Graded Algebra Research Articles

Mackey's obstruction map for discrete graded algebras

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles.Analogous obstruction is suggested here for discrete group graded algebras which are not necessarily saturated, i.e. strongly graded in the discrete context. The discrete obstruction is a map assigning a certain second cohomology class to every equivariance class of absolutely graded-simple modules.The set of equivariance classes of such modules is equipped with an appropriate multiplication, namely a graded product, such that the obstruction map is a homomorphism of abelian monoids. Graded products, essentially arising as pull-backs of bundles, admit many nice properties, including a way to twist graded algebras and their graded modules.The obstruction class turns out to determine the fine part that appears in the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn theorem for graded-simple algebras which are graded Artinian, in case where the base algebras (i.e. the unit fiber algebras) are finite-dimensional over algebraically closed fields.

Non-commutative derived moduli prestacks

We introduce a formalism for derived moduli functors on differential graded associative algebras, which leads to non-commutative enhancements of derived moduli stacks and naturally gives rise to structures such as Hall algebras. Descent arguments are not available in the non-commutative context, so we establish new methods for constructing various kinds of atlases. The formalism permits the development of the theory of shifted bi-symplectic and shifted double Poisson structures in the companion paper [27].

Tensor Products and Crossed Differential Graded Lie Algebras in the Category of Crossed Complexes

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories.

What is the Seiberg–Witten map exactly?

We give a conceptual treatment of the Seiberg–Witten map as a quasi-isomorphism of differential graded algebras. The corresponding algebras have a very simple form, leading to explicit recurrence formulas for the quasi-isomorphism. Unlike most previous papers, our recurrence relations are nonperturbative in the parameter of non-commutativity. Using the language of quasi-isomorphisms, we give a homotopy classification of ambiguities in Seiberg–Witten maps. Possible generalizations to the Wess–Zumino complexes and some other algebras are briefly discussed.

Weighted homological regularities

Let A A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A A -modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones.

Yoneda algebras of the triplet vertex operator algebra

Given a vertex operator algebra V, one can construct two associative algebras, the Zhu algebra A(V) and the C2-algebra R(V). This gives rise to two abelian categories A(V)-Mod and R(V)-Mod, in addition to the category of admissible modules of V. In case V is rational and C2-cofinite, the category of admissible V-modules and the category of all A(V)-modules are equivalent. However, when V is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra W(p), as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra End(⊕L∈IrrPL)op, and the Yoneda algebra Ext⁎(⊕L∈IrrL,⊕L∈IrrL). As a consequence, the category of admissible log-modules for the triplet VOA W(p) has infinite global dimension, as do the Zhu algebra A(W(p)), and the associated graded algebra grA(W(p)) which is isomorphic to R(W(p)). We also describe the Koszul properties of the module categories of W(p), A(W(p)) and grA(W(p)).

Homological properties of 3-dimensional DG Sklyanin algebras

In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let [Formula: see text] be a [Formula: see text]-dimensional DG Sklyanin algebra with [Formula: see text], where [Formula: see text] and [Formula: see text] We systematically study its differential structures and various homological properties. Especially, we figure out the conditions for [Formula: see text] to be Calabi–Yau, Koszul, Gorenstein and homologically smooth, respectively.

Dirac Geometry I: Commutative Algebra

AbstractThe purpose of this paper and its sequel is to develop the geometry built from the commutative algebras that naturally appear as the homology of differential graded algebras and, more generally, as the homotopy of algebras in spectra. The commutative algebras in question are those in the symmetric monoidal category of graded abelian groups, and, being commutative, they form the affine building blocks of a geometry, as commutative rings form the affine building blocks of algebraic geometry. We name this geometry Dirac geometry, because the grading exhibits the hallmarks of spin in that it is a remnant of the internal structure encoded by anima, it distinguishes symmetric and anti-symmetric behavior, and the coherent cohomology of Dirac schemes and Dirac stacks, which we develop in the sequel, admits half-integer Serre twists. Thus, informally, Dirac geometry constitutes a “square root” of $$\mathbb {G}_m$$ G m -equivariant algebraic geometry.

