3
votes
1answer
65 views

Intersection of Maximal Left Ideals with Finite Dimensional Quotient

Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I ...
3
votes
1answer
491 views

The Jacobson radical of an infinite dimensional algebra

Does any one know the Jacobson radical of the path algebra of the following quiver? $$\bullet \leftrightarrows \bullet$$ How many simplerepresentations of it are there? Is there any software that ...
3
votes
3answers
395 views

Gröbner/SAGBI bases for non-commutative setting

It is well known that SAGBI/Gröbner bases are important for commutative and non-commutative algebra. The references for commutative scenery is ample and vast, but I am in trouble to find a good ...
3
votes
2answers
413 views

An example where finitistic dimension does not equal right global dimension?

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...
6
votes
2answers
2k views

Jacobson radical = intersection of all maximal two-sided ideals

I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can ...
9
votes
2answers
1k views

Compute Lie algebra cohomology

Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation? What if the Lie algebra is finite dimensional? In my case I would like to be able to ...
1
vote
2answers
235 views

How canonical is the triangular decomposition of a rational Cherednik algebra?

Introduction: Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, let $G \leq \mathrm{GL}(V)$ be a finite subgroup and let $\kappa:V \times V \rightarrow \mathbb{C}G$ be an alternating bilinear ...
9
votes
2answers
1k views

when are algebras quiver algebras ?

Good Morning from Belgium, I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a ...
7
votes
3answers
875 views

construct scheme from quivers?

I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...
1
vote
1answer
215 views

Formal deformations of algebras over not necessarily commutative rings

In Iain Gordon's survery article "Symplectic reflection algebras" the concept of formal deformations of algebras over semisimple artinian (not necessarily commutative) rings is summarized (chapter 2). ...
7
votes
4answers
629 views

Morita equivalence and moduli problems

Two rings $A$ and $B$ are said to be Morita equivalent if the category of modules over $A$ and $B$ are equivalent as additive categories. (Here I'm considering left modules). Ex: $M_n(R)$ (the algebra ...
11
votes
1answer
645 views

Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complex-valued functions on its ...