On the Differential Geometry of $GL_q(1| 1)$

Structure And Representations Of The Quantum General Linear Supergroup

The structure and representations of the quantum general linear supergroup GLq(m|n) are studied systematically by investigating the Hopf superalgebra Gq of its representative functions. Gq is factorized into $Gq^π Gq^{\barπ}$, and a Peter - Weyl basis is constructed for each factor. Parabolic induction for the quantum supergroup is developed. The underlying geometry of induced representations is discussed, and an analog of Frobenius reciprocity is obtained. A quantum Borel - Weil theorem is proven for the covariant and contravariant tensorial irreps, and explicit realizations are given for cla…

量子一般线性超群的Howe对偶的三种方法

Three approaches to the Howe duality between quantum general linear supergroups

本文提供了量子一般线性超群Howe对偶的两种新方法，分别基于量子微分算子与Beilinson-Lusztig-MacPherson实现，并通过显式作用公式证明了它们与原有方法等价。

Differential geometry of GL$_{p,q}(1| 1)$

We introduce a construction of the differential calculus on the quantum supergroup GL$_{p,q}(1| 1)$. We obtain two differential calculi, respectively, associated with the left and right Cartan-Maurer one-forms. We also obtain the quantum superalgebra of GL$_{p,q}(1| 1)$. Although all of the structures we obtain are derived without an R matrix, they neverthless can be expressed using an R matrix.

Stokes 现象与量子超群 $U_q(\mathfrak{gl}(m|n))$

Stokes phenomenon and quantum supergroup $U_q(\mathfrak{gl}(m|n))$

本文研究与李超代数 $\mathfrak{gl}_{m|n}$ 相关的量子合流超几何超系统的 Stokes 现象，证明其 Stokes 超矩阵满足 Yang-Baxter 方程，从而导出量子超群 $U_q(\mathfrak{gl}(m|n))$。

Covariant Differential Calculi on $SP_q^{1|2}$

A unitary orthosymplectic quantum supergroup is introduced. Two covariant differential calculi on the quantum superspace $SP_q^{1|2}$ are presented. The $h$-deformed symplectic superspaces via a contraction of the $q$-deformed symplectic superspaces are obtained. A new $h$-deformation of the Heisenberg superalgebra is given.

Two-parametric extension of h-deformation of GL$(1| 1)$

The two-parametric quantum deformation of the algebra of coordinate functions on the supergroup GL$(1| 1)$ via a contraction of GL$_{p,q}(1| 1)$ is presented. Related differential calculus on the quantum superplane is introduced.

A differential calculus on Z$_3$-graded quantum superspace ${\mathbb R}_q(2|1)$

We introduce a Z$_3$-graded quantum $(2+1)$-superspace and define Z$_3$-graded Hopf algebra structure on algebra of functions on the Z$_3$-graded quantum superspace. We construct a differential calculus on the Z$_3$-graded quantum superspace, and obtain the corresponding Z$_3$-graded Lie superalgebra. We also find a new Z$_3$-graded quantum supergroup which is a symmetry group of this calculus.

Bicovariant differential calculus on the superspace ${\mathbb R}_q(1|2)$

Super Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.

n=3 Nilpotent differential calculus on some non-commutative (super) spaces

In this paper, we construct a covariant differential calculus on quantum plane with two-parametric quantum group as a symmetry group. The two cases $d^2=0$ and $d^3=0$ are completly established. We also construct differential calculi $n=2$ and $n=3$ nilpotent on super quantum space with one and two-parametric symmetry quantum supergroup.

h-deformation of GL(1|1)

h-deformation of (graded) Hopf algebra of functions on supergroup GL(1|1) is introduced via a contration of GL_q (1|1). The deformation parameter h is odd (grassmann). Related differential calculus on h-superplane is presented.

利用有限维超代数逼近量子queer超群（初步版本）

Approaching quantum queer supergroups using finite dimensional superalgebras (Preliminary version)

本文通过有限维queer q-Schur超代数将BLM的代数逼近方法推广到量子queer超群，建立了其新的代数实现。

Differential geometry of the q-quaternion

Differential calculus on the quantum quaternionic group GL(1,H$_q$) is introduced.

Differential geometry of the Z_3-graded quantum superplane

In this work, differential geometry of the Z$_3$-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.

Differential Geometry of the Lie superalgebra of the quantum superplane

Differential geometry of the quantum Lie superalgebra of the extended quantum superplane and its Z$_2$-graded Hopf algebra structure is obtained. Its Z$_2$-graded dual Hopf algebra is also given.

