Questions tagged [algebras]

For questions about algebras, their properties, and their structures.

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Confused what I did wrong for $\int_{0}^{\infty} \frac{1}{1 + x^4} \, dx$

I did $ x = u\sqrt{i}$ $$\sqrt{i}\int_{0}^{\infty} \frac{1}{1 - u^4} \, du$$ $$\sqrt{i}\int_{0}^{\infty} \frac{1}{1 - u^2} \cdot \frac{1}{1 + u^2} \, du$$ $ v = \tan^{-1}(u)$,$dv = \frac{1}{1 + u^2} ...
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1 vote
1 answer
31 views

Closed Ideal $J$ of $C(X)$ there exists $f\in J$ such that $0\leq f(x)\leq 1$ for all $x\in X$ and $f(x)=1$ for all $x\notin U$

I am having a hard time understanding all the steps in the proof of the following proposition: Proposition: Suppose that $X$ is a compact Hausdorff space and consider the algebra $C(X)$. Let $J$ be a ...
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2 votes
2 answers
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Every finite-dimensional algebra which is not simple contains a maximal ideal whose annihilator is nonzero

The following problem is from Chapter 3 of Drozd and Kirichenko's "Finite-Dimensional Algberas" that I am self-studying. Let $A$ be a finite-dimensional unital algebra that is not simple. ...
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  • 458
0 votes
1 answer
53 views

Proof/disproof of a proposition

I would like to ask for a proof/disproof of a proposition. Proposition 1. Suppose two functions $f(x)=\sum_{i∈M}[a_i/(b_i+x)]$ and $g(x)=\sum_{j∈N}[c_j/(d_j+x)]$ where $a_i,b_i,c_j,d_j>0$, $\min_{i∈...
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  • 167
0 votes
0 answers
35 views

Why were hall cohomological algebras created or what is the motivation behind that algebra?

I can't find a paper that talks about motivation or how they are used in physics, it's that an article by calabi-yau mentions that algebra caught my attention and how I'm learning about topological ...
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1 vote
1 answer
32 views

Generating the algebra $\mathcal{O}[\mathbf{GL}(n,\mathbb{C})]$ (regular functions)

I recently asked a question about the regular functions in $\mathbf{GL}(n,\mathbb{C})$ but now that I have read the appendix I am again confused; here is my past question: Understanding regular ...
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  • 455
1 vote
1 answer
30 views

What are linear coordinate functions?

In the book Symmetry, Representations and Invariants they give the following definition: If $V$ is a finite dimensional $\mathbb{C}$-vector space, a function $f\colon V\to \mathbb{C}$ is a polynomial ...
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  • 455
-2 votes
1 answer
35 views

Algebras with surjection and equal length are isomorphic rings

Let $A, B$ be two $\mathcal{O}$-algebras with a surjective ring homomorphism $A \rightarrow B$, such that $\operatorname{length}_{\mathcal{O}}A = \operatorname{length}_{\mathcal{O}}B$. Can we conclude ...
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  • 467
1 vote
1 answer
51 views

Understanding regular functions of $\mathbf{GL}(n,\mathbb{C})$

I am following the book Symmetry, Representations and Invariants and I have a confusion with the definition of a regular function on $\mathbf{GL}(n,\mathbb{C})$: For the group $\mathbf{GL}(n,\mathbb{...
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  • 455
1 vote
2 answers
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Coalgebra after counit for a monad given by adjunction

Let $L\dashv R$ be an adjunction and $LR$ the associated comonad, with comultiplication $L\eta R\colon LR\to LRLR$ and counit $\varepsilon\colon\mathrm{id}\to LR$. A coalgebra for this comonad is a ...
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  • 844
4 votes
0 answers
55 views

Examples of non-self-induced algebras

Let $A$ be a (possibly non-unital) algebra over $\mathbb C$. We say that $A$ is self-induced if the product map $m:A \otimes_A A \rightarrow A$ is an isomorphism. Here $A \otimes_A A$ is the ...
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  • 2,590
1 vote
2 answers
26 views

Algebraic structures as T-algebras

I'm reading about $T$-algebras, defined as a pair of an object $c$ and a morphism $f:Tc \to c$, where $T$ is an endofunctor. It can be shown that it's a generalization of algebraic structures. For ...
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1 vote
0 answers
29 views

Why do we care about local derivations?

