The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦₂ algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦₂ algebra would be another 𝒦₂ algebra. We show that this is not necessarily the case by constructing a monomial 𝒦₂ algebra for which the corresponding Yoneda algebra is not 𝒦₂.     

Quasi-d-Koszul Modules and Applications

Let R be a Noetherian semiperfect algebra. A necessary and sufficient condition for a finitely generated R-module to be quasi-d-Koszul is given, which improves one of the main results in [1]. The authors also give a necessary and sufficient condition for the Minimal Horseshoe Lemma holding in mod（R）. As an application, it is proved that the ＂Minimal Horseshoe Lemma＂ is true in the category of quasi-d-Koszul modules under certain conditions.     

Notes on δ-Koszul Algebras

This paper is a continuous work of δ-Koszul algebras, which were first introduced by Green and Marcos in 2005 (see Green and Marcos, Commun Algebra 33(6):1753–1764, 2005). Let K^d(A)\mathcal{K}^{\delta}(A) be the category of δ-Koszul modules. It is proved that K^d(A)\mathcal{K}^{\delta}(A) preserves kernels of epimorphisms if and only if the “minimal Horseshoe Lemma” (“MHL” for short) holds. Further, a special class of δ-Koszul algebras named periodic δ -algebras are introduced, which have close connection with Koszul algebras and provide answers to the questions raised by Green and Marcos (Commun Algebra 33(6):1753–1764, 2005). Finally, we construct new periodic δ-algebras from the given ones in terms of one-point extension and sum-extension.     

New criteria for <Emphasis Type="Italic">λ</Emphasis>-Koszul algebras

The main purpose of this paper is to provide some new criteria for a standard graded algebra A = ⊕ _i≥0 A _i to be a λ-Koszul algebra, which was first introduced in [12] and was another class of “Koszul-type” algebras including Koszul and d-Koszul algebras as special examples.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Some Examples of Affine Monoid Algebras

We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K ₀[S] is a monomial algebra.

If K is a field of characteristic p > 0, then we construct a finitely presented group H _p such that the Jacobson radical J of the group algebra K[H _p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H _p ]/J is a domain.     

*-Regular Leavitt path algebras of arbitrary graphs

If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L _K (E). We show that the involution on L _K (E) is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L _K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L _K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.     

An Equivalence of Categories for Lie Color Algebras

Abstract

Let k be a field with char k = p > 0 and G an abelian group with a bicharacter λ on G. For each p-(G,λ)-Lie color algebra L over k the p-universal enveloping algebra u(L) is a G-graded Hopf algebra,i.e.,a Hopf algebra in the category ^kG ? of kG-comodules. In this paper we describe a subcategory of ^kG ? which is equivalent to the category of the finite dimensional p-(G,λ)-Lie color algebras over k.     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.     

Finite vs affine W-algebras

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra Zhu_ΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu_H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu_H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established. “I am an old man, and I know that a definition cannot be so complicated.” I.M. Gelfand (after a talk on vertex algebras in his Rutgers seminar)     

∗-Prime Group Algebras

An associative algebra R over a field K is said to be right ?-prime if for every nonzero r ? R, there exists a finitely generated subalgebra S of R such that rSt = 0 implies t = 0. Clearly, strongly prime implies ?-prime and ?-prime implies prime. A large number of examples of group algebras are given which show that the concept of ?-prime lies strictly between prime and strongly prime. A complete characterization of ?-prime group algebras is given. It is proved that a group algebra KG of the group G over the field K is ?-prime if and only if Λ⁺(G) = (1). Intersection theorems play an important role in the study. In the process, a new intersection theorem for ?-prime group algebras is obtained. Elementwise characterization of the ?-prime radical is given and its relation with some well-known radicals is discussed.     

Quiver Algebras of Coverings and Twisted Tensor Products

Given a covering Γ of a quiver Δ, we show that the quiver algebra K[Γ] of Γ over a field K is a twisted tensor product of the quiver algebra of the fibre of the covering viewed as a trivial quiver and the quiver algebra K[Δ]. To make sense of this, we first extend the theory of twisted tensor products of algebras to include algebras without units.     

Nilpotent n-Lie Algebras

We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I ² nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.     

Cleft extensions of Koszul twisted Calabi–Yau algebras

Let H be a twisted Calabi–Yau (CY) Hopf algebra and σ a 2-cocycle on H. Let A be an N-Koszul twisted CY algebra such that A is a graded H^σ- module algebra. We show that the cleft extension A#_σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U(D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi–Yau algebras.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hyper MV ‐ideals in hyper MV ‐algebras

In this paper we define the hyper operations ?, ∨ and ∧ on a hyper MV ‐algebra and we obtain some related results. After that by considering the notions ofhyper MV ‐ideals and weak hyper MV ‐ideals, we prove some theorems. Then we determine relationships between (weak) hyper MV ‐ideals in a hyper MV ‐algebra (M, ⊕, *, 0) and (weak) hyper K ‐ideals in a hyper K ‐algebra (M, °, 0). Finally we give a characterization of hyper MV ‐algebras of order 3 or 4 based on the (weak) hyper MV ‐ideals (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)