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$\DeclareMathOperator\GK{GK}$Let $k$ be the base field. The Gelfand-Kirillov dimension was introduced by Gelfand and Kirillov in their seminal paper on the Gelfand-Kirillov conjecture. A very famous p …

Let $k$ be any base field and $A$ an affine infinite dimensional $k$-algebra. Let $\mathcal{F}= \{ A_i \}_{i \geq 0}$ be a finite dimensional filtration for $A$: that is, $k \subset A_0$ and each $A_i …

Let $H_{c}$, for simplicity, be the rational Cherednik algebra with parameter $t=1$, with triangular strucuture $\mathbb{C}[h] \otimes \mathbb{C} W \otimes \mathbb{C}[h^*]$, and $(W,h)$ the defining …

Let $A$ be any finitely generated algebra - non necessarely unital neither associative - over a base field $k$. Let us denote the product $*$. Suppose $A$ is finitely generated by $S$, and introduce $ …

Let $\mathbb{k}$ be an algebraically closed field of positive characteristic, $X$ an affine smooth variety over it. Then the ring of crystalline differential operators on $X$ is generated by $\mathcal …

Let $\mathsf{k}$ be a field with zero characteristic, and $q \in \mathsf{k}$ a non-zero elemento which is not a root of unit. The quantum plane $\mathsf{k}_q[x,y]$ is the algebra given by generators $ …

Noether's Problem was introduced by Emmy Noether in [4]: Let $\mathsf{k}$ be a field and $K=\mathsf{k}(x_1,\ldots,x_n)$ be a purely transcendental extension. Let $G<S_n$ be a group acting by permutati …

For the sake of simplicity, my base field will be the complex numbers. My question is simple: what are (preferably natural) examples of infinite dimensional simple Lie algebras? I came up with this qu …

I am not not an specialist in PI-algebras, but I can say I have a rather good understanding on the subject. It is, of course, interesting to discover if an algebra $A$ is a PI-algebra. But it is also …

For the sake of simplicity, I will work over the complex numbers. Let $A$ be a finitely generated algebra and $G$ any finite group of algebra automorphisms. Then, by Noether's Theorem, $A^G$ is also a …

I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra). It is clear from our definition of $\mathcal{C}$ that it is a …

As it was shown by Malcev, unlike the commutative case, in which every domain can be embedded in a field, there are noncommutative domains that can't be embedded in a division ring. For noncommutative …

In his very influential book Infinite dimensional Lie algebras, which is still the main reference for Kac-Moody algebras, in section 0.4 of the introduction, Victor Kac discuss the problem of concrete …

For associative algebras, as your required, see Plotkin, Algebras with the same (algebraic) geometry, Israel J. Math., 96 (2) (1996), 511–522. This is, being more precise, part of this nice relatively …

As mentioned in the answer by user6976, there is the idea of development of algebraic geometry to (essentialy) any general algebraic system. This is carried out(following Plotkin's work) by E. Daniyar …