Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [graded-algebras]

The tag has no usage guidance.

1
vote
1answer
18 views

How can we define a $\mathbb{Z}$-grading on sl2

Let $sl_{2}$ be $2 \times 2$ traceless matrices over field $K$ of characteristic $0$. How can we define a $\mathbb{Z}$-grading on $sl_{2}$? Let consider $h ,e ,f$ as follows respectively: \begin{...
3
votes
0answers
48 views

Explicit formula for the equalizer of coalgebras

The article Limits of Coalgebras, Bialgebras and Hopf Algebras offers two descriptions for the equalizer of two unital coassociative coalgebras over a field. The latter description (Remark 1.2) is ...
1
vote
1answer
74 views

Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$

Algebra $A$ is called graded algebra if it has a direct sum decomposition $A=\bigoplus_{k\in\Bbb Z} A^k$ s.t. product satisfies $(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$ A ...
0
votes
0answers
54 views

Pushout of cdgas and the quotient by an acyclic ideal.

Let $A$ be a cdga over $\mathbb{R}$, $B\subset A$ a dg-subalgebra and $I\subset B$ a dg-ideal. Consider the pushout of cdga's $\require{AMScd}$ \begin{CD} B @>\pi>> B/I\\ @V i V V @VV ...
0
votes
1answer
33 views

Induced map between associated graded algebra?

I have just learned how one can take an filtered algebra and get an associated graded algebra, (here's the wikipedia article for reference). I have also seen in various places, (such as this question),...
0
votes
1answer
43 views

naturally graded algebras

I found a good explanation on graded algebras Understanding of graded algebra But I am confused about difference between "naturally graded" and "graded". May you please clarify the notion of ...
3
votes
0answers
36 views

Categorical product of non-unital associative differential graded coalgebras

Given two non-unital associative dg coalgebras $D$ and $C$, I want to give an explicit construction of the product $C\prod D$, this may follow from the dual construction (coproduct of non-unital ...
2
votes
1answer
31 views

Rational homotopy type of a Lie group

I have to show that Lie groups have the rational homotopy type of a wedge of spheres. Unfortunately, my knowledge of Sullivan models is a bit shaky. The proof I come up with shows that they have the ...
1
vote
1answer
43 views

Projective Resolution of exterior algebra as a module over divided polynomial algebra

Let $\Lambda_\mathbb{Z}[x]$ be an exterior algebra on one generator with $|x|=n$, let $\Gamma_\mathbb{Z}[x]$ be a divided polynomial algebra with $|x_k|= kn$, and suppose that $\Lambda_\mathbb{Z}[x]$ ...
1
vote
1answer
25 views

Graded $C^*$ algebra homomorphism

I am pretty sure I have some definition wrong. But I do not see where. Here is the context: Consider the $C^*$ algebra of continuously compactly supported functions $\Bbb R$ into $\Bbb C$. $$C_0(\...
2
votes
1answer
25 views

The twisted tensor product $BA\otimes_{\tau} A$ as the non-unital Hochschild complex

The twisting universal morphism $\tau: BA\rightarrow A$ induces a differential $\partial_{\tau}$ on $BA\otimes_{\tau}A$, we have: $$\partial_{\tau}(x\otimes y)=\partial x\otimes y+(-1)^{\lvert x\rvert}...
5
votes
1answer
63 views

Associated Graded Algebra

I'm trying to work through Exercise III.27 of Lang's Algebra: Let $A$ be a filtered algebra, $A=\bigcup_{j\geq 0}A_{j}$. For $j\geq 0$, define $R_{j}=A_{j}/A_{j-1}$, with $A_{-1}=\{0\}$. Let $R=\...
2
votes
0answers
103 views

quasi isomorphism of two dg algebras

I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is ...
0
votes
1answer
23 views

Irreducible highest weight representations as a graded algebra

Let $L$ be a semisimple Lie algebra and let $V(\lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $\lambda$. How can we view the sum \begin{align*} \oplus_{n\in\mathbb{N}...
0
votes
0answers
20 views

Trying to understand more about polynomials in noncommuting variables.

