Let$d$be a positive integer, $A={\mathbb{C}} [t_{1}^{\pm1},\ldots ,t_{d}^{\pm1}]$ be the Laurent polynomial algebra, and $W=\operatorname{Der} (A)$ be the derivation Lie algebra of$A$. Then we have the semidirect product Lie algebra$W$⋉$A$which we call the extended Witt algebra of rank$d$. In this paper, we classify all irreducible Harish-Chandra modules over$W$⋉$A$with nontrivial action of$A$.

Let R be a ring. R is called a quasi-Frobenius (QF) ring if R is right artinian and RR is an injective right R-module. In this article, we introduce (weak) fuzzy homomorphisms of modules to obtain a fuzzy characterization of QF rings. We also obtain some fuzzy characterizations of right artinian rings and right CF rings. These results throw new light on the research of QF rings and the related CF conjecture.

We prove the BMV (Bessis, Moussa, Villani, [1]) conjecture, which states that the function $${t \mapsto \mathop{\rm Tr}\exp(A-tB)}$$ , $${t \geqslant 0}$$ , is the Laplace transform of a positive measure on [0,∞) if$A$and$B$are $${n \times n}$$ Hermitian matrices and$B$is positive semidefinite. A semi-explicit representation for this measure is given.

Let $\Lambda$ be a finite dimensional algebra and $G$ be a finite group whose elements act on $\Lambda$ as algebra automorphisms. Under the assumption that $\Lambda$ has a complete set $E$ of primitive orthogonal idempotents, closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that the skew group algebra $\Lambda G$ and $\Lambda$ have the same finitistic dimension and the same strong global dimension if the action of $S$ on $E$ is free. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce that $\Lambda G$ is piecewise hereditary if and only if $S$ acts freely on $E$ and $\Lambda$ is piecewise hereditary as well.

We show that if X is a cocompact G-CW-complex such that each isotropy subgroup Gσ is L(2)-good over an arbitrary commutative ring k, then X satisfies some fixed-point formula which is an L(2)-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper G-CW-complex which relates the equivariant L(2)-Euler characteristic of a fixed point CW-complex Xs and the Euler characteristic of X/G. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji-Mislin in 2009. We also show that if X is a free G-CW-complex such that C∗(X) is chain homotopy equivalent to a chain complex of finitely generated projective Zπ1(X)-modules of finite length and X satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula, then χ(X/G)=χ(2)(X). As an application, we prove that the weak Bass conjecture holds for any finitely presented group G satisfying the following condition: for any finitely dominated CW-complex Y with π1(Y )=G, Y satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula.

We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables.

n this paper we study representations of skew group algebras $\Lambda G$, where $\Lambda$ is a connected, basic, finite-dimensional algebra (or a locally finite graded algebra) over an algebraically closed field $k$ with characteristic $p \geqslant 0$, and $G$ is an arbitrary finite group each element of which acts as an algebra automorphism on $\Lambda$. We characterize skew group algebras with finite global dimension or finite representation type, and classify the representation types of transporter categories for $p \neq 2,3$. When $\Lambda$ is a locally finite graded algebra and the action of $G$ on $\Lambda$ preserves grading, we show that $\Lambda G$ is a generalized Koszul algebra if and only if so is $\Lambda$.

For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.

Let k be a commutative Noetherian ring. In this paper we consider filtered modules of the category FI firstly introduced by Nagpal. We show that a finitely generated FI-module V is filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated FI-modules V whose projective dimension is finite, and describe an upper bound for it. Furthermore, we give a new proof for the fact that V induces a finite complex of filtered modules, and use it as well as a result of Church and Ellenberg to obtain another upper bound for homological degrees of V.

We present a version of a proof by Andy Chermak of the existence and uniqueness of centric linking systems associated with arbitrary saturated fusion systems. This proof differs from the one in [Ch2] in that it is based on the computation of derived functors of certain inverse limits. This leads to a much shorter proof, but one which is aimed mostly at researchers familiar with homological algebra.

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ with irreducible highest weight modules $V(\theta ,h)$ or with irreducible Virasoro modules $\mathrm{Ind}_{\theta}(N)$ defined in Mazorchuk and Zhao (Selecta Math. (N.S.) 20:839–854,2014). We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ defined in Mazorchuk and Weisner (Proc. Amer. Math. Soc. 142:3695–3703,2014). As a by-product we obtain the necessary and sufficient conditions for the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ to be irreducible. We also generalize the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})$ to $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are studied and an open problem in Guo et al. (J. Algebra 387:68–86,2013) is solved. Feigin–Fuchs’ Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.

