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A modern approach to derived functors, that has been shown to be useful in a number of different circunstances is that of a derived category (see the book by Yakutieli, for example, here). However, it …

The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively). Nowdays there are many variations of the t …

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 …

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 …

$\DeclareMathOperator\GK{GK}$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 …

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 …

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 …

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 …

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 …

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 …

We will work over $\mathbb{C}$. Remember that a finite subgroup $G$ of $\operatorname{GL}_n(\mathbb{C})$ is called a complex reflection group if it is generated by complex reflections $r$, which are l …

Complementing Manuel Norman's excelent answer, recently I've found a very nice survey about the Gelfand-Kirillov dimension, from 2015, by Jason Bell, called Growth Functions. This survey discusses man …

Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by algebr …

Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-o …

In his seminal paper, Some open problems on three-dimensional graded domains, M. Artin proposed a very small list of possible division rings of fractions that can appear as 'noncommutative function fi …