arXiv math/01111122001-11-09math.QAmath.AG

On infinite dimensional grassmannians and their quantum deformations

R. Fioresi, C. Hacon

英文摘要：

An algebraic approach is developed to define and study infinite dimensional grassmannians. Using this approach a quantum deformation is obtained for both the ind-variety union of all finite dimensional grassmannians, and the Sato grassmannian introduced by Sato. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite dimensional version of the first theorem of invariant theory is discussed for both the infinite dimensional special linear group and its quantization.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Two parameter deformation of Grassmann matrix group and supergroup

The two parameter quantum deformation of 2x2 Grassmann matrices, Gr(2), and supermatrices, Gr$(1| 1)$, are presented. Gr(2) whose matrix elements are all Grassmannian variables is called the superdual of the genel linear group GL(2), and Gr$(1| 1)$ whose diagonal matrix elements are Grassmannian variables is called the superdual of the supergroup GL$(1| 1)$ whose nondiagonal elements are Grassmannian. Noncentral dual superdeterminant for Grassmann supermatrices belonging to Gr$_{p,q}(1| 1)$ is constructed. As with the 2x2 quantum matrices, the relations satisfied by the matrix elements of the …

2001-12-07math.QA

The vector k-constrained KP hierarchy and Sato's Grassmannian

We use the representation theory of the infinite matrix group to show that (in the polynomial case) the $n$--vector $k$--constrained KP hierarchy has a natural geometrical interpretation on Sato's infinite Grassmannian. This description generalizes the the $k$--reduced KP or Gelfand--Dickey hierarchies.

1996-08-30math.QAq-alg

正性决定奇辛线 Grassmann 簇的量子上同调

Positivity determines the quantum cohomology of the odd symplectic Grassmannian of lines

针对 Picard 秩为 1 的奇辛线 Grassmann 簇 IG(2,2n+1)，本文提出正性条件，证明其量子上同调环在量子参数缩放下是唯一量子形变，并修正 Fulton 猜想。

2025-02-20math.AGmath.QA

Prime ideals in the quantum grassmannian

We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a torus H that acts naturally on the quantum grassmannian and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the non-negative cells in recent stu…

2007-08-06math.COmath.QAmath.RA

Algebraic models of local period maps and Yukawa algebras

We describe some L-infinity model for the local period map of a compact Kaehler manifold. Applications include the study of deformations with associated variation of Hodge structure constrained by certain closed strata of the Grassmannian of the de Rham cohomology. As a byproduct we obtain an interpretation in the framework of deformation theory of the Yukawa coupling.

2018-02-07math.AGmath.QA

Total positivity is a quantum phenomenon: the grassmannian case

The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter $q$ is transcendental, we show that "quantum positroids" are completely prime ideals in the quantum grassmannian $A$. As a consequence, we obtain that torus-invariant prime ideals in the quantum grassmannian are generated by polynormal sequences of quantum Plücker coordinates and give a combinatorial description of these generating sets. We also give a topological description of the poset of torus-i…

2019-06-14math.COmath.QAmath.RA

更多相关论文收起更多论文

The quantum Euler class and the quantum cohomology of the Grassmannians

The Poincare duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical ``characteristic element;'' in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the ``quantum Euler class.'' We prove that the characteristic element of a Frobenius algebra A is a unit if and only if A is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmann…

1997-12-09alg-geommath.AGmath.AT

量子 Grassmann 代数的导子与第一 Hochschild 上同调群

Derivations and the first Hochschild cohomology group of the quantum grassmannian

本文利用分次技术、正规元与去齐次化局部化等式，将量子 Grassmann 代数导子联系到量子矩阵代数导子，并显式确定其第一 Hochschild 上同调群。

2024-01-18math.QAmath.RA

Multivariable Askey-Wilson Polynomials and Quantum Complex Grassmannians

We present a one-parameter family of constant solutions of the reflection equation and define a family of quantum complex Grassmannians endowed with a transitive action of the quantum unitary group. By computing the radial part of a suitable Casimir operator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible representations) as multivariable Askey-Wilson polynomials containing two continuous and two discrete parameters.

1996-03-15math.QAq-alg

超芒福德形式与佐藤格拉斯曼流形

The Super Mumford Form and Sato Grassmannian

本研究推广了Kontsevich等人的经典构造，利用超黎曼曲面模空间与佐藤格拉斯曼流形的关联，证明了特定线丛上平坦全纯联络的存在性及超共形Noether正规化引理。

2022-01-16hep-thmath-phmath.AG

Quantum deformation of planar amplitudes

In maximally supersymmetric four-dimensional gauge theories planar on-shell diagrams are closely related to the positive Grassmannian and the cell decomposition of it into the union of so called positroid cells \cite{A}. We establish that volume forms on positroids used to express scattering amplitudes can be $q$-deformed to Hochschild homology classes of corresponding quantum algebras. The planar amplitudes are represented in \cite {A} as sums of contributions of some set of positroid cells; we quantize these contributions. In classical limit our considerations allow us to obtain explicit for…

2018-02-16hep-thmath-phmath.MP

Graded quantum cluster algebras of infinite rank as colimits

We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, extending an earlier construction of the authors for k=2.

2015-10-14math.QAmath.RT