I've been trying to study the lie algebra of derivations on a given associative, commutative algebra over $\mathbb{R}$ or $\mathbb{C}$. The motivation for this is to study properties of a smooth ...
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  • 384
1 vote
1 answer
53 views

Tensor product of finite-dimensional semisimple algebras over algebraically closed field is semisimple

Let $K$ be an algebraically closed field, and let $A$ and $B$ be semisimple finite-dimensional $K$-algebras. I've seen a claim that the tensor product $A \otimes_K B$ is also a semisimple ring. To ...
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  • 518
0 votes
0 answers
34 views

Geometric interpretation for norms and traces in number fields or matrix algebras

In a simple algebra $A$ over $\mathbb{Q}$ (one might just think of a number field or a matrix algebra), the trace can be interpreted geometrically as giving an "inner product" for $A$ as a ...
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  • 195
1 vote
1 answer
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Ideals in an algebra over a field as subspaces

I was given a homework question about finite dimensional algebras over a field. In my algebra class, we define an $\mathbb{F}-$algebra (where $\mathbb{F}$ is a field) $A$, to be a ring $A$ Together ...
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  • 99
1 vote
1 answer
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Role of finite generation of the ring of invariants in the existence of a categorical quotient

From the Geometric Invariant Theory book [Mumford - Fogarty - Kirwan], we have the following theorem ([MFK,Theorem 1.1) Let $X$ be an affine scheme over a field $k$, let $G$ be a reductive algebraic ...
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  • 2,730
2 votes
1 answer
38 views

Simple question about finitely generated algebras

I have two finitely generated algebras $A$ and $B$ over a field $\mathbb{K}$ such that $B\subseteq A$. Is it true that $A=B[a_1,\ldots,a_n]$ for some $a_1,\ldots,a_n\in A$? Motivation: I am trying to ...
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  • 580
1 vote
0 answers
21 views

$B \in \mathcal{B}, A \Subset G \implies \mathbb{1}_{\{wx_{A^c} \in B\}}$ is $\mathcal{B}_{A^c}$-measurable.

Let $G$ be a countable group and $\mathcal{B}$ be the Borel sigma-algebra of $G$. Suppose that $B \in \mathcal{B}$ and that $A$ is a finite subset of $G$. I want to prove that for any fixed ...
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0 votes
1 answer
36 views

Graded algebras and symbols

Take $A$ a commutative unitary $\mathbb{C}$-algebra and take $A_0\subset A_1\subset...$ a filtration on $A$. If $$GrA=\bigoplus \frac{A_n}{A_{n-1}}$$with $A_{-1}=0$, is the associated graded algebra. ...
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  • 107
2 votes
1 answer
75 views

Let $A$ be a separable $k$-algebra, and $B$ a semisimple $k$-algebra such that $A\otimes_{k}B$ is separable. Is $B$ separable?

I'm pretty confident the answer to this question is yes, but I am struggling to find a proof. The definition I'm using is: a $k$-algebra $A$ is separable if for any field extension $K\supseteq k$ we ...
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0 votes
0 answers
25 views

Associated graded algebra and natural function

Take $A=\bigcup A_m$ a filtered algebra and $\operatorname{gr}A=\bigoplus A_m/A_{m-1}$ the associated graded algebra, so we have $$f: A \rightarrow \operatorname{gr}A$$ that send $a \mapsto a + A_m$, ...
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  • 107
0 votes
0 answers
17 views

Exponential of a sum in a non-commutative graded algebra

Let $a,b$ be two elements of a graded algebra $A$ such that $\deg(a)=1$, $\deg(b)=0$ and $[a,b]\neq 0$. I would like to know whether there exits an explicit expression for the degree 1 component $$\...
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  • 857
1 vote
1 answer
28 views

Uniqueness of measures over a countable family of sets

Considering the following Theorem: If $(X,\mathcal{A})$ a measurable space and $\mathcal{A}=\sigma(\mathcal{G})$ such that $\mathcal{G}$ is stable under finite intersections there exists and ...
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4 votes
0 answers
85 views

The Category of (Lie) Algebras is not $k$-linear, right?

