Gelfand-Kirillov dimension of algebras

Zariski cancellation problem for noncommutative algebras

“…(2) This follows from part (1) and [KL,Proposition 3.11]. (3) This follows from part (1) and [KL,Lemma 6.5].…”

“…We need the following lemma which is [KL,Proposition 3.11 and Lemma 6.5] when k is a field. See the definition of Gelfand-Kirillov dimension, denoted by GKdim, before Proposition 1.3.…”

“…[KL,Lemma 6.5] Let {F i A} i∈Z be a filtration of A in the sense of [KL,p.73]. Let M be a filtered right A-module with filtration {F i M } i∈Z in the sense of [KL,p.74].…”

An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension

“…Let V be a linear space over K; then dim K V denotes the dimension of V over K. The Gelfand-Kirillov dimension of an algebra R is denoted by GKdim(R). For elementary properties of Gelfand-Kirillov dimension we refer to [3].…”

Connected Hopf algebras and iterated Ore extensions

“…Since we are in characteristic 0, a special case of the Cartier-Gabriel-Kostant theorem, [Mon,Theorem 5.6.5], ensures that H is the enveloping algebra of the Lie algebra P (H) = i = 1 n kx i . Conversely, suppose that H is an IHOE which is isomorphic as Hopf algebra to the enveloping algebra of the Lie algebra g. We claim that H can be realised as an IHOE with a basis of g as the set of skew polynomial generators of H. Argue by induction on n := GKdimH = dim k g, the second equality holding by [KL,Example 6.9]. By hypothesis, H = T [x; σ, δ] for an IHOE T , with GKdimT = n−1, by Corollary (2.8).…”