Finiteness conditions on the Yoneda algebra of a monomial algebra

Let A$A$ be a connected graded noncommutative monomial algebra. We associate to A$A$ a finite graph Γ(A)$\Gamma \left(A\right)$ called the CPS graph of A$A$. Finiteness properties of the Yoneda algebra Ext_A(k,k)${\text{Ext}}_A(k,k)$ including Noetherianity, finite GK dimension, and finite generation are characterized in terms of Γ(A)$\Gamma \left(A\right)$. We show that these properties, notably finite generation, can be checked by means of a terminating algorithm.     

Quasi-finiteness and Finiteness of Ring-Extensions

Abstract

A finite ring-homomorphism between rings is quasi-finite. So we investigate what additional conditions are required for a quasi-finite ring-extension to be finite.     

On diameter of the commuting graph of a full matrix algebra over a finite field

It is shown that the commuting graph of a matrix algebra over a finite field has diameter at most five if the size of the matrices is not a prime nor a square of a prime. It is further shown that the commuting graph of even-sized matrices over finite field has diameter exactly four. This partially proves a conjecture stated by Akbari, Mohammadian, Radjavi, and Raja [Linear Algebra Appl. 418 (2006) 161–176].     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

Left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra

The compatible left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra with some natural grading conditions are completely determined. The results of the earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures. As a corollary, any such left-symmetric algebra contains an infinite-dimensional nontrivial subalgebra that is also a submodule of the regular module.     

Extension of a quantized enveloping algebra by a Hopf algebra

Suppose that H is a Hopf algebra,and g is a generalized Kac-Moody algebra with Cartan matrix A =(aij)I×I,where I is an index set and is equal to either {1,2,...,n} or the natural number set N.Let f,g be two mappings from I to G(H),the set of group-like elements of H,such that the multiplication of elements in the set {f(i),g(i)|i ∈I} is commutative.Then we define a Hopf algebra Hgf Uq(g),where Uq(g) is the quantized enveloping algebra of g.     

The planar algebra associated to a Kac algebra

We obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the ‘rotation on 2-boxes’.     

Krull dimension of the enveloping algebra of a semisimple Lie algebra

Let be a complex semisimple Lie algebra and be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of is equal to the dimension of a Borel subalgebra of .

     

Norm conditions for uniform algebra isomorphisms

In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / {0} and T: A → B is a surjective map, not assumed to be linear, satisfying

Symbolic powers of monomial ideals and vertex cover algebras

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.     

On the Factorial Closure of Certain Monomial Modules

Abstract

Let R = K[y ₁,…,y _t ] be an affine domain over a field K and I be a nonzero proper ideal of R. In Sec. 1 of this note, we characterize when (K + I, R) is a Mori pair. In Sec. 2 of this note, we prove the following theorem: Let A ? B be domains such that C/Q is Mori for each subring C of B containing A and for any prime ideal Q of C. Then dim A ? 1 ≤ dim B ≤ dim A + 1 and if dim A > 1 or dim B > 1 then dim A = dim B.     

FI代数同构于一族全序FI代数的直积的子代数的条件

解决了王国俊教授提出的关于FI代数的嵌入问题,并进一步研究了FI代数与一族全序FI代数的直积的子代数同构的条件.主要结果是:交换FI代数可同构嵌入一族全序交换FI代数的直积;分配FI代数可同构嵌入一族全序分配FI代数的直积.     

A quiver presentation for Solomon's descent algebra

The descent algebra Σ(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of S, the set of simple reflections in W. From this construction we obtain some general information about the quiver of Σ(W) and an algorithm for the construction of a quiver presentation for the descent algebra Σ(W) of any given finite Coxeter group W.     

Function representation of a noncommutative uniform algebra

We construct a Gelfand type representation of a real noncommutative Banach algebra satisfying , for all

     

Ergodic sequences in the Fourier-Stieltjes algebra and measure algebra of a locally compact group

Let be a locally compact group. Blum and Eisenberg proved that if is abelian, then a sequence of probability measures on is strongly ergodic if and only if the sequence converges weakly to the Haar measure on the Bohr compactification of In this paper, we shall prove an extension of Blum and Eisenberg's Theorem for ergodic sequences in the Fourier-Stieltjes algebra of We shall also give an improvement to Milnes and Paterson's more recent generalization of Blum and Eisenberg's result to general locally compact groups, and we answer a question of theirs on the existence of strongly (or weakly) ergodic sequences of measures on