Questions tagged [algebras]
For questions about algebras, their properties, and their structures. Use [tag:algebra-precalculus] or [tag:abstract-algebra] if your question is about algebra, not algebras.
492
questions
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A question about the minimal number of operations required to obtain a matrix with generic eigenvalues
Let $A,M \in \operatorname{Mat}_n(\mathbb{C})$ be $n \times n$ matrices such that $M$ is invertible and $MA \neq AM$. Consider the algebra $\mathcal{A}$ generated by the set $\{I,A,MAM^{-1}\}$, where $...
- 107
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1
answer
48
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Homotopy of maps of $A_{\infty}$-algebras
Let $f, g: A \to B$ be maps of $A_{\infty}$-algebras. What is the correct (explicit) notion of a homotopy between $f$ and $g$? This is given in the expository paper of Keller in terms of maps of the ...
- 347
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0
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57
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On the homomorphism as $M_n(A)$-modules.
I'm trying to prove that, if $A$ is an Algebra over a field $F$, and $U,V$ are $A-$modules. Then any element in $Hom_{M_n(A)}(U^n, V^n)$ (recall that $U^n, V^n$ is the set of size $n$ vectors as an $...
- 31
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21
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Turning a subalgebra into an ideal by changing the algebra it is contained in
Let $B$ be an (not necessarily unital) algebra and $A \subseteq B$ a subalgebra. Is there some sort of quotient $C$ of $B$ (or any other interesting construction not necessarily found by taking ...
- 159
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1
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52
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Semisimplicity implies separability for a perfect field
Let $k$ be a field, $A$ a finite-dimensional semisimple $k$-algebra. If $k$ is a perfect field (every finite field extension of $k$ is seperable), then $A$ is separable.
I know a proof that uses the ...
- 1,585
3
votes
1
answer
53
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Does there exist a topological power-associative non-metric unital algebra $V$ such that $\exp x$ is defined for all $x\in V$?
From what I understand, we normally define the operator $\exp$ only on unital Banach algebras, because the triangle inequality ensures that the infinite series converges:
$$\lim_{n\to\infty}\left\|\...
- 3,084
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28
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$\operatorname{Hom}_A(M, N)$ and $M^* \otimes_A N$
If $U$ and $V$ are vector spaces over some field $k$ and $U$ is finite dimensional with basis $\{u_i\}_i$, then we know that $\def\Hom{\operatorname{Hom}}$
$$\Hom_k(U, V) \to U^* \otimes_k V,\quad f \...
- 2,184
22
votes
6
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Proof the quaternions are 4-dimensional?
The quaternions can be defined as
$$\mathbb{R}\langle X,Y\rangle/(X^2+1,Y^2+1,XY+YX)$$
From these relations, it is relatively easy to prove that $1,X,Y,XY$ span the quaternions over $\mathbb{R}$. But ...
- 4,074
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14
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Inferring classification of Clifford algebras from classification of Clifford modules
Let $Cl_n$ be the Clifford algebra (over reals)
$$
Cl_n = T^{*}\mathbb{R}^n/\langle v\otimes v - q(v) \rangle.
$$
There is a periodic table of $K$-representations of $Cl_n$, i.e. $\mathbb{R}$-linear ...
- 766
3
votes
0
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109
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Equivalent definitions of Hopf algebras
Recently, I started to study the book Hopf algebras by Moss Sweedler, in such book, given a coalgebra $(C,\Delta,\epsilon)$ and an algebra $(A,\mu,\eta)$, the autor defines the convolution of two ...
- 536
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1
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91
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Jordan-Holder theorem for group algebras
I'm currently studying the Jordan-Holder theorem for modules and representations of associative algebras over fields. I was wondering if there is a way to prove the Jordan-Holder theorem for finite ...
2
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1
answer
62
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Definition of splitting field: Why do we require centrality?
Let $k$ be a field. Let $D$ be a division algebra over $k$.
Call a field extension $K/k$ a splitting field for $D$ if there exists a positive integer $n$ such that $D\otimes_k K\cong M(n\times n,K).$ ...