双交换多项式环的非交换微分几何

Noncommutative differential geometry of ambiskew polynomial rings

本文研究双交换多项式环的微分光滑性条件，基于Jordan的先前工作，提出充分判定准则。

Differential geometry of the q-superplane

Hopf algebra structures on the extended q-superplane and its differential algebra are defined. An algebra of forms which are obtained from the generators of the extended q-superplane is introduced and its Hopf algebra structure is given

Differential Geometry of the q-plane

Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given.

Z$_3$-graded differential geometry of quantum plane

In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given.

Differential Geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual Hopf algebra. Differential geometry of the quantum Lie algebra of the extended quantum plane and its Hopf algebra structure is obtained. Its dual Hopf algebra is also given.

导出微分几何中移位泊松结构与变形量子化概述

An outline of shifted Poisson structures and deformation quantisation in derived differential geometry

研究如何把导出代数几何结果转化到导出微分几何，涉及NQ -流形等上的移位泊松结构及变形量子化的存在性与分类。

Some aspects of noncommutative differential geometry

We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework.

Noncommutative Differential Geometry and Twisting of Quantum Groups

We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of `quantum manifold' and `quantum Riemannian manifold' based on quantum group principal bundles, a formulation that works over general unital algebras.

On the definition of higher order differential operators in noncommutative geometry

Several definitions of differential operators on modules over noncommutative rings are discussed.

Hopf Modules and Noncommutative Differential Geometry

We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one to one correspondence between anti-Yetter-Drinfeld modules, which serve as coefficients for the Hopf cyclic (co)homology, and modules which admit a flat connection with respect to our differential calculus. Thus we show that these coefficient modules can be regarded as ``flat bundles'' in the sense of Connes' noncommutative differential geometry.

Bicovariant Differential Geometry of the Quantum Group $SL_h(2)$

There are only two quantum group structures on the space of two by two unimodular matrices, these are the $SL_q(2)$ and the $SL_h(2)$ [9-13] quantum groups. One can not construct a differential geometry on $ SL_q(2)$, which at the same time is bicovariant, has three generators, and satisfies the Liebnitz rule. We show that such a differential geometry exists for the quantum group $SL_h(2)$ and derive all of its properties.

A dilemma of nonequivalent definitions of differential operators in noncommutative geometry

In contrast with differential operators on modules over commutative and graded commutative rings, there is no satisfactory notion of a differential operator in noncommutative geometry.

Differential Algebras in Non-Commutative Geometry

We discuss the differential algebras used in Connes' approach to Yang-Mills theories with spontaneous symmetry breaking. These differential algebras generated by algebras of the form functions $\otimes$ matrix are shown to be skew tensorproducts of differential forms with a specific matrix algebra. For that we derive a general formula for differential algebras based on tensor products of algebras. The result is used to characterize differential algebras which appear in models with one symmetry breaking scale.

Instanton Counting, Quantum Geometry and Algebra

The aim of this memoir for "Habilitation à Diriger des Recherches" is to present quantum geometric and algebraic aspects of supersymmetric gauge theory, which emerge from non-perturbative nature of the vacuum structure induced by instantons. We start with a brief summary of the equivariant localization of the instanton moduli space, and show how to obtain the instanton partition function and its generalization to quiver gauge theory and supergroup gauge theory in three ways: the equivariant index formula, the contour integral formula, and the combinatorial formula. We then explore the geometri…

Differential Geometry on Linear Quantum Groups

An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group $GL_{q}(N)$. $SL_{q}(N)$ is then found by constructing matrices with determinant unity, and the induced calculus is found.

The Covariant Picard Groupoid in Differential Geometry

In this article we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.

Noncommutative Differential Geometry and Connections on Simplicial Manifolds

For a simplicial manifold we construct the differential geometry structure and use it to investigate linear connections, metric and gravity. We discuss and compare three main approaches and calculate the resulting gravity action functionals.

Lectures on graded differential algebras and noncommutative geometry

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.

Differential Geometry on Hopf Algebras and Quantum Groups (Ph.D. Thesis)

The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the a…

Quantum kappa-deformed differential geometry and field theory

I introduce in kappa-Minkowski noncommutative spacetime the basic tools of quantum differential geometry, namely bicovariant differential calculus, Lie and inner derivatives, the integral, the Hodge-star and the metric. I show the relevance of these tools for field theory with an application to complex scalar field, for which I am able to identify a vector-valued four-form which generalizes the energy-momentum tensor. Its closedness is proved, expressing in a covariant form the conservation of energy-momentum.