Tomasz Brzeziński, Wojciech Szymański
The quantum flag manifold ${SU_q(3)/\mathbb{T}^2}$ is interpreted as a noncommutative bundle over the quantum complex projective plane with the quantum or Podleś sphere as a fibre. A connection arising from the (associated) quantum principal $U_q(2)$-bundle is described.
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We show that certain homological regularity properties of graded connected algebras, such as being AS-Gorenstein or AS-Cohen-Macaulay, can be tested by passing to associated graded rings. In the spirit of noncommutative algebraic geometry, this can be seen as an analogue of the classical result that, in a flat family of varieties over the affine line, regularity properties of the exceptional fiber extend to all fibers. We then show that quantized coordinate rings of flag varieties and Schubert varieties can be filtered so that the associated graded rings are twisted semigroup rings. This is a …
We establish a multiplication formula for a tridiagonal standard basis element in the idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$ arising from the geometry of affine partial flag varieties of type $C$. We apply this formula to obtain the stabilization algebras $\dot{\mathbf K}^{\mathfrak{c}}_n$, $\dot{\mathbf K}^{\jmath \imath}_{\mathfrak{n}}$, $\dot{\mathbf K}^{\imath \jmath}_{\mathfrak{n}}$ and $\dot{\mathbf K}^{\imath \imath}_η$, which are idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$. The symmetry in the formula leads to an isomorphism of the…
An algebraic framework for noncommutative bundles with (quantum) homogeneous fibres is proposed. The framework relies on the use of principal coalgebra extensions which play the role of principal bundles in noncommutative geometry which might be additionally equipped with a Hopf algebra symmetry. The proposed framework is supported by two examples of noncommutative $\mathbb{C} P_q^1$-bundles: the quantum flag manifold viewed as a bundle with a generic Podleś sphere as a fibre, and the quantum twistor bundle viewed as a bundle over the quantum 4-sphere of Bonechi, Ciccoli and Tarlini.
We show that the Yangian Yn over gl_n possesses some features of the ring of regular functions on GL_n. In particular, we use the theory of quasideterminants to construct noncommutative flags associated to Yn. In so doing, a class of comodule algebras Fl(γ) for Yn (viewed as C[GL_n]) is revealed. As in the classic case, these flag algebras are related to the irreducible representations of Yn (viewed as U(gl_n)). In the course of defining the rings Fl(γ), connections to the new parabolic presentations of Yn given by Brundan and Kleshchev (2005) are uncovered.
We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of $\mathbf{Q}_{G}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over $\mathbf{Q}_{G}$. In order to achieve this, we provide a number of fundamental results on $\mathbf{Q}_{G}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is c…
We study noncommutative differential structures on the group of permutations $S_N$, defined by conjugacy classes. The 2-cycles class defines an exterior algebra $Λ_N$ which is a super analogue of the Fomin-Kirillov algebra $\CE_N$ for Schubert calculus on the cohomology of the $GL_N$ flag variety. Noncommutative de Rahm cohomology and moduli of flat connections are computed for $N<6$. We find that flat connections of submaximal cardinality form a natural representation associated to each conjugacy class, often irreducible, and are analogues of the Dunkl elements in $\CE_N$. We also construct $…
We establish a noncommutative generalisation of the Borel-Weil theorem for the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds $\mathcal{O}_q(G/L_S)$, generalising previous work of a number of authors (including the first and third authors of this paper) on the quantum Grassmannians $\mathcal{O}_q(\mathrm{Gr}_{n,m})$. As a direct consequence we get a novel noncommutative differential geometric presentation of the quantum coordinate rings $S_q[G/L_S]$ of the irreducible quantum flag manifolds. The proof is formulated in terms of quantum principal bundles, and the recently in…
Noncommutative Kähler structures were recently introduced by the second author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic one-forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results a…
In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to the anti-dominant minuscule fundamental weight $- \varpi_{k}$ in the torus-equivariant quantum $K$-group of the partial flag manifold $G/P_{J}$ (where $J = I \setminus \{k\}$) corresponding to the maximal (standard) parabolic subgroup $P_{J}$ of minuscule type in type $A$, $D$, $E$, or $B$. This result is obtained by proving a similar formula in a torus-equivariant $K$-group of the semi-infinite partial flag manifold $\mathbf…
Noncommutative Complex Structures for the Full Quantum Flag Manifold of Quantum SU(3)
本文详细考察了量子 SU(3) 全量子旗流形的非交换复几何，发现近复结构数量从 8 减至 4 且均可积，并证明由于非退化余不变 2-形式的非中心性，这些复结构均不容许协变非交换 Kähler 结构。
We show that tensoring the Laplace and Dolbeault-Dirac operators of a Kähler structure (with closed integral) by a negative Hermitian holomorphic module, produces operators with spectral gaps around zero. The proof is based on the recently established Akizuki-Nakano identity of a noncommutative Kähler structure. This general framework is then applied to the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it is shown that twisting their Dirac and Laplace operators by negative line bundles produces a spectral gap, for q sufficiently close to 1. The main technical challen…
We suggest a possible programme to associate geometric "flag-like" data to an arbitrary simple quantum group, in the spirit of the noncommutative algebraic geometry developed by Artin, Tate, and Van den Bergh. We then carry out this programme for the standard quantum SL(n) of Drinfel'd and Jimbo, where the varieties involved are certain T-stable subvarieties of the (ordinary) flag variety.