The action of a Lie pseudogroup $\mathcal{G}$ on a smooth manifold$M$induces a prolonged pseudogroup action on the jet spaces$J$^{$n$}of submanifolds of$M$. We prove in this paper that both the local and global freeness of the action of $\mathcal{G}$ on$J$^{$n$}persist under prolongation in the jet order$n$. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.

The topological center of the spectrum of the Weyl algebra$W$, i.e. the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i\in\mathbb {N}\}$ , is characterized in a recent paper by$Jabbari$and$Namioka$(Ellis group and the topological center of the flow generated by the map $n\mapsto \lambda^{n^{k}}$ , to appear in$Milan J. Math.$). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra$W$_{$k$}, the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i=0,1,2,\ldots,k\}$ , will be characterized, for all$k$∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.

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Starting with a commutative ring $R$ and an ideal $I$ , it is possible to define a family of rings $R(I)_{a,b}$ , with $a,b \in R$ , as quotients of the Rees algebra $\oplus_{n \geq0} I^{n}t^{n}$ ; among the rings appearing in this family we find Nagata’s idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a$ , $b$ ; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a$ , $b$ . More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.

We give two homological proofs of the Noetherianity of the category FI, a fundamental result discovered by Church, Ellenberg, Farb, and Nagpal.

We prove that there exists uncountably many pairwise disjoint open subsets of the Gelfand space of the measure algebra on any locally compact non-discrete abelian group which shows that this space is not separable (in fact, we prove this assertion for the ideal $M_{0}(G)$ consisting of measures with Fourier-Stieltjes transforms vanishing at infinity which is a stronger statement). As a corollary, we obtain that the spectras of elements in the algebra of measures cannot be recovered from the image of one countable subset of the Gelfand space under Gelfand transform, common for all elements in the algebra.

Let $(R, \frak{m}, k_{R})$ be a regular local$k$-algebra satisfying the weak Jacobian criterion, and such that$k$_{$R$}/$k$is an algebraic field extension. Let $\mathcal{D}_{R}$ be the ring of$k$-linear differential operators of$R$. We give an explicit decomposition of the $\mathcal{D}_{R}$ -module $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}_{R}^{n+1}$ as a direct sum of simple modules, all isomorphic to $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}$ , where certain “Pochhammer” differential operators are used to describe generators of the simple components.

In this paper we consider several homological dimensions of crossed products $A _{\alpha} ^{\sigma} G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^{\sigma} _{\alpha} G$ are classified: global dimension of $A ^{\sigma} _{\alpha} G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^{\sigma} _{\alpha} G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that $A$ is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that $A$ and $A _{\alpha} ^{\sigma} G$ share the same homological dimensions under extra assumptions.

Let$p$be a prime integer, 1≤$s$≤$r$be integers and$F$be a field of characteristic different from$p$. We find upper and lower bounds for the essential$p$-dimension ed_{$p$}( $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ ) of the class $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ of central simple algebras of degree$p$^{$r$}and exponent dividing$p$^{$s$}. In particular, we show that ed($Alg$_{8,2})=ed_{2}($Alg$_{8,2})=8 and ed_{$p$}( $$ Al{{g}_{{{{p}^2},p}}} $$ )=$p$^{2}+$p$for$p$odd.

We study cluster categories arising from marked surfaces (with punctures and nonempty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of Ext1 as intersection numbers of tagged curves and Auslander-Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line $\mathbb X$ over finite fields. These polynomials are then used to define the generic Ringel--Hall algebra of $\mathbb X$ as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein.

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(\Lambda_0)$ of ${\mathscr D}(\Delta_n)$ of the cyclic quiver $\Delta_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $\Delta_n$, we obtain a decomposition of the basic representation $L(\Lambda_0)$ which induces the Kashiwara--Miwa--Stern decomposition of $\bigwedge^\infty$ and a construction of the canonical basis of $\bigwedge^\infty$ defined by Leclerc and Thibon in terms of certain monomial basis elements in ${\mathscr D}(\Delta_n)$.

Let A be a finite dimensional algebra and D^b(A) be the bounded derived category of finitely generated left A-modules. In this paper we consider lengths of compact exceptional objects in D^b(A), proving a sufficient condition such that these lengths are bounded by the number of isomorphism classes of simple A-modules. Moreover, we show that algebras satisfying this condition is bounded derived simple.