The associated graded algebra of a finite dimensional algebra

Question

$\DeclareMathOperator\rad{rad}$Let $A$ be a finite dimensional algebra (we can assume that $A \cong KQ/I$, for a quiver $Q$ and an admissible ideal $I$ if that helps).

Denote by $A_G$ the associated graded algebra $A_G= \bigoplus\limits_{k=0}^{\infty}{\rad^k(A)/\rad^{k+1}(A)}$ where $\rad(A)$ denotes the Jacobson radical of $A$. Note that this is a finite direct sum as $\rad^n(A)=0$ for some $n \geq 1$ and set $\rad^0(A)=A$.

Question 1: Is there a good reference where the basic properties of such algebras $A_G$ are studied in a representation-theoretic context?

Question 2: Given $A$ is there an easy way to obtain $A_G$ using GAP or its package QPA?

Answer 1

Here is how one can compute the associated graded algebra of an admissible quotient of a path algebra in QPA with the newly added functionality:

gap> A;                                                               
<Rationals[<quiver with 3 vertices and 5 arrows>]/
<two-sided ideal in <Rationals[<quiver with 3 vertices and 5 arrows>]>, 
  (5 generators)>>
gap> B := AssociatedGradedAlgebra(A);
<Rationals[<quiver with 3 vertices and 5 arrows>]/
<two-sided ideal in <Rationals[<quiver with 3 vertices and 5 arrows>]>, 
  (4 generators)>>
gap> Dimension(B); Dimension(A);
17
17

Answer 2

It's hard to find references in this generality. But it is worth mentioning the following.

(1) Many properties are lost in forming the associated graded algebra of the radical filtration. For example, the associated graded algebra of a finite dimensional self injective (or even symmetric) algebra does not have to be self injective. Similarly, in the commutative situation, the associated graded algebra of a Gorenstein local ring does not have to be Gorenstein. But at least the simple modules are the same for a finite dimensional algebra and its associated graded for the radical filtration.

(2) On a more positive note, if $P$ is a finite $p$-group and $k$ is a field of characteristic $p$, then the associated graded object of a suitable central series of $P$ has the structure of a $p$-restricted Lie algebra. The restricted universal enveloping algebra of this is isomorphic to the associated graded of the group algebra $kP$ with respect to the radical filtration. So in this situation the property of being symmetric is not lost. This goes by the name of the Jennings-Quillen theorem. This can be used to find the structure of the associated graded algebra in the case where a finite group $G$ has a normal Sylow $p$-subgroup $P$, especially if the quotient $G/PC_G(P)$ is abelian.

(3) If you're interested in cohomology, the May spectral sequence is a computational tool for starting with the cohomology of the associated graded algebra and converges to the cohomology of the original algebra. This is an interesting construction in the case of a finite $p$-group discussed in (2).

(4) I've used the package QPA in Gap, and at least the version I'm using (1.34) is not set up to construct an associated graded. But it shouldn't be that hard to do within this package. On the other hand, Magma has a command AssociatedGradedAlgebra($A$) that takes a basic algebra $A$ as argument and returns its associated graded algebra. Magma will also compute cohomology, so you can directly compare the $E_2$ page and abutment of the May spectral sequence. You can use Magma to a limited extent online for free through an interface, if you want to try it.