Questions tagged [filtered-algebras]

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2 votes
0 answers
67 views

Is the associated algebra of a quadratic filtered algebra again quadratic

Let $A$ be a filtered quadratic algebra. Let $G(A)$ be the associated graded algebra. Will $G(A)$ again be a quadratic algebra or can higher relations appear when passing to the graded seeting? EDIT: ...
3 votes
1 answer
284 views

What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$)

I am reading a paper where they refer to a certain algebra as a PBW algebra. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered ...
3 votes
1 answer
116 views

Properties of a filtered algebra that can be concluded from properties of its associated graded algebra

Let $F$ be a filtered algebra and let $G$ be its associated graded algebra. Some examples of properties of $F$ that can be concluded from properties of $G$: (A) The dimension of $F$ is equal to the ...
1 vote
0 answers
146 views

generators for a graded algebra

I have a graded commutative algebra over a field $K$ and I also know that it is not generated by the degree 1 elements. Somehow I could guess a set of generators, is there a criterion to check that ...
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2 votes
0 answers
32 views

Can one bound the minimal degree of a solution to an inhomogeneous linear equation?

Suppose $A = \cup_{i\in \mathbb N} A_{\leq i}$ is a filtered algebra and $M = \cup_{i\in \mathbb N} M_{\leq i}$ a filtered $A$-module. In my case, I know $A$ to be Noetherian and $M$ to be finitely ...
1 vote
1 answer
143 views

hamilton type Lie algebras

If n be positive integer and for an n-tuple of positive integers m=(m1,...,mn) then p(n,m) is graded and filtered subalgebra of W(n,m).p(n,m) is called non-alternating hamilton lie algebra over GF(2). ...
7 votes
1 answer
486 views

A geometric characterization of Rees algebras in categories without Choice

Before asking my question, a caveat: The category theorist in me would like me to ask this question in more generality, but I will restrict my scope since what I'm really after is some geometric ...
8 votes
1 answer
1k views

Generators of associated graded algebra

Suppose that $A = \bigcup_{n=0}^{\infty} A_n$ is a filtered algebra over a field $k$. The associated graded algebra is $\mathrm{gr} A = \bigoplus_{n=0}^{\infty} A_n/A_{n-1}$, where we define $A_{-1} =...
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8 votes
0 answers
293 views

Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them

I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I have in ...
5 votes
1 answer
576 views

Associated graded of filtered module-algebra over a Hopf algebra

I ran across the following statement in a paper, and it seems fishy to me: Lemma: If $A$ is any Hopf algebra, and if $U$ is an $\mathbb{N}_0$-filtered $A$-module algebra, then $U$ and $\mathrm{gr} (...
  • 8,179
4 votes
0 answers
192 views

Kahler differentials and the m-adic filtration

Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $...