相关论文: Simple algebras of Gelfand-Kirillov dimension two

Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive…

We consider algebras over a field K defined by a presentation K <x_1,..., x_n : R >, where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of…

Let $k$ be an uncountable field. We show that a finitely generated prime Goldie $k$-algebra of quadratic growth is either primitive or satisfies a polynomial identity, answering a question of Small in the affirmative.

Let $k$ be an uncountable algebraically closed field and let $A$ be a countably generated left Noetherian $k$-algebra. Then we show that $A \otimes_k K$ is left Noetherian for any field extension $K$ of $k$. We conclude that all subfields…

We observe that a finitely generated algebraic algebra R (over a field) is finite dimensional if and only if the associated graded ring grR is right noetherian, if and only if grR has right Krull dimension, if and only if grR satisfies a…

We study growth and complexity of \'etale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely generated algebras of arbitrary Gelfand-Kirillov dimension $\ge 2$ and simple finitely…

It is given an example of finitely generated simple algebra over a field k (char k = 0) with arbitrary odd Gel'fand-Kirillov dimension.

We describe simple finitely generated associative conformal algebras of Gel'fand--Kirillov dimension one.

We show that if $k$ is a countable field, then there exists a finitely generated, infinite-dimensional, primitive algebraic $k$-algebra $A$ whose Gelfand-Kirillov dimension is at most six. In addition to this we construct a two-generated…

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension.…

It is shown that over an arbitrary countable field, there exists a finitely generated algebra that is nil, infinite dimensional, and has Gelfand-Kirillov dimension at most three.

Stafford proved that every left or right ideal of the Weyl algebra A_n(K) is generated by two elements. In this paper we prove that every left or right ideal of the ring of differential operators over the field of formal Laurent series…

Let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension less than 3, and let $Q(A)$ denote the quotient division algebra of $A$. We show that if $D$ is a division subalgebra of $Q(A)$ of GK dimension at least 2 then…

Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or…

Smoktunowicz, Lenagan, and the second-named author recently gave an example of a nil algebra of Gelfand-Kirillov dimension at most three. Their construction requires a countable base field, however. We show that for any field $k$ and any…

We show that Golod-Shafarevich algebras can be homomorphically mapped onto infinite-dimensional algebras with polynomial growth, under mild assumptions of the number of relations of given degrees. In case these algebras are finitely…

We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial…

We construct finitely generated nil algebras with prescribed growth rate. In particular, any increasing submultiplicative function is realized as the growth function of a nil algebra up to a polynomial error term and an arbitrarily slow…

Let $\mathfrak{R}$ be a weakly noetherian variety of unitary associative algebras (over a field $K$ of characteristic 0), i.e., every finitely generated algebra from $\mathfrak{R}$ satisfies the ascending chain condition for two-sided…

We classify all noetherian Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero which are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\Ext^1_H(k,k)\neq 0$. The latter condition is…