The Maximal Graded Left Quotient Algebra of a Graded Algebra

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.

Recognizing the Semiprimitivity of N-graded Algebras via Grobner Bases

Let K〈X〉 = K（X1,..., Xn） be the free K-algebra on X = {X1,..., Xn} over a field K, which is equipped with a weight N-gradation （i.e., each Xi is assigned a positive degree）, and let G be a finite homogeneous GrSbner basis for the ideal I = （G） of K（X） with respect to some monomial ordering 〈 on K（X）. It is shown that if the monomial algebra K（X）/（LM（6）） is semiprime, where LM（6） is the set of leading monomials of 6 with respect to 〈, then the N-graded algebra A ： K（X）/I is semiprimitive in the sense of Jacobson. In the case that G is a finite nonhomogeneous Gr6bner basis with respect to a graded monomial ordering 〈gr, and the N-filtration FA of the algebra A = K（X）/I induced by the N-grading filtration FK（X） of K（X） is considered, if the monomial algebra K（X）/（LM（6）） is semiprime, then it is shown that the associated N-graded algebra G（A） and the Rees algebra A of A determined by FA are all semiprimitive.

Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

In this paper, we first prove for two differential graded algebras （DGAs） A, B which are derived equivalent to k-algebras A, F, respectively, that ：D（Ak B） ≈D（Ak Г）. In particular, Hp^b（Ak B） ≈ Hb（proj-A k Г）. Secondly, for two quasi-compact and sepa- rated schemes X, Y and two algebras A, B over k which satisfy ：D（Qcoh（X）） ≈：D（A） and ：D（Qcoh（Y）） ≈D（B）, we show that ：D（Qcoh（X × Y）） ≈ 79（AB） and ：Db（Coh（X × Y）） ≈Db（mod-（A B））. Finally, we prove that if X is a quasi-compact and separated scheme over k, then ：D（Qcoh（X ~ pl）） admits a recollement relative to D（Qcoh（X））, and we de- scribe the functors in the recollement explicitly. This recollement induces a recollement of bounded derived categories of coherent sheaves and a recollement of singularity categories. When the scheme X is derived equivalent to a DGA or algebra, then the recollement which we get corresponds to the recollement of DGAs in [14] or the recollement of upper triangular algebras in [7].

A theorem of Fong's type for graded algebras

Let G be a finite p-solvable group and k be an algebraic closed field of characteristic p. It is proved that any projective indecomposable module of a G-graded k-algebra is an induced module of a module of the subalgebra graded by a Hall p′-subgroup. A necessary and sufficient condition for the indecomposability of an induced module from a Hall p′-subgroup is obtained.

Character Sheaves for Classical Graded Lie Algebras

In this note we study character sheaves for graded Lie algebras arising from inner automorphisms of special linear groups and Vinberg’s type Ⅱ classical graded Lie algebras.

A COMPLETE CLASSIFICATION OF FINITE-DIMENSIONAL SIMPLE NOVIKOV ALGEBRAS OF CHARACTERISTIC p>2

In the paper we will give a complete classification of finite-dimemsional simple Novikov algebras over an algebraically closed field with prime characteristic p>2.

CONSTRUCTING POINTED WEAK HOPF ALGEBRAS BY ORE-EXTENSION

The main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra from a Clifford monoid S =[Y; Gα. φα,β]by Ore-extensions, and to obtain a co-Frobenius semilattice graded weak Hopf algebra H（S, n, c, x, a, b） through factoring At by a semilattice graded weak Hopf ideal.

Super-Shuffle Product and Cut-Box Coproduct on (0,1)-Matrices

In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple graphs. In this paper, we generalize the super-shuffle product and the cut-box coproduct from labeled simple graphs to (0,1)-matrices. Then we prove that the vector space spanned by (0,1)-matrices with the super-shuffle product is a graded algebra and with the cut-box coproduct is a graded coalgebra.

一类Gerstenhaber代数的构造

For any K-algebra A,based on Hochschild complex and Hochschild coho-mology of A,we construct a new Gerstenhaber algebra,and give Gerstenhaber algebra epimorphism from the new Gerstenhaber algebra to the Gerstenhaber algebra of the Hochschild cohomology of A.

On the Structure of Graded λ-Hopf Algebras

Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras （coalgebras）, we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded λ-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded A-Hopf algebra is invertible.

Structures of Not-finitely Graded Lie Algebras Related to Generalized Heisenberg–Virasoro Algebras

In this paper, we study the structure theory of a class of not-finitely graded Lie alge- bras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.

Graded Hochschild Cohomology of a Path Algebra with Oriented Cycles

The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.

Modules of the Intermediate Series over Not-finitely Graded Lie Algebras W(Г)

Let W(Г)be a class of not-finitely graded Lie algebras related to generalized Virasoro algebras with basis{L_(α,i),C|α∈Г,i∈Z_(+)},which satisfies relations[L_(α,i),L_(β,j)]=L_(α+β,i+j)+(j-i)L_(α+β,i+j+1)+δ_(α+β,0)δ_(i+j),0α^(3)-α/12 C and[C,L_(α,i)]=0.In this paper,W(Г)-modules of the intermediate series satisfying some conditions are constructed and classified.We also obtain modules of the intermediate series over the related Lie superalgebra.

On the Structure of Graded Leibniz Algebras

We study the structure of graded Leibniz algebras with arbitrary dimension and over an arbitrary base field K. We show that any of such algebras ￡ with a symmetric G-support is of the form ￡ = U-∑jIj with U a subspace of ￡1, the homogeneous component associated to the unit element 1 in G, and any Ij a well described graded ideal of ￡, satisfying [Ij, Ik]= 0 if j≠ k. In the case of ￡ being of maximal length, we characterize the gr-simplicity of the algebra in terms of connections in the support of the grading.

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.

Differential Graded Structures on Path Categories

In this paper, we consider the graded path category associated to a quiver. We investigate all n-differentials on such a category, and also study the associated graded Lie algebra. Moreover, a necessary and sufficient condition is given for the graded path categorv to admit a DG category structure.

On Crossed Product Algebras over Henselian Valued Fields

Let D be a tame central division algebra over a Henselian valued field E,D be the residue division algebra of D,E be the residue field of E,and n be a positive integer.We prove that M_(n)(D)has a strictly maximal subfield which is Galois(resp.,abelian)over E if and only if M_(n)(D)has a strictly maximal subfield K which is Galois(resp.,abelian)and tame over E withГ_(K)■Г_(D),whereГ_(K)andГ_(D)are the value groups of K and D,respectively.This partially generalizes the result proved by Hanke et al.in 2016 for the case n=1.

On ε-Derivations and Local ε-Derivations

In this paper, we describe ∈-derivations in certain graded algebras by their actions on elements satisfying some special conditions. One of the main results is applied to local ∈-derivations on some certain graded algebras.

Connected Graphs Arising from Products of Veronese Varieties

Given a Segre squarefree Veronese configuration, following Bernd Sturmfels, we improve the study of the graphs associated to the configuration. We determine two special families of toric ideals and a finite set of moves for each of them, which still guarantee simultaneously the connection of all graphs arising from each family of moves.

The Jacobian Dual of Certain Mixed Product Ideals

We consider the symmetric algebra of a class of monomial ideals generated by s-sequences.For these ideals with linear syzygies,we determine their Jacobian dual modules and study their duality properties.