Israel Gelfand, Sergei Gelfand, Vladimir Retakh
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The algebra $Q_n$ is closely related to factorizations of a generic polynomial of degree $n$ over a division algebra into linear factors.
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We expose the notion of noncommutative CW (NCCW) complexes, define noncommutative (NC) mapping cylinder and NC mapping cone, and prove the noncommutative Approximation Theorem. The long exact homotopy sequences associated with arbitrary morphisms are also deduced.
We introduce in this paper the notion of noncommutative Serre fibration (shortly, NCSF) and show that up to homotopy, every NCCW complex morphism is some noncommutative Serre fibration. We then deduce a six-term exact sequence for the periodic cyclic homology and for K-theory of an arbitrary noncommutative Serre fibration. We also show how to use this technique to compute K-theory and cyclic theory of some noncommutative quotients.
In this article some noncommutative topological objects such as NC mapping cone and NC mapping cylinder are introduced. We will see that these objects are equipped with the NCCW complex structure of [PEDERSEN]. As a generalization we introduce the notions of NC mapping cylindrical and conical telescope. Their relations with NC mappings cone and cylinder are studied. Some results on their K0 and K1 groups are obtained and the cyclic six term exact sequence theorem for their k-groups are proved. Finally we explain their NCCW coplex structure and the conditions in which these objects admit NCCW c…
We prove in this paper a noncommutative version of Leray Spectral Sequence Theorem and then Leray-Serre Spectral Theorem for noncommutative Serre fibrations: for NC Serre fibration there are converging spectral sequences with $\E^2$ terms as $\E^2_{p,q} = \HP_p(A; \HP_q(B,A)) \Longrightarrow \HP_{p+q}(B)$ and $\E^2_{p,q} = \HP_p(A;\K_q(B,A)) \Longrightarrow \K_{p+q}(B)$.
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 study a noncommutative deformation of the commutative Hopf algebra of rooted trees which was shown by Connes and Kreimer to be related to the mathematical structure of renormalization in quantum field theories. The requirement of the existence of an antipode for the noncommutative deformation leads to a natural extension of the algebra. Noncommutative deformations of the Connes-Kreimer algebra might be relevant for renormalization of field theories on noncommutative spaces and there are indications that in this case the extension of the algebra might be linked to a mixing of infrared and ul…
We introduce a new type of noncommutative Poisson structure on associative algebras. It induces Poisson structures on the moduli spaces classifying semisimple modules. Path algebras of doubled quivers and preprojective algebras have noncommutative Poisson structures given by the necklace Lie algebra.
Using the corepresentation of the quantum group $ SL_q(2)$ a general method for constructing noncommutative spaces covariant under its coaction is developed. The method allows us to treat the quantum plane and Podleś' quantum spheres in a unified way and to construct higher dimensional noncommutative spaces systematically. Furthermore, we extend the method to the quantum supergroup $ OSp_q(1|2).$ In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed.
In this paper we extend the standard differential geometric theory of Hamiltonian dynamics to noncommutative spaces, beginning with symplectic forms. Derivations on the algebra are used instead of vector fields, and interior products and Lie derivatives with respect to derivations are discussed. Then the Poisson bracket of certain algebra elements can be defined by a choice of closed 2-form. Examples are given using the noncommutative torus, the Cuntz algebra, the algebra of matrices, and the algebra of matrix valued functions on $\Bbb R^2$.
A one parameter set of noncommutative complex algebras is given. These may be considered deformation quantisation algebras. The commutative limit of these algebras correspond to the algebra of polynomial functions over a manifold or variety. The topology of the manifold or variety depends on the parameter, varying from nothing, to a point, a sphere, a certain variety and finally a torus. The irreducible adjoint preserving representations of the noncommutative algebras are studied. As well as typical noncommutative sphere type representations and noncommutative torus type representations, a new…
We study the bialgebra structure of the Hall algebra of two-periodic complexes recently introduced by Bridgeland. We introduce coproduct on Bridgeland's Hall algebra, and show that in the hereditary case the resulting bialgebra structure coincides with that on Drinfeld double of the ordinary Hall algebra
We review the notion of submanifold algebra, as introduced by T. Masson, and discuss some properties and examples. A submanifold algebra of an associative algebra $A$ is a quotient algebra $B$ such that all derivations of $B$ can be lifted to $A$. We will argue that in the case of smooth functions on manifolds every quotient algebra is a submanifold algebra, derive a topological obstruction when the algebras are deformation quantizations of symplectic manifolds, present some (commutative and noncommutative) examples and counterexamples.
