Selecta Mathematica最新文献_第6页

The distribution of Weierstrass points on a tropical curve 热带曲线上魏尔斯特拉斯点的分布

Selecta Mathematica Pub Date : 2024-02-29 DOI: 10.1007/s00029-024-00919-5

David Harry Richman

引用次数: 0

The birational geometry of $$overline{{mathcal {R}}}_{g,2}$$ and Prym-canonical divisorial strata $$overline{/mathcal {R}}_{g,2}$ 和 Prym-canonical divisorial strata 的二元几何图形

Selecta Mathematica Pub Date : 2024-02-28 DOI: 10.1007/s00029-023-00907-1

引用次数: 0

The cyclic open–closed map, u-connections and R-matrices 循环开闭图、u 连接和 R 矩阵

Selecta Mathematica Pub Date : 2024-02-27 DOI: 10.1007/s00029-024-00925-7

Kai Hugtenburg

引用次数: 0

Galois closures and elementary components of Hilbert schemes of points 点的希尔伯特方案的伽罗瓦闭包和基本组成部分

Selecta Mathematica Pub Date : 2024-02-27 DOI: 10.1007/s00029-024-00915-9

Matthew Satriano, Andrew P. Staal

引用次数: 0

On the Goncharov depth conjecture and polylogarithms of depth two 论冈察洛夫深度猜想和深度为 2 的多项式

Selecta Mathematica Pub Date : 2024-02-22 DOI: 10.1007/s00029-024-00918-6

Steven Charlton, Herbert Gangl, Danylo Radchenko, Daniil Rudenko

引用次数: 0

Davydov–Yetter cohomology and relative homological algebra 戴维多夫-叶特尔同调与相对同源代数

Selecta Mathematica Pub Date : 2024-02-21 DOI: 10.1007/s00029-024-00917-7

M. Faitg, A. M. Gainutdinov, C. Schweigert

{"title":"Davydov–Yetter cohomology and relative homological algebra","authors":"M. Faitg, A. M. Gainutdinov, C. Schweigert","doi":"10.1007/s00029-024-00917-7","DOIUrl":"https://doi.org/10.1007/s00029-024-00917-7","url":null,"abstract":"Davydov–Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category ({mathcal {C}}) are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center ({mathcal {Z}}({mathcal {C}})) relative to ({mathcal {C}}). From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of ({mathcal {Z}}({mathcal {C}})). Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras (Lambda {mathbb {C}}^k rtimes {mathbb {C}}[{mathbb {Z}}_2]), the Taft algebras and the small quantum group of (mathfrak {sl}_2) at a root of unity.","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139954802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian 格拉斯曼切向束的等变量子微分方程的单色性

Selecta Mathematica Pub Date : 2024-02-15 DOI: 10.1007/s00029-024-00916-8

Vitaly Tarasov, Alexander Varchenko

引用次数: 0

Koszul modules with vanishing resonance in algebraic geometry 代数几何中具有消失共振的科斯祖尔模块

Selecta Mathematica Pub Date : 2024-02-13 DOI: 10.1007/s00029-023-00912-4

引用次数: 0

Macdonald Duality and the proof of the Quantum Q-system conjecture 麦克唐纳对偶性与量子 Q 系统猜想的证明

Selecta Mathematica Pub Date : 2024-02-06 DOI: 10.1007/s00029-023-00909-z

{"title":"Macdonald Duality and the proof of the Quantum Q-system conjecture","authors":"","doi":"10.1007/s00029-023-00909-z","DOIUrl":"https://doi.org/10.1007/s00029-023-00909-z","url":null,"abstract":"<h3>Abstract</h3> The ({textsf{SL}}(2,{{mathbb {Z}}})) -symmetry of Cherednik’s spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a torus Dehn twist in type A. We prove for all twisted and untwisted affine algebras of type ABCD that the time-evolved q-difference Macdonald operators, in the (trightarrow infty ) q-Whittaker limit, form a representation of the associated discrete integrable quantum Q-systems, which are obtained, in all but one case, via the canonical quantization of suitable cluster algebras. The proof relies strongly on the duality property of Macdonald and Koornwinder polynomials, which allows, in the q-Whittaker limit, for a unified description of the quantum Q-system variables and the conserved quantities as limits of the time-evolved Macdonald operators and the Pieri operators, respectively. The latter are identified with relativistic q-difference Toda Hamiltonians. A crucial ingredient in the proof is the use of the “Fourier transformed” picture, in which we compute time-translation operators and prove that they commute with the Pieri operators or Hamiltonians. We also discuss the universal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations, for which we prove a duality relation, which simplifies the proofs further. ","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139763757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Hall Lie algebras of toric monoid schemes 环状单元方案的霍尔李代数

Selecta Mathematica Pub Date : 2024-02-04 DOI: 10.1007/s00029-023-00913-3

Jaiung Jun, Matt Szczesny

{"title":"Hall Lie algebras of toric monoid schemes","authors":"Jaiung Jun, Matt Szczesny","doi":"10.1007/s00029-023-00913-3","DOIUrl":"https://doi.org/10.1007/s00029-023-00913-3","url":null,"abstract":"We associate to a projective n-dimensional toric variety (X_{Delta }) a pair of co-commutative (but generally non-commutative) Hopf algebras (H^{alpha }_X, H^{T}_X). These arise as Hall algebras of certain categories ({text {Coh}}^{alpha }(X), {text {Coh}}^T(X)) of coherent sheaves on (X_{Delta }) viewed as a monoid scheme—i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When (X_{Delta }) is smooth, the category ({text {Coh}}^T(X)) has an explicit combinatorial description as sheaves whose restriction to each (mathbb {A}^n) corresponding to a maximal cone (sigma in Delta ) is determined by an n-dimensional generalized skew shape. The (non-additive) categories ({text {Coh}}^{alpha }(X), {text {Coh}}^T(X)) are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff–Kapranov. The Hall algebras (H^{alpha }_X, H^{T}_X) are graded and connected, and so enveloping algebras (H^{alpha }_X simeq U(mathfrak {n}^{alpha }_X)), (H^{T}_X simeq U(mathfrak {n}^{T}_X)), where the Lie algebras (mathfrak {n}^{alpha }_X, mathfrak {n}^{T}_X) are spanned by the indecomposable coherent sheaves in their respective categories. We explicitly work out several examples, and in some cases are able to relate (mathfrak {n}^T_X) to known Lie algebras. In particular, when (X = mathbb {P}^1), (mathfrak {n}^T_X) is isomorphic to a non-standard Borel in (mathfrak {gl}_2 [t,t^{-1}]). When X is the second infinitesimal neighborhood of the origin inside (mathbb {A}^2), (mathfrak {n}^T_X) is isomorphic to a subalgebra of (mathfrak {gl}_2[t]). We also consider the case (X=mathbb {P}^2), where we give a basis for (mathfrak {n}^T_X) by describing all indecomposable sheaves in ({text {Coh}}^T(X)).\u0000","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"222 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139678524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0