Questions tagged [noncommutative-algebra]
For questions about rings which are not necessarily commutative and modules over such rings.
2
votes
1answer
33 views
An example of a ring homomorphism
Before stating my question, let me recall some preliminaries in rings (especially noncommutative).
Recall that for a noncommutative ring $R$, $\textbf{B}(R)=\{e\in Z(R): e^2=e\}$.
$\textbf{...
3
votes
0answers
25 views
Deriving the projection onto the isotypic component
Let $k$ be a field, and $G$ a finite group such that the characteristic of $k$ does not divide $|G|$. Then $kG$ is a semisimple $k$-algebra, and the representation theory of $G$ over $k$ is semisimple....
0
votes
1answer
35 views
Proving that the first Weyl Algebra is simple
The first Weyl algebra over the complex numbers $\mathbb{C}[x]$ is defined to be the set $A_1 = {\{\sum_{i = 0}^{n} f_i(x) \delta^i : f_i(x) \in \mathbb{C}[x] }\}$. So it is the set of all linear ...
0
votes
1answer
34 views
What are some simple examples of algebras? [closed]
So an algebra $A$ over a field $K$ is a ring under the operations $+$ and $x$ and also a vector space under a scalar operation and the operation $+$.
What are some examples of algebras? When rings ...
0
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0answers
24 views
Why does “noncommutative probability” capture quantum probability?
In this article, Terry Tao states that non-commutative probability can be used for quantum probability. However, he then goes on to explain non-commutative probability without explaining how it ...
1
vote
1answer
58 views
Confusion regarding the definition of the Weyl Algebra
I've been studying algebraic structures recently; rings, fields, vector spaces and so on. I've recently just started learning what an algebra is, which, from what I can tell is a ring-like structure ...
0
votes
1answer
31 views
$S_4$ is not nilpotent but has central lower central series.
The lower central series of $S_4$ is given by :
$$\gamma_1 =S_4\ge \gamma_2=A_4\ge\gamma_3=A_4\ge \gamma_4 =A_4\ge... $$
This series is clearly central as each $\gamma_i/\gamma_{i+1}$ is central in $...
0
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0answers
18 views
Definition of a $H$- stable ideal
Let $H$ be an Hopf algebra and let $A$ be an $H$-module algebra. What's the definition of a $H$-stable ideal?
0
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1answer
34 views
The possible representations of a monoid derived from four “standard” or “regular” presentations and their relation
Given some monoid $M$ we can form the algebra $\mathbb C[M]$ by considering all formal sums and lineary extending the given multiplication in $M$.
Let $f = \sum_{x\in M} \lambda_x x$ and $m\in M$. ...
3
votes
1answer
100 views
Matrices over noncommutative rings?
In chapter 1, section 2 of Categories for the Working Mathematician, Mac Lane says:
For each commutative ring $K$, the set $\mathbf{Matr_K}$ of all rectangular matrices with entries in $K$ is a ...
1
vote
0answers
29 views
Non-constant coefficient matrix in first order linear differential equations
I want to solve a differential equation of the following form
$$
\frac{d}{dt}x=A(t)x\, ,
$$
where $A(t)$ does not commute at different times. This equation holds on the interval $(a,b)$. Hence, the ...
1
vote
0answers
21 views
Classifying irreducible real representations
Let $G$ be a group and say $V$ is an irreducible representation over $\mathbb{R}$. Then $End_G(V) = End_{\mathbb{R}[G]}(V)$ must be a division algebra, since $V$ is a simple $\mathbb{R}[G]$-module. ...
2
votes
0answers
19 views
Direct sum of reals and quaternions is not a semigroup algebra for some semigroup
For a given semigroup $S$, the semigroup algebra over some field $F$ is the set of formal sum with the convolution product, and is denoted by $F[S]$.
If we built the direct sum
$$
\mathbb R \oplus \...
1
vote
0answers
32 views
Direct sum of Matrix algebras is not isomorphic to some semigroup algebra
An $n \times n$ matrix unit is any matrix which has zeros every, except at one position where it has one. By $E_{ij}^{(n)}$ we denote the $n \times n$ matrix unit which has its one at the $i$-th row ...
1
vote
1answer
46 views
Every algebra could be decomposed into a part with unit, and without a unit. Question on uniqueness proof.
By an algebra $A$ over some field $F$ I mean a finite dimensional vector space over $F$ with an $F$-bilinear multiplication. That $A$ has a unit with respect to its multiplication is not assumed. A ...
0
votes
1answer
52 views
Complement of a maximal direct factor is indecomposable
Reading a book I saw the following assertion:
Let $R$ be a ring (not necessarilly commutative) and $\varepsilon$ be the poset (w.r.t. inclusion) of all internal direct factors of an $R$-module $M$. ...
5
votes
2answers
138 views
On the radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices
Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set
$$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
0
votes
1answer
28 views
Question on proof that matrix algebra over given algebra is semisimple iff original algebra is semisimple
Let $A$ be a finite-dimensional linear associative algebra over some field $F$. Then denote by $M_n(A)$ the set of $n \times n$ matrices with entries in $A$ and the usual operations. Then $M_n(A)$ ...
1
vote
0answers
66 views
Find the Jacobson radical of a matrix ring
Find the Jacobson radical of matrix ring $R=\mathbf{M}_2(K\otimes_kK)$ where $k =\mathbb{F}_2[x]$ and
$K = \mathbb{F}_2[x^{1/2}]$
I tried to find the radical of this matrix ring. I have also found ...
