Simple Algebras of Gelfand - Kirillov Dimension Two
نویسنده
چکیده
Let k be a field. We show that a finitely generated simple Goldie k-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed.
منابع مشابه
The Gelfand-kirillov Dimension of Quadratic Algebras Satisfying the Cyclic Condition
We consider algebras over a field K presented by generators x1, . . . , xn and subject to (n 2 ) square-free relations of the form xixj = xkxl with every monomial xixj , i = j, appearing in one of the relations. It is shown that for n > 1 the Gelfand-Kirillov dimension of such an algebra is at least two if the algebra satisfies the so-called cyclic condition. It is known that this dimension is ...
متن کاملLie Algebras with Finite Gelfand-kirillov Dimension
We prove that every finitely generated Lie algebra containing a nilpotent ideal of class c and finite codimension n has Gelfand-Kirillov dimension at most cn. In particular, finitely generated virtually nilpotent Lie algebras have polynomial growth.
متن کاملLeavitt Path Algebras of Finite Gelfand–kirillov Dimension
Groebner–Shirshov Basis and Gelfand–Kirillov dimension of the Leavitt path algebra are derived.
متن کاملGelfand-Kirillov Dimension of Commutative Subalgebras of Simple Infinite Dimensional Algebras and their Quotient Division Algebras
Throughout this paper, K is a field, a module M over an algebra A means a left module denoted AM , ⊗ = ⊗K . In contrast to the finite dimensional case, there is no general theory of central simple infinite dimensional algebras. In some sense, structure of simple finite dimensional algebras is ‘determined’ by their maximal commutative subalgebras (subfields)[see [18] for example]. Whether this s...
متن کاملRe-filtering and Exactness of the Gelfand–kirillov Dimension
– We prove that any multi-filtered algebra with semi-commutative associated graded algebra can be endowed with a locally finite filtration keeping up the semi-commutativity of the associated graded algebra. As consequences, we obtain that Gelfand–Kirillov dimension is exact for finitely generated modules and that the algebra is finitely partitive. Our methods apply to algebras of current intere...
متن کامل1 7 M ar 2 00 4 Nichols algebras of U q ( g ) - modules ⋆
A technique is developed to reduce the investigation of Nichols algebras of integrable Uq(g)-modules to the investigation of Nichols algebras of braided vector spaces with diagonal braiding. The results are applied to obtain information on the Gelfan’dKirillov dimension of these Nichols algebras and their defining relations if the braiding is of a special type and q is not a root of unity. For ...
متن کامل