Flag integrable models and generalized graded algebras

We introduce new classes of integrable models that exhibit a structure similar to that of flag vector spaces. We present their Hamiltonians, R-matrices and Bethe-ansatz solutions. These models have a new type of generalized graded algebra symmetry.

Number of generators of ideals in Jordan cells of the family of graded Artinian algebras of height two

We let A=R/I be a standard graded Artinian algebra quotient of R=k[x,y], the polynomial ring in two variables over a field k by an ideal I, and let n be its vector space dimension. The Jordan type Pℓ of a linear form ℓ∈A1 is the partition of n determining the Jordan block decomposition of the multiplication on A by ℓ – which is nilpotent. The first three authors previously determined which partitions of n=dimk⁡A may occur as the Jordan type for some linear form ℓ on a graded complete intersection Artinian quotient A=R/(f,g) of R, and they counted the number of such partitions for each complete intersection Hilbert function T[1].We here consider the family GT of graded Artinian quotients A=R/I of R=k[x,y], having arbitrary Hilbert function H(A)=T. The Jordan cell V(EP) corresponding to a partition P having diagonal lengths T is comprised of all ideals I in R whose initial ideal is the monomial ideal EP determined by P. These cells give a decomposition of the variety GT into affine spaces. We determine the generic number κ(P) of generators for the ideals in each cell V(EP), generalizing a result of [1]. In particular, we determine those partitions for which κ(P)=κ(T), the generic number of generators for an ideal defining an algebra A in GT. We also count the number of partitions P of diagonal lengths T having a given κ(P). A main tool is a combinatorial and geometric result allowing us to split T and any partition P of diagonal lengths T into simpler Ti and partitions Pi, such that V(EP) is the product of the cells V(EPi), and Ti is single-block: GTi is a Grassmannian.

Hochschild homology of twisted crossed products and twisted graded Hecke algebras

Hochschild homology of twisted crossed products and twisted graded Hecke algebras

The differential graded Verlinde formula and the Deligne Conjecture

AbstractA modular category gives rise to a differential graded modular functor, that is, a system of projective mapping class group representations on chain complexes. This differential graded modular functor assigns to the torus the Hochschild chain complex and, in the dual description, the Hochschild cochain complex of . On both complexes, the monoidal product of induces the structure of an ‐algebra, to which we refer as the differential graded Verlinde algebra. At the same time, the modified trace induces on the tensor ideal of projective objects in a Calabi–Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of with a second ‐structure. Our main result is that the action of a specific element in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second ‐structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of . In general, these two versions of the result are inequivalent. In the case of Hochschild chains, we obtain a block diagonalization of the Verlinde algebra through the action of the mapping class group element . In the semisimple case, both results reduce to the Verlinde formula. In the non‐semisimple case, we recover after restriction to zeroth (co)homology earlier proposals for non‐semisimple generalizations of the Verlinde formula.

Affine Noetherian algebras, filtrations and presentations

Resco and Small gave the first example of an affine Noetherian algebra which is not finitely presented. It is shown that their algebra has no finite-dimensional filtrations whose associated graded algebras are Noetherian, affirming their prediction. A modification of their example yields countable fields over which ‘almost all’ (that is, a co-countable continuum of) affine Noetherian algebras lack such a filtration, and an answer to a question suggested by Irving and Small is derived.

Cohomology Algebras of a Family of DG Skew Polynomial Algebras

Let A be a connected cochain DG algebra such that its underlying graded algebra A# is the graded skew polynomial algebra k⟨x1,x2,x3⟩/x1x2+x2x1x2x3+x3x2x3x1+x1x3,|x1|=|x2|=|x3|=1. Then the differential ∂A is determined by ∂A(x1)∂A(x2)∂A(x3)=Mx12x22x32 for some M∈M3(k). When the rank r(M) of M belongs to {1,2,3}, we compute H(A) case by case. The computational results in this paper give substantial support for the research of the various homological properties of such DG algebras. We find some examples, which indicate that the cohomology graded algebras of such kind of DG algebras may be not left (right) Gorenstein.