1. Context In one of my exercise classes I am asked to do the following: "Verify that Lie algebras (resp. associative algebras) over a field F form an F-linear category LieAlgF (resp. AssocAlgF) ...
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3 votes
1 answer
51 views

Finding dual of matrix algebra

Consider $A=M_n(K)$, the algebra of matrices over the field $K$. Let $\{e_{ij}\}$ be the standard basis for $A$ and $\{X_{ij}\}$ be the basis of $A^*$ dual to $\{e_{ij}\}$. I need to find the ...
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0 votes
0 answers
31 views

About the definition of graded algebra

I'm looking for a definition of graded algebra in a general context. I find only definition in the case of algebras over a field or about graduation indexed by natural or integer numbers. This is my ...
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  • 107
0 votes
2 answers
90 views

Is every vector a function on the basis of the vector space?

Let $B= \left\{ a_ 1 , \dots a_n \right\}$ a basis of a vector space $V$ on a field $k$. Let $v \in V$ a vector such that $v = \sum_{i=1}^n \alpha_i a_i$. Is it true that $v$ corresponds to a well ...
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  • 1,045
4 votes
2 answers
71 views

Non-commutative algebra $A$ with a non-trivial maximal ideal $M$ such that $A/M$ is not a division algebra

By "ideal", I mean "two-sided ideal". I'm looking for an example of a real/complex algebra $A$ which is non-commutative and has some maximal ideal $M$ with $\{0\}\subsetneq M\...
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  • 12.7k
5 votes
1 answer
73 views

Tensor product of rings: well-definedness of multiplication

Let $R$ and $S$ be rings, and let $T$ be a ring with ring homomorphisms $\alpha: T \to R$, $\beta: T \to S$. (None of these rings are assumed to be commutative.) Since $R$ and $S$ contain subrings ...
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  • 518
1 vote
1 answer
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Supposse $U, V$ and $W$ are subspaces of a coalgebra $C$. Show that $\Delta(U) \subseteq V\otimes W$ implies $U\subseteq V\cap W$.

I'm new to coalgebras and this is a question from section 2.1 of the book "Hopf Algebras" from Davied E Radford. I tried to pick an element $u \in U$, so $\Delta(u) = u_1\otimes u_2 \in V\...
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  • 151
1 vote
1 answer
33 views

Example of finite-dimensional module over a K-algebra without a composition series

Let $A$ be a $K$-algebra. It is well-known that if $A$ is finite-dimensional as a $K$-vector space then every finite-dimensional $A$-module $M$ has a composition series (more generally, the same is ...
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3 votes
0 answers
72 views

Relation between an "algebra" in Analysis and in Algebra

While studying some measure theory, I came across the following two definitions, given a set $\mathbb{X}$: A ring is a non-empty subset of $\mathcal{P}(\mathbb{X})$ such that it is closed for unions ...
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  • 1,497
1 vote
0 answers
49 views

Integral commutative algebra's definition

I was reading a text where I found the expression Take an integral commutative $\mathbb{C}$-algebra. It' doesn't give a definition so I don't know what it means for $\textbf{integral}$. Now I don't ...
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  • 748
1 vote
1 answer
35 views

Minimal right ideals in semisimple algebra

This question concerns Proposition on page 43 of Associative Algebras by R. S. Pierce. Lemma c. Let $N$ be a right ideal of the algebra $A$ that satisfies $N^{k}=0$. If $P$ is a simple $A$-module, ...
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2 votes
1 answer
96 views

Multiplication of a real algebra is a smooth map

Let $A$ be a finite dimensional real algebra, and let $F:A\times A\to A$ be the multiplication map $F(a,b)=ab$. Is it true that the map $F$ is smooth? (Here I am considering the canonical smooth ...
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  • 1,720
0 votes
0 answers
26 views