The aim of the paper is twofold. First, we introduce analogs of (partial) derivatives on certain Noncommutative algebras, including some enveloping algebras and their "braided counterparts", namely, the so-called modified Reflection Equation algebras. By using these derivatives we construct an analog of the de Rham complex on these algebras. Second, we discuss deformation property of some quantum algebras and show that contrary to a commonly held view, in the so-called q-Witt algebra there is no analog of the PBW theorem. In this connection, we discuss different forms of the Jacobi condition r…
We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, $θ$-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice $(\Z_2)^n$ and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell's equations. We exactly quantize noncommutative U(1)-Yang-Mills theory on $\Z_2\times\Z_2$ in a path integral approac…
A survey on Zariski cancellation problems for noncommutative and Poisson algebras
本文综述了非交换代数与Poisson代数背景下Zariski消去问题的最新研究进展。
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.
The quadratic algebras Q_n are associated with pseudo-roots of noncommutative polynomials. We compute the Hilbert series of the algebras Q_n and of the dual quadratic algebras Q_n^!
We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T^*B, which is a basic example of noncommutative symplectic manifold. Applying Hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras, P=P(B), that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction…
We define ''convergence'' for noncommutative power series and construct two topologies on the algebra of power series, convergent with respect to a positive radius. We indicate all finite dimensional continuous representations of this algebra and prove completeness for both topologies.
Given a Lie algebra, there uniquely exists a Poisson algebra which is called a Lie-Poisson algebra over the Lie algebra. We will prove that given a Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra over the Loday algebra. The noncommutative Poisson algebras are called the Loday-Poisson algebras. In the super/graded cases, the Loday-Poisson bracket is regarded as a noncommutative version of classical (linear) Schouten-Nijenhuis bracket. It will be shown that the Loday-Poisson algebras form a special subclass of Aguiar's dual-prePoisson algebras. We also study a proble…
The first part of these notes gives an introduction to noncommutative projective geometry after Artin--Zhang. The second part provides an overview of the work of Polishchuk that reconciles noncommutative two-tori having real multiplication with the Artin--Zhang setting.
Quadratic algebras associated to pseudo-roots of noncommutative polynomials have been introduced by I. Gelfand, Retakh, and Wilson in connection with studying the decompositions of noncommutative polynomials. Later they (with S. Gelfand and Serconek) shown that the Hilbert series of these algebras and their quadratic duals satisfy the necessary condition for Koszulity. It is proved in this note that these algebras are Koszul.
We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving Z_2-graded complexes of quiver representations.
The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra $\gg$ sitting inside an associative algebra $A$ and any associative algebra $\FF$ we introduce and study the algebra $(\gg,A)(\FF)$, which is the Lie subalgebra of $\FF \otimes A$ generated by $\FF \otimes \gg$. In many examples $A$ is the universal enveloping algebra of $\gg$. Our description of the algebra $(\gg,A)(\FF)$ has a striking resemblance to the commutator expansions of $\FF$ used by M. Kapranov in his approach to noncommutative geometry. To each algebra $(\gg, …
A noncommutative algebra $A$, called an algebraic noncommutative geometry, is defined, with a parameter $ε$ in the centre. When $ε$ is set to zero, the commutative algebra $A^0$ of algebraic functions on an algebraic manifold $M$ is obtained. This $A^0$ is a subalgebra of $C(M)$, which is dense if $M$ is compact. The generators of $A$ define an immersion of $M$ into $R^n$, and $M$ inherits a Poisson structure as the limit of the commutator. Thus $A$ is a quantisation of a Poisson manifold. If an ordering convention is prescribed for $A$ then a star product on $M$ is obtained. Homomorphism and …
A.A. Kirillov introduced the family algebras in 2000. In this paper we study the noncommutative Poisson bracket P on the classical family algebra. We show that P is the first-order deformation from the classical family algebra to the quantum family algebra. We will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary therefore the deformation is infinitesimally trivial. In the last part of this paper we also talk about Mackey's analogue and the quantization problem of the family algebras.
