Abstract: If $G$ is a reductive algebraic group acting rationally on a smooth affine variety $X$ then it is generally believed that $D(X)^G$ has properties very similar to those of enveloping algebras of semisimple Lie algebras. In this paper we show that this is indeed the case when $G$ is a torus and $X=k^r\times (k^*)^s$. We give a precise description of the primitive ideals in $D(X)^G$ and we study in detail the ring theoretical and homological properties of the minimal primitive quotients of $D(X)^G$. The latter are of the form $D(X)^G/(\g-\chi(\g))$ where $\g=\Lie(G)$, $\chi\in \g^\ast$ and $\g-\chi(\g)$ is the set of all $v-\chi(v)$ with $v\in \g$. They occur as rings of twisted differential operators on toric varieties. As a side result we prove that if $G$ is a torus acting rationally on a smooth affine variety then $D(X\quot G)$ is a simple ring.
Abstract: In this paper we will think of certain abelian categories with favorable properties as non-commutative surfaces. We show that under certain conditions a point on a non-commutative surface can be blown up. This yields a new non-commutative surface which is in a certain sense birational to the original one. This construction is analogous to blowing up a Poisson surface in a point of the zero-divisor of the Poisson bracket. By blowing up $\le 8$ points in the elliptic quantum plane one obtains global non-commutative deformations of Del-Pezzo surfaces. For example blowing up six points yields a non-commutative cubic surface. Under a number of extra hypotheses we obtain a formula for the number of non-trivial simple objects on such non-commutative surfaces.
Abstract: Let G be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let W, U two finite dimensional representations of G. In this paper we give a survey on the computation of the local cohomology of (U tensor SW)^G.
Abstract: To a pair consisting of an elliptic curve and a point on it, Odeskii and Feigin associate certain quadratic algebras (``Sklyanin algebras''), having the Hilbert series of a polynomial algebra. In this paper we show that Sklyanin algebras have good homological properties and we obtain some information about their so-called linear modules. We also show how the construction by Odeskii and Feigin may be generalized so as to yield other ``Sklyanin-type'' algebras.
Abstract: For a 3-dimensional Artin-Schelter-regular algebra A with Hilbert series 1/(1-t)^3 we study central extensions; that is, graded algebras D with a regular central element z in degree 1, such that D/(z)=A. We classify such D and we also classify certain D-modules (point modules and line modules) which proved to be important in the study of 3-dimensional Artin-Schelter-regular algebras.
Abstract: In this paper we prove a translation principle for the central quotients of four-dimensional Sklyanin algebras, which is analogous to the translation principle for semi-simple Lie algebras. In the course of the proof we construct an ``elliptic'' analog of the sheaf of differential operators on P^1. The translation principle may be used to construct the fat points of a four-dimensional, non-PI, Sklyanin algebra.
Abstract: It has been conjectured that the ring of differential operators of the algebraic quotient of a connected smooth affine variety under a reductive group action is simple. This is known in the case that the group in question is the extension of a finite group with a torus and in the case of classical representation of classical groups. In these notes we present some tools relevant to this conjecture. In particular we show that it is true for some representations of Sl2.
Abstract: Let G be a connected non-special semisimple algebraic group and let W be a finite dimensional G-representation such that W has trivial generic stabilizer. Let g=Lie(G). Then the semi-direct product g+W is a counter example to the Gel'fand-Kirillov conjecture.
Abstract: The simplest non-trivial division algebras that can be constructed over a rational function field in two variables are those that ramify along a smooth cubic curve. In this note we show that these division algebras are cyclic if they have odd index.
Abstract: Let $A$ be a three dimensional Artin-Schelter regular algebra. We give a description of the category of finitely generated $A$-modules of Gelfand-Kirillov dimension one (modulo those of finite dimension over the ground field). The proof is based upon a result by Gabriel which says that locally finite categories can be described by module categories over topological rings.
Abstract: In this note we prove existence theorems for dualizing complexes over graded and filtered rings, thereby generalizing some results by Zhang, Yekutieli and J{\o}rgensen.
Abstract: Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a polynomial ring over a perfect field of characteristic p>0 and if the inclusion R-> S splits, then D_k(R) is a simple ring. In the last section of the paper, we discuss how one might try to deduce the characteristic zero case from this result. As yet, however, this is a subtle problem and the answer to the question of Levasseur and Stafford remains open in characteristic zero.
