We study admissible subcategories of the derived categories of smooth noncommutative (nc) curves as classified by Reiten–van den Bergh. We prove that any admissible subcategory of the derived category of a smooth nc curve is again the derived category of a smooth nc curve. We use this result to classify semiorthogonal decompositions in derived categories of nc curves. The results obtained imply that phantom categories do not exist in these cases. As a further application, we prove an extension of the Bondal–Orlov reconstruction theorem to the case of orbifold curves.
In this paper, we investigate noncommutative resolutions of AS-Gorenstein isolated singularities. Noncommutative resolutions in graded case are achieved as the graded endomorphism rings of some finitely generated graded modules, which are seldom N-graded algebras but bounded-below Z-graded algebras. So the paper works on locally finite bounded-below Z-graded algebras. We first define and study noncommutative projective schemes after Artin-Zhang and define noncommutative quasi-projective spaces as the base spaces of noncommutative projective schemes. The equivalences between noncommutative quasi-projective spaces are proved to be induced by so-called modulo-torsion-invertible bimodules, which are in fact a Morita-like theory at the quotient category level. Based on the equivalences, we propose a definition of noncommutative resolutions of AS-Gorenstein isolated singularities and prove that such noncommutative resolutions are generalized AS-regular algebras. The center of any noncommutative resolution is isomorphic to the center of the original AS-Gorenstein isolated singularity. In the final part, we prove that a noncommutative resolution of an AS-Gorenstein isolated singularity of dimension d is given by an MCM generator M if and only if M is a (d-1)-cluster tilting module. A noncommutative version of the Bondal-Orlov conjecture is also proved to be true in dimensions 2 and 3.
We introduce an algebraic extension of the Turaev cobracket as a family of maps parametrised by a positive integer. It is defined by constructing the generalised version of the divergence maps for an arbitrary associative algebra with the help of a connection in non-commutative geometry, and we recover the above extension by applying the construction to the case of the group algebra of the fundamental group of a compact connected oriented surface with boundary. If the connection is flat, the generalised divergence maps define classes in the Lie algebra cohomology of the space of derivations, and in the case of the free associative algebra, we show that they are canonically identified with the standard generators of the cohomology ring of the matrix Lie algebra g(n)(l).
In this paper, we study p-adic Hodge theory for non-commutative algebraic varieties. Firstly, we propose a conjecture that for K a complete discretely valued nonarchimedean extension K of \mathbb{Q}_{p} with perfect residue field k and \mathcal{T} an \mathcal{O}_{K}-linear idempotent-complete, small smooth proper stable \infty-category, there exists a B_{\mathrm{crys}}-coefficient isomorphism preserving additional structures between the K(1)-local K-theory on the generic fiber of \mathcal{T} and the topological periodic cyclic homology on the special fiber. This conjecture can be regarded as a non-commutative analog of the crystalline comparison theorem. We then proceed to prove the following results: the topological negative cyclic homology \pi_{i}\mathrm{TC}<^>{-}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_{p}) admits a Breuil-Kisin module structure, and the non-commutative analog of Bhatt-Morrow-Scholze's comparison theorems holds. Additionally, we demonstrate that the \mathbb{Z}_{p}[G_{K}]-module obtained from the topological negative cyclic homology is a \mathbb{Z}_{p}-lattice of a crystalline representation. Finally, we show that when the generic fiber of \mathcal{T} admits a geometric realization in the sense of Orlov, the non-commutative analog of the crystalline comparison theorem proposed by the author holds.
We calculate the derivations and the first Hochschild cohomology group of the quantum grassmannian over a field of characteristic zero in the generic case when the deformation parameter is not a root of unity. Using graded techniques and two special homogeneous normal elements of the quantum grassmannian, we reduce the problem to computing derivations of the quantum grassmannian that act trivially on these two normal elements. We then use the dehomogenisation equality which shows that a localisation of the quantum grassmannian is equal to a skew Laurent extension of quantum matrices. This equality is used to connect derivations of the quantum grassmannian with those of quantum matrices. More precisely, again using graded techniques, we show that derivations of the quantum grassmannian that act trivially on our two normal elements restrict to homogeneous derivations of quantum matrices. The derivations of quantum matrices are known in the square case, and technical details needed to deal with the general case are given in the appendix. This allows us to explicitly describe the first Hochschild cohomology group of the quantum grassmannian.
In this paper, we use the higher derived bracket to give the controlling algebra of pre-LieDer pairs. We give the cohomology of pre-LieDer pairs by using the twist L-infinity-algebra of this controlling algebra. In particular, we define the cohomology of regular pre-LieDer pairs. We study infinitesimal deformations of pre-LieDer pairs, which are characterized by the second cohomology group of pre-LieDer pairs. We also define the cohomology of regular pre-LieDer pairs with coefficients in an arbitrary representation and using the second cohomology group to classify abelian extensions of regular pre-LieDer pairs.
We discuss Poisson structures on a weighted polynomial algebra A := k[x, y, z] defined by a homogeneous element Omega is an element of A, called a potential. We start with classifying potentials Omega of degree deg(x) + deg(y) + deg(z) with any positive weight (deg(x), deg(y), deg(z)) and list all with isolated singularity. Based on the classification, we study the rigidity of A in terms of graded twistings and classify Poisson fraction fields of A=(Omega) for irreducible potentials. Using Poisson valuations, we characterize the Poisson automorphism group of A when Omega has an isolated singularity extending a nice result of Makar-Limanov-Turusbekova-Umirbaev. Finally, Poisson cohomology groups are computed for new classes of Poisson polynomial algebras.
