S. Majid
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.
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
We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e. we establish a noncommutative correspondence principle from *-Poisson brackets to *-commutator…
This is the first of series of talks presented at a permanent Rutgers workshop on noncommutative algebra and geometry. We study here quadratic and quadratic-linear algebras defined by factorizations of noncommutative polynomials and differential polynomials. Such algebras posses a natural derivation and give us a new understanding of a nature of noncommutative symmetric functions.
We show that the crossed modules and bicovariant different calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisation of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra $\C[x]\bicross_{\hbar,\grav}\C[p]$. This is a quantum group which has an $\hbar\to 0$ limit as the functions on a classical but non-Abelian group and a $\grav\to 0$ limit as flat spac…
This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.
By twisting the commutation relations between creation and annihilation operators, we show that quantum conformal invariance can be implemented in the 2-d Moyal plane. This is an explicit realization of an infinite dimensional symmetry as a quantum algebra.
We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules.We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors generalizes the theory of theta-functions. The paper is self-contained; it can be used also as an introduction to the theory of noncommutative spaces with simplest space of this kind thoroughly analyzed as a basic example.
We show that a class of braided Hopf algebras, which includes the braided $SU_q(2)$ is obtained by twisting. We show further examples and demonstrate that twisting of bicovariant differential calculi gives braided bicovariant differential calculi.
Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We extend the cochain twist framework to connections and Riemannian structures and provide examples including twist of the $S^7$ coordinate algebra to a nonassociative hyperbolic geometry in the same category as that of the octonions.
The real sphere $S^{N-1}_\mathbb R$ appears as increasing union, over $d\in\{1,...,N\}$, of its "polygonal" versions $S^{N-1,d-1}_\mathbb R=\{x\in S^{N-1}_\mathbb R|x_{i_0}... x_{i_d}=0,\forall i_0,...,i_d\ {\rm distinct}\}$. Motivated by general classification questions for the undeformed noncommutative spheres, smooth or not, we study here the quantum isometries of $S^{N-1,d-1}_\mathbb R$, and of its various noncommutative analogues, obtained via liberation and twisting. We discuss as well a complex version of these results, with $S^{N-1}_\mathbb R$ replaced by the complex sphere $S^{N-1}_\m…
For differential calculi on noncommutative algebras, we construct a twisted de Rham cohomology using flat connections on modules. This has properties similar, in some respects, to sheaf cohomology on topological spaces. We also discuss generalised mapping properties of these theories, and relations of these properties to corings. Using this, we give conditions for the Serre spectral sequence to hold for a noncommutative fibration. This might be better read as giving the definition of a fibration in noncommutative differential geometry. We also study the multiplicative structure of such spectra…
We consider differential operators over a noncommutative algebra $A$ generated by vector fields. These are shown to form a unital associative algebra of differential operators, and act on $A$-modules $E$ with covariant derivative. We use the repeated differentials given in the paper to give a definition of noncommutative Sobolev space for modules with connection and Hermitian inner product. The tensor algebra of vector fields, with a modified bimodule structure and a bimodule connection, is shown to lie in the centre of the bimodule connection category ${}_A\mathcal{E}_A$, and in fact to be an…
We give an explicit description of the set of all factorization structures, or twisting maps, existing between the algebras k^2 and k^2, and classify the resulting algebras up to isomorphism. In the process we relate several different approaches formerly taken to deal with this problem, filling a gap that appeared in a recent paper by Cibils. We also provide a counterexample to a result concerning the Hochschild (co)homology appeared in a paper by J.A. Guccione and J.J. Guccione.
This paper proposes a new notion of smoothness of algebras, termed differential smoothness, that combines the existence of a top form in a differential calculus over an algebra together with a strong version of the Poincaré duality realized as an isomorphism between complexes of differential and integral forms. The quantum two- and three-spheres, disc, plane and the noncommutative torus are all smooth in this sense. Noncommutative coordinate algebras of deformations of several examples of classical orbifolds such as the pillow orbifold, singular cones and lens spaces are also differentially sm…
We study the 10 noncommutative spheres obtained by liberating, twisting, and liberating+twisting the real and complex spheres $S^{N-1}_\mathbb R,S^{N-1}_\mathbb C$. At the axiomatic level, we show that, under very strong axioms, these 10 spheres are the only ones. Our main results concern the computation of the quantum isometry groups of these 10 spheres, taken in an affine real/complex sense. We formulate as well a proposal for an extended formalism, comprising 18 spheres.
We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a connection with respect to a differential calculus and consider questions of existence and uniqueness. At the end these constructions are applied to basic examples of noncommutative bundles over a coquasitriangular Hopf algebra.
Elements of noncommutative differential geometry of ${\mathbb Z}$-graded generalized Weyl algebras ${\mathcal A}(p;q)$ over the ring of polynomials in two variables and their zero-degree subalgebras ${\mathcal B}(p;q)$, which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of ${\mathcal A}(p;q)$ are constructed, and three-dimensional first-order differential calculi induced by these derivations are described. The associated integrals are computed and it is shown that the dimension of the inte…
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.
