Koszul Duality for Semidirect Products and Generalized Takiff Algebras

We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones.     

From Lie algebra crossed modules to tensor hierarchies

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.     

Extensions of Schrödinger–Virasoro conformal modules

In this paper, we study extensions between two finite irreducible conformal modules over the Schrödinger–Virasoro conformal algebra and the extended Schrödinger–Virasoro conformal algebra. Also, we classify all finite nontrivial irreducible conformal modules over the extended Schrödinger–Virasoro conformal algebra. As a byproduct, we obtain a classification of extensions of Heisenberg–Virasoro conformal modules.     

Generic Flatness and the Jacobson Conjecture

A result of Artin, Small, and Zhang is used to show that a Noetherian algebra over a commutative, Noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, Noetherian associated graded ring. This result is extended to show that if an algebra over a commutative Noetherian ring has a locally finite, Noetherian associated graded ring, then the intersection of the powers of the Jacobson radical is nilpotent. The proofs rely on a weak generalization of generic flatness and some observations about G-rings.     

Self-tests for freeness over commutative artinian rings

We prove that the Auslander-Reiten conjecture holds for commutative standard graded artinian algebras, in two situations: the first is under the assumption that the modules considered are graded and generated in a single degree. The second is under the assumption that the algebra is generic Gorenstein of socle degree 3.     

The lower and upper bounds of the degree distribution of a graded algebra

For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.     

Koszul differential graded modules

The concept of Koszulity for differential graded (DG, for short) modules is introduced. It is shown that any bounded below DG module with bounded Ext-group to the trivial module over a Koszul DG algebra has a Koszul DG submodule (up to a shift and truncation), moreover such a DG module can be approximated by Koszul DG modules (Theorem 3.6). Let A be a Koszul DG algebra, and Dc(A) be the full triangulated subcategory of the derived category of DG A-modules generated by the object AA. If the trivial DG module...     

Algebras Graded by Discrete Doi–Hopf Data and the Drinfeld Double of a Hopf Group-Coalgebra

We study Doi–Hopf data and Doi–Hopf modules for Hopf group-coalgebras. We introduce modules graded by a discrete Doi–Hopf datum; to a Doi–Hopf datum over a Hopf group coalgebra, we associate an algebra graded by the underlying discrete Doi–Hopf datum, using a smash product type construction. The category of Doi–Hopf modules is then isomorphic to the category of graded modules over this algebra. This is applied to the category of Yetter–Drinfeld modules over a Hopf group coalgebra, leading to the construction of the Drinfeld double. It is shown that this Drinfeld double is a quasitriangular ${\mathbb{G}}$ -graded Hopf algebra.     

Whittaker Modules for Graded Lie Algebras

In this paper, we study Whittaker modules for graded Lie algebras over ℂ. We define Whittaker modules for a class of graded Lie algebras and obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra. As a consequence of this, we obtain a classification of simple Whittaker modules for such algebras. Also, we discuss some concrete algebras as examples.     

Derived categories of graded gentle one-cycle algebras

Let A be a graded algebra. It is shown that the derived category of dg modules over A (viewed as a dg algebra with trivial differential) is a triangulated hull of a certain orbit category of the derived category of graded A-modules. This is applied to study derived categories of graded gentle one-cycle algebras.     

Representations of the virasoro-like algebra and its q-Analog

In this paper, some irreducible graded modules with 1-dimensional homogeneous spaces over the Virasoro-like algebra and its q-analogs are constructed. The unitarizability of these modules, and the conditions under which two of such irreducible graded modules are ismorphic are determined. Some other kinds of irreducible graded modules with 1-dimensional homogeneous spaces over the Virasorolike algebra and its q-analogs are also given.     

一类特殊的Koszul Calabi-Yau DG代数

假设一个连通上链DG代数A的基分次代数A~#或者同调分次代数H(A)是由一次元素x,y生成的代数kx,y/(xy+yx).本文证明A是Koszul Calabi-Yau DG代数.     

Schemes of Line Modules I

It is proved that there exists a scheme that represents thefunctor of line modules over a graded algebra, and it is calledthe line scheme of the algebra. Its properties and its relationshipto the point scheme are studied. If the line scheme of a quadratic,Auslander-regular algebra of global dimension 4 has dimension1, then it determines the defining relations of the algebra. Moreover, the following counter-intuitive result is proved.If the zero locus of the defining relations of a quadratic (notnecessarily regular) algebra on four generators with six definingrelations is finite, then it determines the defining relationsof the algebra. Although this result is non-commutative in nature,its proof uses only commutative theory. The structure of the line scheme and the point scheme of a 4-dimensionalregular algebra is also used to determine basic incidence relationsbetween line modules and point modules.     

One-Point Extensions of t-Koszul Algebras

We introduce the category of t-fold modules which is a full subcategory of graded modules over a graded algebra. We show that this subcategory and hence the subcategory of t-Koszul modules are both closed under extensions and cokernels of monomorphisms. We study the one-point extension algebras, and a necessary and sufficient condition for such an algebra to be t-Koszul is given. We also consider the conditions such that the category of t-Koszul modules and the category of quadratic modules coincide.     

New results on the lower central series quotients of a free associative algebra

We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated in Feigin and Shoikhet (2007) [FS]. We establish a linear bound on the degree of tensor field modules appearing in the Jordan–Hölder series of each graded component. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.     

Class and rank of differential modules

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class – a substitute for the length of a free complex – and on the rank of a differential module in terms of invariants of its homology. These results specialize to basic theorems in commutative algebra and algebraic topology. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over commutative noetherian rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings.     

Graded cellular bases for Temperley–Lieb algebras of type A and B

We show that the Temperley–Lieb algebra of type A and the blob algebra (also known as the Temperley–Lieb algebra of type B) at roots of unity are \(\mathbb{Z}\) -graded algebras. We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.     

Weakly Koszul-like modules

The Koszul-like property for any finitely generated graded modules over a Koszul-like algebra is investigated and the notion of weakly Koszul-like module is introduced. We show that a finitely generated graded module M is a weakly Koszul-like module if and only if it can be approximated by Koszul-like graded submodules, which is equivalent to the fact that G(M) is a Koszul-like module, where G(M) denotes the associated graded module of M. As applications, the relationships between minimal graded projective resolutions of M and G(M), and Koszul-like submodules are established. Moreover, the Koszul dual of a weakly Koszul-like module is proved to be generated in degree 0 as a graded E(A)-module.     

A class of non‐graded left‐symmetric algebraic structures on the Witt algebra

We classify the compatible left‐symmetric algebraic structures on the Witt algebra satisfying certain non‐graded conditions. It is unexpected that they are Novikov algebras. Furthermore, as applications, we study the induced non‐graded modules of the Witt algebra and the induced Lie algebras by Novikov‐Poisson algebras’ approach and Balinskii‐Novikov's construction.