- 1,585
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1
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42
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Complete Subalgebras and Dense Subalgebras of Complete Boolean Algebras are Regular Subalgebras
In Thomas Jech's Set Theory book, he states that
If $A$ is a complete subalgebra of a complete Boolean algebra $B$ then $A$ is a regular subalgebra of $B$.
Also, if $A$ is a dense subalgebra of $B$ ...
- 435
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112
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Finding the discriminant of a quaternion algebra
Consider the totally real number field $ F=\mathbb{Q}(\zeta_{10}+\zeta_{10}^*) $. Consider the quaternion algebra $ Q=(\frac{-1,-1}{F}) $. How do I compute the discriminant of this algebra?
I gave ...
- 7,124
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1
answer
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On the property of a ring modulo its Jacobson radical being a division ring
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $A$ be a finitely generated associative $R$-algebra. Let $x\in \mathfrak m$ be a non-zero-divisor on $A$ such that $xA\neq A$. If $A/J(...
- 1,388
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0
answers
60
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Definition of finite-dimensional 'self-dual algebra' over a field
In some informal notes that are not publicly available and for which I do not have permission to reproduce here, there is a reference to a 'finite-dimensional self-dual algebra over a field $K$'.
I ...
- 3,304
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0
answers
81
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What is precisely an anti-linear anti-involution?
Given an arbitrary $\mathbb{C}$-algebra $A$, what should be required from a function $f:A\to A$ for it to be an anti-linear anti-involution?
As I understand “anti-linear” for $A$ as a vector space ...
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48
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A sort of "minimal presentation " for a local ring essentially of finite type over a field
Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
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In what sense is the Temperley-Lieb algebra related to the Braid group?
Question:
In what sense is the Temperley-Lieb algebra $ TL_n $ related to (a representation of?) the braid group $ B_n $?
For example, is $ TL_n $ the algebra of matrices generated by the image of $ ...
- 7,124
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0
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51
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What is the dual Hopf algebra of T(V)?
I am reading "Geometric versus Non-Geometric Rough Paths" by Martin Hairer and David Kelly. I don't know much about Hopf algebras but $T(V)$ is the one example I'm comfortable with so far ...
- 1,100
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49
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I need help to prove this inequality, I'm not really sure what the question means.
Question
Let n be a positive integer and $x>y$. Prove that
$$\frac{x^n-y^n}{x-y}>ny^{n-1}$$
By choosing suitable values of x and y, further prove than
$$\left(1+\frac{1}{n}\right)^n<\left(1+\...
- 121
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0
answers
22
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An associative k-algebra whose enveloping algebra $A \otimes_k A^{op}$ is not iso to $A \otimes_k A$
I'm trying to find an example of a finite-dimensional $k$-algebra $A$ for some field $k$, ideally $\mathbb R$, such that $A \otimes_k A \not\cong A \otimes_k A^{op}$.
A lot of algebras have $A \cong A^...
- 8,155
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0
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23
views
Let a, b, c, d be complex numbers satisfying $a+b+c+d=a^3+b^3+c^3+d^3=0$. Prove that a pair of the a, b, c, d must add up to 0 [duplicate]
When doing this I tried using the identity
$x^3+y^3+z^3=3xyz$ if $x+y+z=0$
I take $x=a$, $y=b$, and $z=c+d$
So $a+b+(c+d)=0$
$a^3+b^3+(c+d)^3=3ab(c+d)$
$a^3+b^3+c^3+d^3+3cd(c+d)=3ab(c+d)$
$(a^3+b^3+c^...
- 121
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vote
1
answer
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Is the Galois action on $k$-algebra homomorphisms transitive?
I am reading the proof of Proposition 2.19 in Liu's book on Algebraic Geometry and Arithmetic Curves. The proposition is as follows:
Let $X$ be an algebraic variety over $k$, and let $K / k$ be a ...
- 1,643
0
votes
1
answer
42
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Is $x\mapsto 1_A\otimes x:V\rightarrow A\otimes V$ injective for $K$-algebra $A$ and $K$-vector space $V$?
Let $K$ be a field, $A$ be a commutative unital $K$-algebra and $V$ be a $K$-vector space. Is the $K$-linear map $$x\mapsto 1_A\otimes x:V\to A\otimes V$$ injective?
The specific case that I am ...