Motivated by the form of the noncommutative *-product in a system of open strings and Dp-branes with constant nonzero Neveu-Schwarz 2-form, we define a deformed multiplication operation on a quasitriangular Hopf algebra in terms of its R-matrix, and comment on some of its properties. We show that the noncommutative string theory *-product is a particular example of this multiplication, and comment on other possible Hopf algebraic properties which may underlie the theory.
We extend the Adler-Manin trace on the algebra of pseudodifferential symbols to a twisted setting.
We show that the Hall algebra of two-periodic complexes, which is recently introduced by T.Bridgeland, coincides with the Drinfeld double of the ordinary Hall bialgebra.
We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic geometry, etc.). These algebras have finite presentations and are completely characterized and classified by their (noncommutative) volume forms.
This is the text of the lecture given by the author in Naples at "Giornata IndAM", June 7, 2005. The lecture is addressed at the general mathematical audience and reviews several topics in deformation theory of associative algebras.
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.
Let $\A$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes. Secondly, we characterize the Hall algebra of complexes of fixed size by generators and relations, and relate it to the derived Hall algebra of $\A$. Finally, we give the integration map on the Hall algebra of $2$-term complexes over projectives.
We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative two-torus with real multiplication. These results are based on the techniques of math.QA/0211262 allowing to interpret all the data in terms of autoequivalences of the deriv…
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.
We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of transverse index theory for foliations.
Noncommutative lattices have been recently used as finite topological approximations in quantum physical models. As a first step in the construction of bundles and characteristic classes over such noncommutative spaces, we shall study their $K$-theory. We shall do it algebraically, by studying the algebraic $K$-theory of the associated algebras of `continuous functions' which turn out to be noncommutative approximately finite dimensional (AF) $C^*$-algebra . We also work out several examples.
We establish an embedding of the quantum enveloping algebra of a symmetric generalized Kac--Moody algebra into a localized Hall algebra of $\mathbb Z_2$-graded complexes of representations of a quiver with (possible) loops. To overcome difficulties resulting from the existence of infinite dimensional projective objects, we consider the category of finitely-presented representations and the category of $\mathbb Z_2$-graded complexes of projectives with finite homology.
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.
This is a brief introduction to the world of Noncommutative Algebra aimed at advanced undergraduate and beginning graduate students.
Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or {\it noncommutative lattices}, since they can be realized as structure spaces of noncommutative $C^*$-algebras. These noncommutative …
A well-known conjecture says that every one-relator group is coherent. We state and partly prove an analogous statement for graded associative algebras. In particular, we show that every Gorenstein algebra $A$ of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line $\PP^1$ as a noncommutative scheme based on the coherent noncommutative spectrum $\cohp A$ of such an algebra $A$, that is, the category of coherent $A$-modules modulo the torsion ones. This category is always abelian Ext-finite hereditary with Serre duality, like the categ…
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.
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…
Recently V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry. In this note we generalize this argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg.
In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O-operators on noncommut…
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 derive the Kac-Paljutkin finite-dimensional Hopf algebras as finite fibrations of the quantum double torus and generalize the construction for quantum multiple tori.
For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a com…
We introduce the new notion of epsilon-graded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras. We define and study the associated notion of epsilon-derivation-based differential calculus, which generalizes the derivation-based differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of epsilon-graded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also t…
We generalize the notion, introduced by Henri Cartan, of an operation of a Lie algebra $\mathfrak g$ in a graded differential algebra $Ω$. We define the notion of an operation of a Hopf algebra $\mathcal H$ in a graded differential algebra $Ω$ which is refered to as a $\mathcal H$-operation. We then generalize for such an operation the notion of algebraic connection. Finally we discuss the corresponding noncommutative version of the Weil algebra: The Weil algebra $W(\mathcal H)$ of the Hopf algebra $\mathcal H$ is the universal initial object of the category of $\mathcal H$-operations with con…
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