Properties of the Peirce decomposition

Let $U$ be an alternative algebra over a field $\mathbb{F}$, and let $U= \bigoplus_{i,j=0}^{t} U_{ij}$ be a Peirce decomposition of $U$ relative to pairwise orthogonal idempotents $e_1, e_2,...,e_t$. ...
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  • 848
0 votes
0 answers
56 views

A system of equations in an algebraic structure

Does there exist an algebraic structure $A$ (for instance, an algebra over a field or an algebra over a ring) such that for a fixed positive integer $m$ and for every non-zero and pairwise different $...
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0 votes
0 answers
23 views

Show that an algebra $A$ is cyclic under some conditions

For a field $\mathbb{F}$ with $\mathrm{char}\ \mathbb{F} \nmid n$ and $Gal(\mathbb{F}/\mathbb{E})=\left<\sigma\right>$ of order $n$ and $a\in \mathbb{F}^\times$ the cyclic algebra is defined to ...
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1 vote
0 answers
107 views

Good examples of modules and algebras?

I'm going through Atiyah-Macdonald on my own, and I'm feeling a distinct lack of examples. With rings I can just about get by with quotients of polynomial rings and $\mathbb{Q}$ and so forth. ...
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4 votes
1 answer
95 views

What does "associative up to homotopy" mean for $A_\infty$-algebras?

I'm reading Keller's Introduction to A-infinity algebras and modules to learn about $A_\infty$-algebras. For reference, an $A_\infty$-algebra $A$ is a graded $k$-vector space $A = \bigoplus_{i\in\...
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  • 1,148
3 votes
0 answers
43 views

Question about homomorphism of algebras

Let $f: \Bbb{Q}[[x]]\rightarrow\Bbb{Q}[[x]]$ be a $\Bbb{Q}$ homomorphism of algebras and $f$ sends every inverse element in inverse elements. a)Show that $f(\langle x \rangle)\subset \langle x \...
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  • 777
2 votes
0 answers
60 views

Show that $\operatorname{Hom}(Ae_{1,1}, Ae_{2,2}) = 0$

Question If $K$ is a field, fix $A = \left \{ \begin{pmatrix} a & 0\\ b & c \end{pmatrix} : a,b,c \in K \right \}$ and $e_{1,1} = \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}$, $e_{2,...
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  • 742
1 vote
1 answer
49 views

Does an $R$ algebra always contain $R$?

In an associative algebra with unit over a commutative ring $R$ it's true that $R$ is inside the algebra? And, is $1$ is the unit in the algebra, is this inclusion $R\cdot1$?
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  • 748
0 votes
0 answers
88 views

Showing algebra is division algebra

Suppose $A$ is a $k$-subalgebra of $M_n(k)$ containing the identity of $M_n(k)$, where $k$ is a field. Suppose $A$ is a domain. I would like to show that $A$ is a division algebra and $\dim_k(A) \mid ...
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4 votes
2 answers
160 views

When does a Path Algebra give a unique Quiver?

I am studying the text An introduction to Quiver Representations by Derksen. Exercise 1.5.4 asks to prove that if the path algebras $\mathbb{C}Q$ and $\mathbb{C}Q'$ are isomorphic $\mathbb{C}$-...
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1 vote
0 answers
42 views

Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal

Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
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  • 115
0 votes
0 answers
32 views

Simple modules corresponding to vertices of a quiver with potential.

Given a quiver with reduced potential $(Q,W)$. The simple dg module over the complete Jacobian algebra $J(Q,W)$ are in bijection to the set of vertices of Q ? How do the simple modules look like ? And ...
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0 votes
1 answer
35 views

Does taking covariant Hom commute with taking (co-)kernel?

Let $A$ be a Commutative ring and $R$ be a commutative $A$-algebra. So every $R$-module has a natural $A$-module structure, and for every $A$-module $W$, and $R$-module $M$, the $A$-module $\text{Hom}...
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  • 3,583
-3 votes
1 answer
113 views

definition of $C^*$-subalgebra

Every time I come across something with a $C^*$-subalgebra, I am confused. There is no definition in my course notes for this. Could someone please help me with the definition, i.e. the things I have ...
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