In this survey article we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on noncommutative motives. We propose a motivic measure with values in a motivic ring. This enables us to introduce certain zeta functions of noncommutative spaces.
Derivation based noncommutative geometry of quiver path algebras
本文研究箭图路代数的基于导子的非交换几何，引入分离箭图概念得到微分形式模为投射双模的等价条件，可构造Levi-Civita联络等非交换几何结构。
Several definitions of differential operators on modules over noncommutative rings are discussed.
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。
Quantum jet Hopf algebroids by cotwist
通过余扭构造方法，将经典基础的喷流代数转化为非交换霍普夫代数体，揭示了喷流代数的霍普夫代数体结构。
We study (quasi-)twilled pre-Lie algebras and the associated $L_\infty$-algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maurer-Cartan equations of the associated differential graded Lie algebras ($L_\infty$-algebras). Furthermore, we show that $\mathcal{O}$-operators and twisted $\mathcal{O}$-operators are solutions of the Maurer-Cartan equations. As applications, we study (quasi-)pre-Lie bialgebras using the associated differential graded Lie algebras ($L_\infty$-algeb…
In our previous publications we have introduced a differential calculus on the algebra $U(gl(m))$ based on a new form of the Leibniz rule which differs from that usually employed in Noncommutative Geometry. This differential calculus includes partial derivatives in generators of the algebra $U(gl(m))$ and their differentials. The orresponding differential algebra $Ω(U(gl(m)))$ is a deformation of the commutative algebra $Ω({\rm Sym}(gl(m)))$.A similar claim is valid for the Weyl algebra ${\cal W}(U(gl(m)))$ generated by the algebra $U(gl(m))$ and the mentioned partial derivatives. In the parti…
This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of as category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considere…
We survey some aspects of the theory of noncommutative manifolds focusing on the noncommutative analogs of two-dimensional tori and low-dimensional spheres. We are particularly interested in those aspects of the theory that link the differential geometry and the algebraic geometry of these spaces.
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.
We define a noncommutative algebra of four basic objects within a differential calculus on quantum groups: functions, 1-forms, Lie derivatives and inner derivations, as the cross-product algebra associated with Woronowicz's (differential) algebra of functions and forms. This definition properly takes into account the Hopf algebra structure of the Woronowicz calculus. It also provides a direct proof of the Cartan identity.
The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on $\mathbb R^n$, one may derive a similar pseudo-differential calculus on noncommutative $n$ tori $\mathbb T_θ^n$, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for $n=2,4$, as shown in the groundbreaking paper in which t…
We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers.
This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential graded algebra. This is compared to current ideas on noncommutative algebraic 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.
Comparison between two differential graded algebras in Noncommutative Geometry
本文比较了非交换几何中Connes和Fröhlich等人定义的两种微分格代数，发现Connes的dga在信息性上更具优势。
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.
We discuss the construction of finite noncommutative geometries on Hopf algebras and finite groups in the `quantum groups approach'. We apply the author's previous classification theorem, implying that calculi in the factorisable case correspond to blocks in the dual, to classify differential calculi on the quantum codouble $D^*(G)=kG\lcocross k(G)$ of a finite group $G$. We give $D^*(S_3)$ as an example including its exterior algebra and lower cohomology. We also study the calculus on $D^*(\CA)$ induced from one on a general Hopf algebra $\CA$ in general and specialise to $D^*(G)=U(\cg)\lcocr…
Symbols in Noncommutative Geometry
本文通过反对称化复合运算将经典李括号推广到非交换环境，依赖微分算子可表示性，给出jet模表示条件，并发展符号理论以保证括号封闭性。
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 construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.
We introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded algebras and show that the category of flat connections on such modules forms a Tannakian category. As such this category can be realised as the category of representations of an affine group scheme $G$, which in the classical case is (the pro-algebraic completion of) the usual fundamental group. This motivates us to define $G$ to be the fundamental group of the n…
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.
We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space.
We show that the quantisation of a connected simply-connected Poisson-Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre-Lie algebra structure. As an example, we find a pre-Lie algebra structure underlying the standard 3D differential structure on $\C_q[SU_2]$. At the noncommutative geometry level we show that the enveloping algebra $U(\cm)$ of a Lie algebra $\cm$, viewed as quantisation of $\cm^*$, admits a connected differential exterior algebra of classical dimension if and only if $\cm$…
A complete Fock space representation of the covariant differential calculus on quantum space is constructed. The consistency criteria for the ensuing algebraic structure, mapping to the canonical fermions and bosons and the consequences of the new algebra for the statistics of quanta are analyzed and discussed. The concept of statistical transmutation between bosons and fermions is introduced.