Abstract: Let ``$HH$'' stand for Hochschild (co)homology. In this note we show that for many rings $A$ there exists $d\in\NN$ such that for an arbitrary $A$-bimodule $N$ we have $HH^i(N)=HH_{d-i}(N) $. Such a result may be viewed as an analog of Poincare duality. Combining this equality with a computation of Soergel allows one to compute the Hochschildt homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo. (Also see erratum below.)
Abstract: Assume that S is a semigroup generated by {x_1,...,x_n}, and let U be the multiplicative free commutative semigroup generated by {u_1,...,u_n}. We say that S is of I-type if there is a bijection v:U->S such that for all a in U, {v(u_1a),....,v(u_na)}={x_1v(a),....,x_nv(a)}. This condition appeared naturally in the work on Sklyanin algebras by John Tate and the second author. In this paper we show that the condition for a semigroup to be of I-type is related to various other mathematical notions found in the literature. In particular we show that semigroups of I-type appear in the study of the settheoretic solutions of the Yang-Baxter equation, in the theory of Bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author.
Abstract: We classify regular algebras of global dimension four that map surjectively onto the two-Veronese of a regular algebra of global dimension three on two generators. We also study the point modules.
Abstract: A conjecture of Kac states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we give a combinatorial reformulation of Kac's conjecture in terms of a property of $q$-multinomial coefficients. As a side result we give a formula for certain inverse Kostka-Foulkes polynomials.
Abstract: We show that it is possible to define reflection isomorphisms on the double of the (twisted) Hall algebra of a quiver. Combining these reflections with Fourier transform yields an alternative construction of Lusztig's braid group action on a quantum enveloping algebra.
Abstract: Let G be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let W, U two finite dimensional representations of G. In this paper we compute the local cohomology of (U tensor SW)^G provided a certain relatively weak technical condition is true.
Abstract: We show that critical modules of Gelfand-Kirillov dimension 2 and multiplicity $d$ over an elliptic algebra have (up to modules of lower GK-dimension and shifting) a presentation by $d\times d$-matrices of linear forms. In the language of non-commutative algebraic geometry this amounts to a generic description of ``curves'' of degree $d$ in a projective quantum plane.
Abstract: We show that the Gelfand-Kirillov holds for Lie algebras of dimension at most eigth. Recall that in dimension nine the authors have constructed a counter example.
Abstract: Assume that $X$ is a surface over an algebraically closed field $k$. Let $\tilde{X}$ be obtained from $X$ by blowing up a smooth point and let $L$ be the exceptional curve. Let $\coh(X)$ be the category of coherent sheaves on $X$. In this note we show how to recover $\coh({X})$ from $\coh(\tilde{X})$, if we know the object $\Oscr_L(L)$.
Abstract: The representations of dimension vector $\alpha$ of the quiver $Q$ can be parametrised by a vector space $R(Q,\alpha)$ on which an algebraic group $\Gl(\alpha)$ acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi--invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate to a suitable map between projective representations a semi--invariant polynomial function that describes when this map is inverted on the representation and we show that these semi--invariant polynomial functions form a spanning set of all semi--invariant polynomial functions in characteristic $0$. If the quiver has no oriented cycles, we may replace consideration of inverting maps between projective representations by consideration of representations that are left perpendicular to some representation of dimension vector $\alpha$. These left perpendicular representations are just the cokernels of the maps between projective representations that we consider.
Abstract: In this paper we show that the Hall algebra of a quiver, as defined by Ringel, is the positive part of the quantived enveloping algebra of a generalized Kac-Moody Lie algebra. We give a potential application of this result to a conjecture of Kac which states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra.
Abstract: Let $\Cscr$ be a connected noetherian hereditary abelian category with Serre functor over an algebraically closed field $k$, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from \cite{RV2}, we prove that if $\Cscr$ has some object of infinite length, then the Grothendieck group of $\Cscr$ is finitely generated if and only if $\Cscr$ has a tilting object.
Abstract: In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.
Abstract: In this paper we classify $\Ext$-finite noetherian hereditary abelian categories over an algebraically closed field $k$ satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories. As a side result we show that when our hereditary abelian categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.
Abstract: We give alternative proofs of some theorems.
Abstract: The paper "A relation between Hochschild homology and cohomology for Gorenstein rings" contains an error in the sense that Theorem 1 (the "duality theorem") is false in the generality stated. As a result the same is true for its corollaries: Proposition 3 and Corollary 6. The main conclusion, which is an affirmative answer to a question by Patrick Polo, remains valid however
Abstract: We introduce the notion of a ``non-commutative crepant'' resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.
