intrinsic Generalized Sine-Gordon equation ($n\geq 2$).

math.DG↗
arXiv2025

Infinite-dimensional Siegel disc as symplectic and Kaehler quotient

In this paper, we construct the restricted infinite-dimensional Siegel disc as a Marsden-Weinstein symplectic reduced space and as Kaehler quotient of a weak Kaehler manifold. The obtained symplectic form is invariant with respect to the left action of the infinite-dimensional restricted symplectic group and coincides with the Kirillov-Kostant-Souriau symplectic form of the restricted Siegel disc obtained via the identification with an affine coadjoint orbit of the restricted symplectic group, or equivalently with a coadjoint orbit of the universal central extension of the restricted symplectic group.

math.SG↗
arXiv2025

Hyperkaehler Marriage of the two sphere with the hyperbolic space

The Eguchi-Hanson metric is a natural metric on the total space of the cotangent bundle $T^*\mathbb{CP}(1)$ of the complex projective line $\mathbb{CP}(1) \simeq \mathbb{S}^2$, which extends the Fubini-Study metric of $\mathbb{CP}(1)$. By virtue of the Mostow decomposition theorem, $T^*\mathbb{CP}(1)$ is isomorphic, as $SU(2)$-equivariant fiber bundle over $\mathbb{CP}(1)$, to a complex (co-)adjoint orbit of $SL(2, \mathbb{C})$. In fact, this complex (co-)adjoint orbit is fibered over $\mathbb{CP}(1)\simeq \mathbb{S}^2$ with each fiber isomorphic to the hyperbolic disc $\mathbb{H}^2$. In this paper, we are interested in the complex structure inherited on the hyperbolic disc $\mathbb{H}^2$ by the hyperkähler extension of the $2$-sphere. Contrary to what is generally believed, we show that it differs from the natural complex structure of $\mathbb{H}^2\subset \mathbb{C}$ inherited from its embedding in $\mathbb{C}$. In other words, the embedding of $\mathbb{H}^2$ with its Hermitian-symmetric structure into the hyperkähler manifold $T^*\mathbb{CP}(1)$ is not holomorphic.

math.DG↗
arXiv2024

Nijenhuis operators on homogeneous spaces related to $C^*$-algebras

For a unital non-simple $C^*$-algebra $\mathcal A$ we consider its Banach--Lie group $G$ of invertible elements. For a given closed ideal $\mathfrak k$ in $\mathcal A$, we consider the embedded Banach--Lie subgroup $K$ of $G$ of elements differing from the unit element by an element in $\mathfrak k$. We study vector bundle maps of the tangent space of the homogeneous space $G/K$, induced by an admissible bounded operator on $\mathcal A$. In particular, we discuss when this vector bundle map is a Nijenhuis operator in $G/K$. The special case of almost complex structures in $G/K$ is also addressed. Examples for particular classes of $C^*$-algebras are presented, including the Toeplitz algebra and crossed products by $\mathbb Z$.

math.DG↗
arXiv2024

Nijenhuis operators on Banach homogeneous spaces

For a Banach--Lie group $G$ and an embedded Lie subgroup $K$ we consider the homogeneous Banach manifold $\mathcal M=G/K$. In this context we establish the most general conditions for a bounded operator $N$ acting on $Lie(G)$ to define a homogeneous vector bundle map $\mathcal N:T\mathcal M\to T\mathcal M$. In particular our considerations extend all previous settings on the matter and are well-suited for the case where $Lie(K)$ is not complemented in $Lie(G)$. We show that the vanishing of the Nijenhuis torsion for a homogeneous vector bundle map $\mathcal N:T\mathcal M\to T\mathcal M$ (defined by an admissible bounded operator $N$ on $Lie(G)$) is equivalent to the Nijenhuis torsion of $N$ having values in $Lie(K)$. As an application, we consider the question of integrability of an almost complex structure $\mathcal J$ on $\mathcal M$ induced by an admissible bounded operator $J$, and we give a simple characterization of integrability in terms of certain subspaces of the complexification of $Lie(G)$ (which are not eigenspaces of the complex extension of $J$).

math.DG↗
arXiv2023

3 methods to put a Riemannian metric on Shape Space

In many applications, one is interested in the shape of an object, like the contour of a bone or the trajectory of joints of a tennis player, irrespective of the way these shapes are parameterized. However for analysis of these shape spaces, it is sometimes useful to have a parameterization at hand, in particular if one is interested in deforming shapes. The purpose of the paper is to examine three different methods that one can follow to endow shape spaces with a Riemannian metric that is measuring deformations in a parameterization independent way.

math.DG↗