Some refinements of the Deligne–Illusie theorem

We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the de Rham complex in $\max(p-1, 2)$ consecutive degrees can be reconstructed as objects of the derived category in terms of its truncation in degrees at most one (or, equivalently, in terms the obstruction class to lifting modulo $p^2$). Consequently, these truncations are decomposable if $X$ admits a lifting to $W_2(k)$, in which case the first nonzero differential in the conjugate spectral sequence appears no earlier than on page $\max(p,3)$ (these corollaries have been recently strengthened by Drinfeld, Bhatt-Lurie, and Li-Mondal). Without assuming the existence of a lifting, we describe the gerbes of splittings of two-term truncations and the differentials on the second page of the conjugate spectral sequence, answering a question of Katz. The main technical result used in the case $p>2$ belongs purely to homological algebra. It concerns certain commutative differential graded algebras whose cohomology algebra is the exterior algebra, dubbed by us "abstract Koszul complexes", of which the de Rham complex in characteristic $p$ is an example. In the appendix, we use the aforementioned stronger decomposition result to prove that Kodaira-Akizuki-Nakano vanishing and Hodge-de Rham degeneration both hold for $F$-split $(p+1)$-folds.

Flat Families of Point Schemes for Connected Graded Algebras

We study truncated point schemes of connected graded algebras as families over the parameter space of varying relations for the algebras, proving that the families are flat over the open dense locus where the point schemes achieve the expected (i.e., minimal) dimension. When the truncated point scheme is zero-dimensional, we obtain its number of points counted with multiplicity via a Chow ring computation. This latter application in particular confirms a conjecture of Brazfield that a generic two-generator two-relation algebra has seventeen truncated point modules of length six.

Augmentations and immersed Lagrangian fillings

AbstractFor a Legendrian link with or , immersed exact Lagrangian fillings of can be lifted to conical Legendrian fillings of . When is embedded, using the version of functoriality for Legendrian contact homology (LCH) from Pan and Rutherford [J. Symplectic Geom. 19 (2021), no. 3, 635–722], for each augmentation of the LCH algebra of , there is an induced augmentation . With fixed, the set of homotopy classes of all such induced augmentations, , is a Legendrian isotopy invariant of . We establish methods to compute based on the correspondence between MCFs and augmentations. This includes developing a functoriality for the cellular differential graded algebra from Rutherford and Sullivan [Adv. Math. 374 (2020), 107348, 71 pp.] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary , we give examples of Legendrian torus knots with distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when and , every ‐graded augmentation of can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of ‐graded augmented Legendrian cobordism.

A vectorial Darboux transformation for the Fokas–Lenells system

We first reveal that there is a Riccati-type Miura transformation from the Fokas–Lenells (FL) system to the first member of the negative part of the AKNS hierarchy. We derive a vectorial Darboux transformation for the FL system by using the bidifferential graded algebra approach. Two types of reductions lead to the FL equation and nonlocal FL (nFL) equation, respectively. From vanishing seed solutions, N-soliton solutions for the FL equation and nFL equation are provided, with spectral matrices of the form of diagonal. Finally, as an application, we perform an asymptotic analysis for the bright–bright solitons of the nonlocal case.

A topological characterisation of the Kashiwara–Vergne groups

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in R 4 \mathbb {R}^4 , which can be finitely presented algebraically as a circuit algebra, or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups K V \mathsf {KV} and K R V \mathsf {KRV} —the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group G R T 1 \mathsf {GRT}_1 as automorphisms of arrow diagrams.

Universal quantum semigroupoids

We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a weak bialgebra that coacts on a (not necessarily connected) graded algebra A universally while preserving grading. We restrict our attention to algebraic structures with a commutative base so that the UQSGds under investigation are face algebras (due to Hayashi). The UQSGd construction generalizes the universal quantum linear semigroups introduced by Manin in 1988, which are bialgebras that coact on a connected graded algebra universally while preserving grading. Our main result is that when A is the path algebra k Q of a finite quiver Q , each of the various UQSGds introduced here is isomorphic to the face algebra attached to Q . The UQSGds of preprojective algebras and of other algebras attached to quivers are also investigated.