- 3,304
-1
votes
1
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Is there more than one notion of algebras? [duplicate]
I know that an algebra is an algebraic structure, that can be seen as a vector space with a multiplication operation or as a ring with a vector space structure. However, in measure theory we define an ...
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Question about a proof that simple algebras are semi-simple
In the book Clifford Algebras: An Introduction by D. J. H. Garling, Theorem 2.7.2 states that a finite-dimensional simple unital algebra $A$ is semi-simple. My question is specifically about the proof ...
- 1,054
1
vote
1
answer
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$R[x]$-algebra structure that on $R[y]$ seen as quotient of $R[x,y]$
In a broader exercise about the fibered product of schemes, I'm given a ring (commutative and unitary) $R$ and the following ring homomorphism:
$\begin{align*}
R[x]&\to R[y]\\
x&\mapsto y^2\\
\...
- 304
3
votes
1
answer
81
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Tensor Products of R-Algebras from Atiyah and Macdonald
I know this part of Atiyah and Macdonald has a typo, but that is not what this question is about.
Let $R$ be a commutative ring and $S$ and $T$ be $R$ algebras. I am trying to show that
$S\otimes_R T$ ...
- 2,174
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votes
0
answers
37
views
Enumerating monomorphisms of finite-dimensional $\mathbb{F}_2$-algebras
I want to enumerate the monomorphisms of finite-dimensional $\mathbb{F}_2$-algebras.
Of course, each such monomorphism is a linear map between finite-dimensional $\mathbb{F}_2$-vector spaces, and so ...
- 3,304
0
votes
0
answers
20
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conditions that an annihilator banach algebra will be a dual banach algebra
a banach algebra $\frak{A}$ is called an annihilator algebra if, for arbitrary closed left ideal $\frak{L}$ and closed right adeal $\frak{R}$ in $\frak{A}$, both of the following conditions are ...
4
votes
1
answer
197
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Finite order automorphisms of semi simple lie algebras (Kac Lemma 8.1)
I am currently reading Kac's book on infinite dimensional Lie algebras and have some trouble with Lemma 8.1. The setup is as follows:
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra ...
- 172
1
vote
1
answer
157
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What is the difference between an $FG$-module and a group algebra?
I am beginning the study of representation theory, and am having trouble understanding the difference between an $FG$-module and a group algebra of $G$. This seems to be the only other question on ...
- 2,812
1
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1
answer
44
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Voight's Quaternion Algebras, Corollary 7.1.2
I am reading the proof that if $B$ is a central simple $F$-algebra of dimension $4$ ($\operatorname{char}(F) \neq 2$), then $B$ is a quaternion algebra in John Voight's book Quaternion Algebras (...
- 1,643
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1
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If $G$ is cyclic of order n, $\mathbb{Q}[G]\cong \oplus_{d|n}\mathbb{Q[\xi_d]}$
I am trying to find a proof of this result.
If $G$ is cyclic of order n, $\mathbb{Q}[G]\cong \oplus_{d|n}\mathbb{Q[\xi_d]}$.
I think the proof will involve the use of characters of finite abelian ...
- 874
1
vote
1
answer
37
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Why quotient ring $ k[x_1,...x_n]/I(Y)$ is k - algebra?
Let $Y \subseteq \mathbb{A}^n $ and $f,g \in k[x_1,...,x_n]$ have the same restriction to $T$ if and only if $f-g \in I(Y)$. Why the quotient ring $ k[x_1,...x_n]/I(Y)$ is k - algebra?
Definition k-...
- 25
1
vote
1
answer
45
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Examples of rings by their relation to projective coverings
A projective covering of an $R$-module $M$ is an epimorphism $\pi:P\rightarrow M$ s.t. $P$ is a projective $R$-module and $\textrm{Ker}(\pi)$ is co-essential in $P.$ The existence theorem for ...
- 1,303
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0
answers
36
views
$𝑅$ a $\mathbb C$- algebra and $𝑀$ a simple $𝑅$-module with a countable basis. Then $\textrm{End}_R(M)\cong \mathbb C.$ [duplicate]
Sсhur's lemma states:
Lemma. Let $R$ be a finite-dimensional algebra over an algebraically closed field $K$ and let $M$ be a simple $R$-module. Then $\textrm{End}_R(M)\cong K.$
Proof. Since $M$ is ...