Abstract: We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.
Abstract: A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors.
Abstract: {For $Y,Y^+$ three-dimensional smooth varieties related by a flop, Bondal and Orlov conjectured that the derived categories $D^b(\coh(Y))$ and $D^b(\coh(Y^+))$ are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this paper is to give a partially new proof of Bridgeland's result using non-commutative rings. The new proof also covers some mild singular and higher dimensional situations (including the one in the recent paper by Chen: ``Flops and Equivalences of derived Categories for Threefolds with only Gorenstein Singularities''). }
Abstract: In this paper we classify graded reflexive ideals, up to isomorphism and shift, in certain three dimensional Artin-Schelter regular algebras. This classification is similar to the classification of right ideals in the first Weyl algebra, a problem that was completely settled recently. The situation we consider is substantially more complicated however.
Abstract: We give a proof avoiding spectral sequences of Deligne's decomposition theorem for objects in a triangulated category admitting a Lefschetz homomorphism.
Abstract: We determine the possible Hilbert functions of graded rank one torsion free modules over three dimensional Artin-Schelter regular algebras. It turns out that, as in the commutative case, they are related to Castelnuovo functions. From this we obtain an intrinsic proof that the space of torsion free rank one modules on a non-commutative P^2 is connected. A different proof of this fact, based on deformation theoretic methods and the known commutative case has recently been given by Nevins and Stafford. For the Weyl algebra it was proved by Wilson
Abstract: In this paper we give a direct proof of the properties of the $\ZZ D_\infty$ category which was introduced in the classification of noetherian, hereditary categories with Serre duality by Idun Reiten and the author.
Abstract: This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer-Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf.
Abstract: In this paper we develop the basic infinitesimal deformation theory of \emph{abelian categories}. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems.
Abstract: In this paper we discuss some of the recent developments on derived equivalences in algebraic geometry.
Abstract: The Hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated Hilbert series. In general, the precise inclusion relation between the closures of the strata is still unknown. Guerimand studied this problem for strata whose Hilbert series are as close as possible. Preimposing a certain technical condition he obtained necessary and sufficient conditions for the incidence of such strata. In this paper we present a new approach, based on deformation theory, to Guerimand's result. This allows us to show that the technical condition is not necessary.
Abstract: quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology.
Abstract: In this paper we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by Crawley-Boevey and Shaw carry a natural Poisson structure. This follows the fact that appropriately localized path algebras of double quivers carry a certain kind of non-commutative quasi-Hamiltonian structure.
Abstract: Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated k-algebra has rational singularities. In particular if a finitely generated normal commutative k-algebra has a noncommutative crepant resolution, as introduced by the second author, then it has rational singularities.
Abstract: In this paper we introduce non-commutative analogues for the quasi-Hamiltonian $G$-spaces introduced by Alekseev, Malkin and Meinrenken. We outline the connection with the non-commutative analogues of quasi-Poisson algebras which the author had introduced earlier.
Abstract: In this paper we provide a proof of a result announced by Kontsevich that the HKR-morphism, twisted by the square root of the Todd genus, gives an isomorphism between the Hochschild cohomology of a smooth algebraic variety and the cohomology of poly-vector fields, both considered as Gerstenhaber algebras. Our proof is set in the framework of Lie algebroids and so applies in more general settings as well.
Abstract: This is a companion note to ``Hochschild cohomology and Atiyah classes'' by Damien Calaque and the author. We compute the Kontsevich weight of a wheel with spokes pointing outward. The result is in terms of modified Bernouilli numbers. Our computation uses Stokes theorem together with some basic properties proved by Kontsevich.
Abstract: These are some notes I wrote many years ago. I occasionally get requests for them so I decided to tidy them up and put them here. The main motivation is that there are two possible definitions for the notion of a G-equivariant D-module. The first one is the standard category theoretic definition in terms of the schemes GxX and GxGxX. The second, more convenient one, is in terms of the Lie algebra of G. In these notes we show that these two definitions are equivalent for a connected group. This is certainly well-known but I haven't been able to locate an elementary proof.
Abstract: These are notes on de Jong's proof of the period$=$index theorem over fields of transcendence degree two. They are actually about the ``simplified'' proof sketched by de Jong in the last section of his paper. These notes were meant as support for my lectures at the summer school ``Central Simple Algebras over Function Fields of Surfaces'' at the Universit\"at Konstanz between August, 26 and September, 1 2007 but I did not finish them in time.