We show that Villadsen algebras, which are not \mathcal{Z}-stable, are singly generated. More generally, we show that any simple unital AH algebra with diagonal maps is singly generated.
We present a classification framework for saturated Fell bundles over groups, utilizing data associated with their base group and unit fiber. This framework provides a unified perspective on the structure and properties of such bundles and yields key insights into their classification. In the case of discrete groups, we obtain a complete and transparent classification in terms of generalized factor systems, leading to a cohomological description of equivalence classes. For general locally compact groups, the situation is more delicate: While our construction extends to this setting, the classification depends on choices of topology on the underlying Banach bundle.
Given a connected semisimple Lie group G and an arithmetic subgroup Gamma , it is wellknown that each irreducible representation pi of G occurs in the discrete spectrum L-disc(2)(pi\G) of L-2 (Gamma\G) with at most a finite multiplicity m Gamma(pi). While m(Gamma)(pi) is unknown in general, we areinterested in its limit as Gamma is taken to be in a tower of lattices Gamma(1) superset of Gamma(2 )superset of . . . For a bounded measur-able subset X of the unitary dualOG, we let m Gamma(k )(X) be the integration of the multiplicity m Gamma(k )(pi) over all pi in X , which can be proved finite. Let H-X be the direct integral of the irreducible representations in X with respect to the Plancherel measure of G, which is also a module over the groupvon Neumann algebra L(Gamma(k)). Based on the work of Sauvageot and Finis-Lapid-M & uuml;ller, we prove lim k -> infinity m Gamma(k )(X)/dim L(Gamma(k)) H-X =1 for any bounded subset X of G when (i) {Gamma(k)} k >= 1 are cocompact or (ii) G = SL(n, R) and {Gamma(k)} are principal congruence subgroups.
We study a class of Gorenstein isolated singularities which are the quotients of generic and unimodular representations of the one-dimensional torus, or of the product of the one-dimensional torus with a finite abelian group. Based on the works of Spenko and van den Bergh [Invent. Math. 210 (2017), 3-67] and Mori and Ueyama [Adv. Math. 297 (2016), 54-92], we show that the graded singularity categories of these varieties admit tilting objects, and hence are triangulated equivalent to the perfect categories of some finite-dimensional algebras.
Let (X, G) be a d-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator Delta(G), and let Pi(X) be the C*-algebra obtained by locally transferring the C*-algebra generated by multiplication operators and Riesz transforms on Rd. Denote by sym(X)the principal symbol mapping of Pi(X). For any S is an element of Pi(X), we prove that, in the framework of C*-algebra, lim(t ->infinity), t(1/p)mu(1 + Delta(G))(-d/2p)) = (2 pi (d)root d)(-1/p) ||sym(X) (S) || L-p (T*X,e(-qG) d lambda), 100 where 0 < p <00, e-96 is the canonical weight on X, and da is the Liouville measure on the cotangent bundle T*X.
Punctual noncommutative Hilbert schemes are projective varieties parametrizing finite codimensional left ideals in noncommutative formal power series rings. We determine their motives and intersection cohomology, by constructing affine pavings and small resolutions of singularities.
Given a finite directed acyclic graph R , we construct from it two graphs E_{R} and F_{R} , one by adding a loop at every vertex of R and one by replacing every arrow of R by countably infinitely many arrows. We show that the graph C^{*} -algebra C^{*}(F_{R}) is isomorphic to the AF core of C^{*}(E_{R}) . Examples include C^{*} -algebras of quantum flag manifolds and quantum teardrops. We discuss in detail the quantum Grassmannian \operatorname{Gr}_{q}(2,4) and use our description as AF core to study its CW-structure.
We study the index homomorphism of even K-groups arising from a class in even KK-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the C^*-algebras in question such a class in KK-theory can always be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even K-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator D, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does however reflect the interplay between the K-theoretic input and the abstract Dirac operator. Since the spectral projections for D are not available in the general context of Hilbert C^*-modules we instead rely on certain continuous compactly supported functions applied to D to construct the spectral localizer. In the special case where even KK-theory coincides with even K-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.
To address the need for a unified framework that incorporates Lie algebroid connections on both vector and principal bundles, this paper investigates a generalized Atiyah algebroid structure and its short exact sequence. Building on this generalization, we describe Atiyah-type extensions and sequences that represent Atiyah classes through three explicit constructions designed to encode Lie algebroid connections compatible with specified sub-structures. As illustrative examples, we work out an enriched Atiyah algebroid construct, providing a systematic tool to characterize certain key properties of holomorphic connections and invariant connections.
This article investigates the two-parameter quantum matrix algebra at roots of unity. In the roots of unity setting, this algebra becomes a Polynomial Identity (PI) algebra and it is known that simple modules over such algebra are finite-dimensional with dimension at most the PI degree. We determine the center, compute the PI degree, and classify simple modules for two-parameter quantum matrix algebra, up to isomorphism, over an algebraically closed field of arbitrary characteristics.
Let L-v C Z(D) be a suitable cone semigroup and U-v its reduced semigroup C*-algebra. In this paper, we compute the L-v-invariant measures in the transversal hull of the semigroup L-v that exhibit regularity in the boundaries of L-v. These measures enable the construction of a trace per-unit hypersurface for observables in U-v supported near the boundaries of L-v, leading to the construction of appropriate Chern cocycles in the "boundary" ideals of U-v. Our approach applies to both finitely and non-finitely generated cone semigroups. Applications for the bulk-defect correspondence of lattice models of topological insulators are also provided.