- 1,303
3
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1
answer
45
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The pre-image sigma algebra of the greatest integer function [x].
The first exercise in my Measure and Integration book is to find the pre-image sigma algebra of various functions, namely $f(x)=x^3$, $x^2$ and $[x]$. You can correct me if I'm wrong but I'm fairly ...
- 305
16
votes
1
answer
708
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Does the product rule imply the chain rule?
Let $\mathbb{F}$ be a field, and consider $\mathbb{F}^\mathbb{F}$ as an algebra over $\mathbb{F}$ with the standard function multiplication. Let $D$ be a derivation on a subalgebra of $\mathbb{F}^\...
- 3,905
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1
answer
275
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Einstein notation with three indexes
I met in a book a mention of Sklyanin algebra for $(S_0,S_1,S_2,S_3)$:
$$\{S_0,S_{\alpha}\}=\varepsilon_{\alpha\beta\gamma}S_{\beta}S_{\gamma}(J_{\beta}-J_{\gamma})$$
$$\{S_{\alpha},S_{\beta}\}=\...
- 23
2
votes
1
answer
191
views
Is there a finite-dimensional algebra over $\mathbb C$ with no zero divisors?
Every non-trivial finite dimensional associative algebra over an algebraically closed field has zero divisors. Is there a non-trivial finite dimensional non-associative algebra over $\mathbb C$ or ...
2
votes
1
answer
110
views
A division quaternion algebra in which the integral elements don't form a ring
I wish to find a division quaternion algebra $B$ over $\mathbb{Q}$ and elements $\alpha, \beta\in B$ such that $\alpha, \beta$ are integral over $\mathbb{Z}$ but both $\alpha + \beta$ and $\alpha \...
- 1,187
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votes
1
answer
109
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Module over algebra is free iff it has a basis
$
\newcommand{\cn}{\colon}
\newcommand{\<}{\leqslant}
\newcommand{\>}{\geqslant}
\newcommand{\ss}{\subset}
\newcommand{\k}{\mathrm{k}}
\newcommand{\gr}{\mathrm{gr}}
\newcommand{\R}{\mathrm{R}}
\...
- 145
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0
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94
views
Rees algebra is finitely generated if associated graded algebra is
$
\newcommand{\cn}{\colon}
\newcommand{\<}{\leqslant}
\newcommand{\>}{\geqslant}
\newcommand{\ss}{\subset}
\newcommand{\k}{\mathrm{k}}
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\...
- 145
2
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0
answers
118
views
Algebra is noetherian if associated graded algebra is noetherian
$
\newcommand{\cn}{\colon}
\newcommand{\<}{\leqslant}
\newcommand{\>}{\geqslant}
\newcommand{\ss}{\subset}
\newcommand{\gr}{\mathrm{gr}}
$
Let $A$ be associative commutative algebra with unity ...
- 145
1
vote
0
answers
103
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Uniqueness of direct product of two $A$-modules
$
\newcommand{\cn}{\colon}
\newcommand{\<}{\leqslant}
$
Let $k$ be a field.
Def. Algebra $A$ is associative ring with unity and a ring homomorphism $f\cn k\to A$ such that $f(1_k)=1_A$ and $f(a)x=...
- 145
0
votes
0
answers
35
views
When an Isomorphism between two algebras is an equivalence?
I'm currently involved in the study of algebraic structures, and there's a concept that seems to appear every so often.
Given an Algebra $ A=<A,F_i> $ an isomorphism $ f $ is a bijective ...
1
vote
1
answer
43
views
Subring is integral over finitely generated subalgebra
Let $R \subseteq S$ be unital rings and let $G \leq \mathrm{Aut}_R(S)$ be a finite group of automorphisms of $S$ as an $R$-algebra.
We define the invariant subring: $$S^G = \{a \in S \mid \forall \...
1
vote
1
answer
81
views
The forgetful functor $U:T-Alg\rightarrow C$ preserves limits
Suppose $F$ is an endofunctor of a complete category $C$.
Let $F-Alg$ be the category that has objects the pairs $(X, \ a:FX\rightarrow X)$ where $X$ is an object in $C$, and that has morphisms $f:(X,...
- 312