The category O of BGG can be thought of as a category of sheaves over the flag variety F in the sense that the algebra E of self-extensions of the trivial object of O is isomorphic to the cohomology algebra of the flag variety. A deformation of O' - giving rise to a "new" algebra E' - can be thought of as a (possibly noncommutative) deformation F' of F. The mythic variety F', being a deformation of F, should have the same homotopy type as F, and E' should therefore be isomorphic to E.
We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere $S^3$ defined by basic K-theoretic equations. We find a 3-parameter family of deformations $S^3_{\mathbf{u}}$ of the standard 3-sphere $S^3$ and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space $\mathbb R^4$. For generic values of the deformation parameters we show that the o…
Quantum Segre maps via cocycle twists
本研究提出一种对幺半群分次代数的张量积与态射进行余循环扭量子化的通用方法，构造了射影空间经典Segre嵌入的形变，且已有非交换Segre映射为其可因子余循环特例。
We propose to define $\mathbb{F}_1$-algebras as integral bi-rings with the co-ring structure being the descent data from $\mathbb{Z}$ to $\mathbb{F}_1$. The coordinate bi-ring of $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$ is then the co-ring of integral linear recursive sequences equipped with the Hadamard product. We associate a noncommutative moduli space to this setting, show that it is defined over $\mathbb{F}_1$, and has motive $\prod_{n \geq 0} \frac{s-n}{2 π}$.
Lusztig's Quantum Root Vectors and a Dolbeault Complex for the A-Series Full Quantum Flag Manifolds
本文证明 Lusztig 正根向量在好约化分解下构成 A 系列全量子旗流形的量子切空间，其微分演算为经典反全纯 Dolbeault 复形的 q-形变，并由此建立量子 Borel-Weil 定理，实现所有有限维 type-1 不可约表示。
CGL extensions, named after G. Cauchon, K. Goodearl, and E. Letzter, are a special class of noncommutative algebras that are iterated Ore extensions of associative algebras with compatible torus actions. Examples of CGL extensions include quantum Schubert cells and quantized coordinate rings of double Bruhat cells. CGL extensions have been studied extensively in connection with quantum groups and quantum cluster algebras. For a field $\mathbf{k}$ of characteristic $0$, let $L=\mathbf{k}[q^{\pm 1}]$ be the $\mathbf{k}$-algebra of Laurent polynomials in the single variable $q$ and let $\mathbb{K…
Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localization…
The first part of this thesis deals with certain properties of the quantum symmetric and exterior algebras of Type 1 representations of $U_q(g)$ defined by Berenstein and Zwicknagl. We define a notion of a commutative algebra object in a coboundary category, and we prove that the quantum symmetric algebra of a module is the universal commutative algebra generated by that module. That is, the functor assigning to a module its quantum symmetric algebra is left adjoint to a forgetful functor. We also prove a conjecture of Berenstein and Zwicknagl, stating that the quantum symmetric and exterior c…
We construct by geometric methods a noncommutative model E of the algebra of regular functions on the universal (2-fold) cover M of certain nilpotent coadjoint orbits O for a complex simple Lie algebra g. Here O is the dense orbit in the cotangent bundle of the generalized flag variety X associated to a complexified Cartan decomposition g=(p^+)+k+(p^-) where p^+- are Jordan algebras by the TKK construction. We obtain E as the algebra of g-finite differential operators on a smooth Lagrangian subvariety in M where g is given by differential operators twisted according to a critical parameter. Af…
Let $G_k$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be a quasi-split symmetric subgroup of $G_k$ with respect to an involution $θ_k$ of $G_k$. The classification of such involutions is independent of the characteristic of $k$ (provided not $2$). We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of the Chevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G}, \mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any algebraically closed field of characteri…
Compact Quantum Homogeneous Kähler Spaces
本文研究紧量子齐性凯勒空间，探讨其与Woronowicz量子群理论的兼容性，分析Dolbeault-Dirac算子的分析行为及扭后的Fredholm性与欧拉特征数的关系。
We propose a simple but effective framework for producing examples of covariant faithfully flat (generalised) Hopf-Galois extensions from a nested pair of quantum homogeneous spaces. Our construction is modelled on the classical situation of a homogeneous fibration $G/N \to G/M$, for $G$ a group, and $N \subseteq M \subseteq G$ subgroups. Variations on Takeuchi's equivalence and Schneider's descent theorem are presented in this context. Quantum flag manifolds and their associated quantum Poisson homogeneous spaces are taken as motivating examples. Moreover, a large collection of noncommutative…
Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on noncommutative manifolds
本文结合非交换黎曼几何与相干2-群理论，将经典U(1)-规范理论推广至非交换情形，建立了厄米线双模与可微量子U(1)-主丛的等价关系，并求解了非交换黎曼结构的提升问题。
We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type $C$. These identities express the product $e^μ \, \mathrm{gch} \, V_{x}^{-}(λ)$ of the (one-dimensional) character $e^μ$, where $μ$ is a (not necessarily dominant) minuscule weight, with the graded character $\mathrm{gch} \, V_{x}^{-}(λ)$ of the level-zero Demazure submodule $V_{x}^{-}(λ)$ over the quantum affine algebra $U_{\mathsf{q}}(\mathfrak{g}_{\mathrm{af}})$ as an explicit finite linear combination of the graded char…
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