arXiv subjects

Arlo Caine

Publications and source records attributed to Arlo Caine.

7 recordsLinked to original sources

arXiv2017

On Toric Poisson Structures of Type $(1,1)$ and their Cohomology

We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in each of the distinguished holomorphic coordinate charts determined by the open cones of the associated simplicial fan. As an approximation to the smooth cohomology problem in each ${\mathbb C}^n$ chart, we consider the Poisson differential on the complex of polynomial multi-vector fields. For the algebraic problem, we compute $H^0$ and $H^1$ under the assumption that the Poisson structure is generically non-degenerate. The paper concludes with numerical investigations of the higher degree cohomology groups of $({\mathbb C}^2,π_B)$ for various $B$.

math.DG↗

arXiv2015

Loops in noncompact groups and factorization

In [11] we showed that a loop in a simply connected compact Lie group $\dot{U}$ has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group). In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact type semisimple Lie group $\dot G_0$ of inner type. In [4] we showed that for an element of $\dot G_0$, i.e. a constant loop, there is a unique Birkhoff factorization if and only if there is a root subgroup factorization. However for loops in $\dot G_0$, while a root subgroup factorization implies a unique Birkhoff factorization, there are several obstacles to the converse. As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.

math.RT↗

arXiv2015

Non-compact groups of inner type and factorization

We investigate Birkhoff (or triangular) factorization and (what we propose to call) root subgroup factorization for elements of a noncompact simple Lie group $G_0$ of inner type. For compact groups root subgroup factorization is related to Bott-Samelson desingularization, and many striking applications have been discovered by Lu (\cite{Lu}). In this paper, in the inner noncompact case, we obtain parallel characterizations of the Birkhoff components of $G_0$ and an analogous construction of root subgroup coordinates for the Birkhoff components. As in the compact case, we show that the restriction of Haar measure to the top Birkhoff component is a product measure in root subgroup coordinates.

math.RT↗

arXiv2015

Equivalence of Demazure and Bott-Samelson Resolutions via Factorization

Let $G$, $B$, and $H$ denote a complex semi-simple algebraic group, a Borel subgroup of $G$, and a maximal complex torus in $B$, respectively. Choose a compact real form $K$ of $G$ such that $T=K\cap H$ is a maximal torus in $T$. Then there are two models for the flag space of $G$: the complex quotient $X=G/B$ and the real quotient $K/T$. These models are smoothly equivalent via the map $\tilde{\mathbf k}\colon G/B\to K/T$ induced by factorization in $G$ relative to the Iwasawa decomposition $G=KAN$, where $N$ is the nilradical of $B$ and $H=TA$. Likewise, there are two models for resolutions of the Schubert subvarieties $\overline{X_w}\subset X$: the Demazure resolution of $\overline{X_w}$ which is constructed via a complex algebraic quotient and the Bott-Samelson resolution of $\mathbf k(\overline{X_w})$ which is constructed as a real quotient of compact groups. This paper makes explicit the equivalence and compatibility of these two resolutions using factorization. As an application, we can compute the change of variables map relating the standard complex algebraic coordinates on $X_w$ to Lu's real algebraic coordinates on $\tilde{\mathbf k}(X_w)$.

math.DG↗

arXiv2009

Toric Poisson Structures

Let X(Σ) be a smooth projective toric variety for a complex torus T_\C. In this paper, a real T_\C-invariant Poisson structure Π_Σis constructed on the complex manifold X(Σ), the symplectic leaves of which are the T_\C-orbits in X(Σ). It is shown that each leaf admits a Hamiltonian action by a sub-torus of the compact torus T\subset T_\C. However, the global action of T_\C on (X(Σ),Π_Σ) is Poisson but not Hamiltonian. The main result of the paper is a lower bound for the first Poisson cohomology of these structures. For the simplest case, X(Σ)=\CP^1, the Poisson cohomology is computed using a Mayer-Vietoris argument and known results on planar quadratic Poisson structures and in the example the bound is optimal. The paper concludes with the example of \CP^n, where the modular vector field with respect to a particular Delzant Liouville form admits a curious formula in terms of Delzant moment data. This formula enables one to compute the zero locus of this modular vector field and relate it to the Euclidean geometry of the moment simplex.

math.SG↗

arXiv2008

Homogeneous Poisson structures on symmetric spaces

We calculate, in a relatively explicit way, the Hamiltonian systems which arise from the Evens-Lu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. A corollary is that the Hamiltonian system arising in the noncompact case is isomorphic to the generic Hamiltonian system arising in the compact case. In the group case these systems are also isomorphic to those arising from the Bruhat Poisson structure on the flag space, and hence, by results of Lu, can be completely factored.

math.SG↗

arXiv2006

Compact symmetric spaces, triangular factorization, and Poisson geometry

Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let \g denote the complexification of the Lie algebra of U, \g=\u^\C. Each \u-compatible triangular decomposition \g=\n_- + \h + \n_+ determines a Poisson Lie group structure π_U on U. The Evens-Lu construction produces a (U,π_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of \g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that the action is Hamiltonian and the momentum map is computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.

math.SG↗