Journal of Pure and Applied Algebra最新文献

Using a $Z\left[\frac{1}{2}\right]$-form of Virasoro vertex operator algebra $L(\frac{1}{2},0)$ with central charge $\frac{1}{2}$, we obtain a modular vertex operator algebra over any field $F$ of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra.

We construct the finite-dimensional continuous complex representations of ${\mathrm{SL}}_2$ over compact discrete valuation rings of even residual characteristic, assuming the level is large enough compared to the ramification index, in the mixed characteristic case. We also prove that the complex group algebras of ${\mathrm{SL}}_2$ over finite quotient rings of such compact discrete valuation rings depend on the characteristic of the ring. In particular, we prove that the group algebras $C\left[{\mathrm{SL}}_2\right(Z/2^{r}Z\left)\right]$ and $C\left[{\mathrm{SL}}_2\right(F_2\left[t\right]/\left(t^r\right)\left)\right]$ are not isomorphic for any $r\ge 4$.

We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit p-basis. These extensions are called Galois extensions of exponent one. For such an extension $A\subset C$, we introduce an A-scheme, called the Yuan scheme, which parametrizes subextensions $A\subset B\subset C$ such that $B\subset C$ is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers.

Let Λ be an Artin algebra and $\mathrm{mod}\text{-}\left(\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda \right)$ the category of finitely presented functors over the stable category $\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda$ of finitely generated Gorenstein projective Λ-modules. This paper deals with those algebras Λ in which $\mathrm{mod}\text{-}\left(\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda \right)$ is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence (called representation equivalence in the terminology of Auslander), i.e. a full and dense functor that reflects isomorphisms, from the stable category of Gorenstein projective representations $\underset{\_}{\mathrm{Gprj}}(Q,\Lambda )$ of a finite acyclic quiver $Q$ to the category of representations $\mathrm{rep}(Q,\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda )$ over $\underset{\_}{\mathrm{Gprj}}\text{-}\Lambda$, provided Λ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra $\Lambda Q$ of the G-semisimple algebra Λ is $\mathrm{CM}$-finite if and only if $Q$ is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within $\mathrm{Gprj}(A_n,\Lambda )$ of the linear quiver $A_n$ over a G-semisimple algebra Λ. We also determine almost split sequences in $\mathrm{Gprj}(A_n,\Lambda )$ with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver $\mathrm{Gprj}(A_n,\Lambda )$.

We investigate the structure of power-closed ideals of the complex polynomial ring $R=C[x_1,\dots ,x_d]$ and the Laurent polynomial ring $R^{\pm }=C{[x_1,\dots ,x_d]}^{\pm }=S^{-1}C[x_1,\dots ,x_d]$, where S is the multiplicatively closed semigroup $S=[x_1,\dots ,x_d]$. Here, an ideal I is power-closed if $f(x_1,\dots ,x_d)\in I$ implies $f(x_1^i,\dots ,x_d^i)\in I$ for each natural number i. Important examples of such ideals are provided by the ideals of relations in Minkowski rings of convex polytopes. We investigate related closure and interior operators on the set of ideals of R and $R^{\pm }$ and we give a complete description of principal power-closed ideals and of radicals of general power-closed ideals of R and $R^{\pm }$.

Let M be a monoid, $C$ a category with pullbacks and X an object of $C$. We introduce the notion of a partial action α of M on X and study the globalization question for α. If α admits a reflection in the subcategory of global actions, then we reduce the problem to the verification that a certain diagram is a pullback in $C$. We then give a construction of such a reflection in terms of a colimit of a certain functor with values in $C$. We specify this construction to the case of categories admitting certain coproducts and coequalizers.

In this paper, we investigate the rigidity of the stable comodule category of a specific class of Hopf algebroids known as finite Adams, shedding light on its Picard group. Then, we establish a reduction process through base changes, enabling us to effectively compute the Picard group of the

It is known that unstable Steenrod module structure on the polynomial algebra $F_2[t_0,\dots ,t_{N-1}]\cong H^{⁎}({\left(R{P}^{\infty }\right)}^N;F_2)$ obtained by forgetting the multiplication is isomorphic to that arising from a twisted action of ${\mathrm{Sq}}^1$. We show that the same theorem holds for tensor algebras. As in the abelian case, the result is applied to produce a decomposition of the tensor algebra into “weight spaces”.

This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/policies/article-withdrawal).

This article has been retracted at the request of the Author.

There is an error in [1, Prop. 2.2] and, as a result, the classification of monogenic bialgebras as provided in Theorem 1 is incomplete. Therefore, the Author requested retraction and the Managing Editors agreed with this request. The Author regrets this error.

We introduce a graphical calculus for the representation theory of the quantized enveloping algebra of type $F_4$. We do this by giving a diagrammatic description of the category of invariant tensors on the 26-dimensional fundamental representation.