I'm working on a project and I will need to look at some theory regarding polynomials in noncommuting variables. I have a 1st ed. copy of Rotman's Advanced Modern Algebra, which I've been getting some ...
1
vote
0answers
38 views

Tensor product of graded algebras 3

Let $A$ and $B$ be $\mathbb{Z}_2$-graded algebras, i.e. $A=A_0 \oplus A_1$, $B= B_0 \oplus B_1$. I am trying to show that the graded tensor product $A \otimes B = (A \otimes B)_0 \oplus (A \otimes B)...
1
vote
0answers
35 views

When a homogeneous ideal is written as a product of two ideals, then each of two ideals is homogeneous?

I know that in a graded domain, if a homogeneous element is written as a product of two elements, then each of two elements is also homogeneous. That is, the set of all the homogeneous elements of ...
0
votes
0answers
41 views

Is every (left) graded-Noetherian graded ring (left) Noetherian?

I call a graded (non-commutative) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) Noetherian as a ring. In the ...
0
votes
0answers
23 views

Can you have supersymmetry inside $E_8$?

$E_8$ which is of size 248 has an SO(16) sub-algebra (120 generators, M) as well as 128 spinor generators (Q). The M's can be thought of as the even part of the algebra and the Q's as the odd part. ...
0
votes
1answer
38 views

In a $\mathbb Z$-graded ring we have $IR \cap R_0 = I$

I have a problem with an exercise from Tom Marley which is: Let $R$ be a $\mathbb Z$-graded ring and $I$ an ideal of $R_0$. Prove that $IR \cap R_0 = I$. For $I \subset IR \cap R_0 $, we can ...
0
votes
1answer
65 views

Units are homogeneous in $\mathbb Z$-graded domains

I am confused by an exercise from Tom Marley which is: Let $R$ be an arbitrary $\mathbb Z$-graded domain: $1)$ Prove that all units in $R$ are homogeneous. $2)$ By using $1$, if $R$ is a field, ...
0
votes
1answer
78 views

Direct sum and the inclusion property

Let $R$ be a ring and if $R= \bigoplus R_i$ as additive subgroups for each $i \in I$ where $I $ is a finite group Is that implies $R_i R_j $ must be contained in $R_p$ for any $p \in I$ ?? In ...
0
votes
2answers
76 views

On graded fields

Tom Marly here noted that if R is a graded field then $R$ is concentrated in degree 0, i.e., $R=R_0$ and $R_n =0$ for all $n \neq 0$. Is this proposition mentioned in any book or paper ?? ...
1
vote
0answers
97 views

Is the algebra dual to a graded coalgebra graded?

Given a graded coalgebra $C = \bigoplus_{n\geq 0} C_n $ with coproduct $$\Delta : C_n \to \bigoplus_{i=0}^n C_i\otimes C_{n-i} $$ must we have that the dual $C^* = \bigotimes_{n\geq 0}C_n^*$ is a ...
3
votes
1answer
75 views

What is the notation $\mathcal{A}_\infty$ for a sheaf of graded $\mathcal{O}_X$-algebras mean?

Today I encountered Hartshorne's condition $(\dagger)$ for quasi-coherent sheaves of algebras. This states that gives a graded $\mathcal{O}_X$-module $\mathcal{A}$ which has the structure of a graded ...
1
vote
0answers
54 views

The associated graded of a filtered coalgebra

Given a coalgebra $C$ with a filtration $F$ such that $\Delta(F^n C)\subset \sum_{i=0}^n F^i C\otimes F^{n-i} C$, how does the coproduct manifest in the associated graded? Do we get something to the ...
1
vote
0answers
30 views

Morphisms between the graded algebras of forms of vector bundles?

For a vector bundle $\pi_A:A\longrightarrow M$ and a non-negative integer $k$ let us define $$\Omega^k(A):=\left\{\begin{array}{clc} C^\infty(M) &\textrm{if}& k=0\\ \displaystyle\mathsf{Hom}_{...