A notion of Cartan pairs as an analogy of vector fields in the realm of noncommutative geometry has been proposed in q-alg/9609011 In this paper we give an outline of the construction of a noncommutative analogy of the algebra of partial differential operators as well as its natural (Fock type) representation. We shall also define co-universal vector fields and covariant derivatives.
Let $K$ be any unital commutative $\mathbb Q$-algebra and $z=(z_1, ..., z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by $\kzz$ and $\kttzz$ the formal power series algebras of $z$ over $K$ and $K[[t]]$, respectively. For any $α\geq 1$, let $\cDazz$ be the unital algebra generated by the differential operators of $\kzz$ which increase the degree in $z$ by at least $α-1$ and $ \ataz $ the group of automorphisms $F_t(z)=z-H_t(z)$ of $\kttzz$ with $o(H_t(z))\geq α$ and $H_{t=0}(z)=0$. First, for a…
Noncommutative differential calculus on quantum Minkowski space is not separated with respect to the standard generators, in the sense that partial derivatives of functions of a single generator can depend on all other generators. It is shown that this problem can be overcome by a separation of variables. We study the action of the universal L-matrix, appearing in the coproduct of partial derivatives, on generators. Powers of he resulting quantum Minkowski algebra valued matrices are calculated. This leads to a nonlinear coordinate transformation which essentially separates the calculus. A com…
In the context of a noncommutative differential calculus on the algebra of real valued functions of an $n$-dimensional manifold $M$, a commutative and associative product of 1-forms is naturally defined. Ordinary differential calculus corresponds to this product being trivially zero. We consider the minimal generalization in which the algebra of 1-forms with this product is nilpotent of degree 3. Basic tensor analysis and differential geometry are developped in this context and applied to the formulation of symplectic geometry and Hamiltonian dynamics. It is shown that the corresponding Liouvi…
We carry over to a quite general noncommutative setting some of the basic tools of differential geometry, using from the very beginning the setting of convenient vector spaces developed by Froelicher and Kriegl, which allows to carry all of multilinear algebra into this kind of functional analysis with suitably completed tensor products. In the first section we give a short description of the setting of convenient spaces elaborating those aspects which are needed later. Then we repeat the usual construction of noncommutative differential forms for convenient algebras. Next they show that the b…
Deformation quantization and intrinsic noncommutative differential geometry
本文提出非交换微分几何的内蕴表述，给出协变导数、曲率及Bianchi恒等式的几何定义，证明特定条件下Ricci曲率等价，并计算球对称等距嵌入的量子涨落与曲率，表明引力量子化可重整化，最后定义准连接并证明第一Bianchi恒等式。
There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for any two functions f and g. We notice that this non-commutative differential algebra has a geometrical realization as a convolution algebra of the symplectic groupoid integrating the Poisson manifold. This quantization is just a part of a quantization of the odd symplectic category (where objects are odd symplectic supermanifolds and morphisms are Lagrangian relation) in terms of Z_2-graded chain complexes. It is a straightforward consequence of the theory of BV…
Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum homogeneous vector bundles are classified and explicitly constructed by using the theory of projective modules.
These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results, in particular, noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative differential forms and double-tangent bundles.
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck's notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DDer(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain `Fock space' associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in …
We find an infinite number of noncommutative geometries which posses a differential structure. They generalize the two dimensional noncommutative plane, and have infinite dimensional representations. Upon applying generalized coherent states we are able to take the continuum limit, where we recover the punctured plane with non constant Poisson structures.
We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on $S^3$.
We construct a period mapping for deformations of a differential graded algebra, that generalizes Griffiths' period mapping. It is constructed as a morphism between differential graded Lie algebras which has a moduli-theoretic interpretation, where the domain of the morphism is the shifted Hochschild cocohain complex. We then use this period mapping to give some applications such as unobstructedness of deformations.
We study covariant derivatives on a class of centered bimodules $\mathcal{E}$ over an algebra A. We begin by identifying a $\mathbb{Z} ( A ) $-submodule $ \mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ \mathcal{X} ( A ) $ is proven to be a Lie algebra. Connections on $\mathcal{E}$ are in one to one correspondence with covariant derivatives on $ \mathcal{X} ( A ). $ We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is als…
We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by certain Hopf quiver data as familiar in the context of path algebras. We explore a duality between geometry on the function algebra vs geometry on the group algebra, i.e. on the dual Hopf algebra, illustrated by the noncommutative Riemannian geometry of the group algebra of $S_3$. We show how quiver geometries arise naturally in the context of quantum princi…
We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The class of noncommutative spaces studied is very rich. Non-anticommutative superspaces are also briefly considered. The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein's equations for gravity on noncommutative manifolds.
Noncommutative Riemannian geometry of Kronecker algebras
本文研究Kronecker箭图路径代数的非交换黎曼几何，构造无挠双模联络及Levi-Civita联络，并探讨其与厄米形式的相容性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。