# Questions tagged [filtered-algebras]

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### Filtrations and Koszul algebras

I was looking at this question and asked my self the following: Let $A$ be graded algebra, which is also an $\mathbb{N}_0$-filtered algebra. If its associated graded algebra $\mathrm{gr}(A)$ is ...
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### Is there a coalgebraic definition of filtered algebras?

If $M$ is a monoid, then an $M$-graded algebra over $k$ is the same thing as a $k[M]$ comodule algebra. To see this, if $\delta$ is a coaction of $k[M]$ on an algebra $A$, for each $m \in M$ define ...
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### 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: ...
344 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 ...
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### 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 ...
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1 vote
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### 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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### 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 ...
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1 vote
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### 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). ...
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### 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 ...
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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 $A_{-1} =... • 8,399 8 votes 0 answers 301 views ### Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I have in ... • 2,517 5 votes 1 answer 590 views ### Associated graded of filtered module-algebra over a Hopf algebra I ran across the following statement in a paper, and it seems fishy to me: Lemma: If$A$is any Hopf algebra, and if$U$is an$\mathbb{N}_0$-filtered$A$-module algebra, then$U$and$\mathrm{gr} (...
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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 \$...