0
votes
0answers
17 views
A question about semiregularity
Let $R$ be a unital ring such that any cyclic right ideal is the direct sum of a ring direct summand $eR$ and a right ideal $S$ of $R$ with $S\subseteq J(R)$, where $e=e^2$ and $ J(R)$ is Jacobson ...
5
votes
2answers
83 views
Is the $\mathbb{Z}$-grading of a Clifford algebra basis independent?
Let $V$ be a finite dimensional vector space over a field $K$ of characteristic $\neq 2$, and let $q \colon V \to K$ be a quadratic form. One of the first things to show when learning the theory of ...
7
votes
0answers
151 views
Associative, non-commutative, non-trivial, analytic binary operation
There was a question whether associative, but non-commutative binary operation over the real numbers exist. A trivial answer is the binary operation $x\circ y = x$ or $x\circ y = y$.
As a followup, ...
0
votes
0answers
12 views
Is there a Grassman equivalent of orthogonal functions?
Orthogonal functions such as the Hermite functions work with commutative variables. Is there a similar thing that works with Grassman variables?
1
vote
1answer
57 views
Noncommutative rings and prime/maximal ideals
Let $R$ a non-simple noncommutative ring, and let $\mathcal{I}$ the set of non-trivial (right, left) ideals of $R$, with the following property: "Every element $I \in \mathcal{I}$ is prime and/or ...
1
vote
1answer
28 views
Computing the left annihilator $\text{Ann}_R(1_R-a)$ of a ring $R$.
I was computing the left annihilators of the elements $x$ of a ring $R$ with $1_R$ (denoted by Ann$_R(x)$) and encountered the following scenario:
For any $a\in R$, $$\text{Ann}_R(1_R-a)=\{r\in R: r(...
0
votes
1answer
49 views
Is it true to say that : Every submodule of a module M contains in a maximal submodule?
Let $R$ be an arbitrary ring with $1\neq 0$ and $_RM$ a left $R$-module. Is it true to say that : Every proper submodule of a module $M$ is contained in a maximal submodule?
5
votes
1answer
96 views
Determining the structure of a finite ring
I'm working on the following problem and would like some guidance
$R$ is a finite ring with $x^5 = x$ for every $x$ in $R$. Determine the structure of $R$.
My first thoughts were to factor and ...
2
votes
0answers
32 views
Definition of socle of a module
For me, the socle of an $R$-module $M$ is the unique maximal semisimple submodule of $M$. If $(R,\mathfrak m)$ is a local ring, this is equivalent to the maximal submodule annihilated by $\mathfrak m$....
1
vote
1answer
32 views
Semisimple subalgebras of $M_4(\mathbb{C})$
I'm working on the following problem and I'd like some guidance.
Describe up to isomorphism all semisimple $\mathbb{C}$-subalgebras of $M_4(\mathbb{C})$ (4 by 4 matrices over $\mathbb{C}$). Note ...
1
vote
1answer
31 views
Boson operator algebra - unitary transformation
What is the simple way to evaluate
$$ e^{i \alpha n(n-1)}a^{\dagger} e^{-i \alpha n(n-1)}$$
and
$$ e^{i \alpha n(n-1)}a e^{-i \alpha n(n-1)}$$
where $n=a^\dagger a$ and $a$ and $a^\dagger$ are ...
0
votes
0answers
22 views
Computations in matrix space over noncommutative semiring
Let $M^{n\times n}$ is a matrix space over a noncommutative idempotent finite semiring $S$. $X\in M,L\in M,R \in M$. Whether exists a way to compute a sequence of the form $((((((X*L)^T * R^T)^T * L)^...
2
votes
2answers
63 views
Complicated differential operator and application of the Zassenhaus formula
I have a differential operator
$$D_x = \frac{1}{x} \left[ (x^2 - a^2) \frac{d}{dx} \left( \frac{1}{x} \frac{d}{dx} \right) + 2 \frac{d }{dx} \right], $$
and I would like to compute
$$ e^{i \...
2
votes
1answer
33 views
Is $\operatorname{Hom}(E,M)$ simple as an $\operatorname{End}(M)$-module if $E$ is simple?
Let $R$ be a ring, let $M$ be an $R$-module and let $R' := \operatorname{End}_R(M)$.
Let $E$ be a simple $R$-module with $\operatorname{Hom}_R(E,M) \neq 0$.
Question:
Is $\operatorname{Hom}_R(E,M)...
0
votes
1answer
53 views
Matrix trace equality [closed]
Suppose $A$ and $B$ are positive definite matrices of the same size. Prove that
\begin{align}
\operatorname{Tr}( A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2}) \leq \operatorname{Tr}( (A^{1/2} B A^{1/2}...
0
votes
0answers
57 views
Is pullback of non-commutative rings well defined?
I know that pullback is defined for commutative ring, but what about non-commutative case?
Let's consider the following diagram, where $R_i,\bar{R}$ are rings and $R$ is the pullback:
Then $1\in R$ ...
1
vote
1answer
26 views
Constructing an integral domain with a specific subring
What tools are available for constructing various (noncommutative) domains with a given subdomain? That is, how can I begin to examine various domains which contain the domain $S$, other than ...
2
votes
1answer
70 views
Reference: Every Scheme is Derived Affine
I'm trying to track down a reference for the following claim, found in Kaledin's lectures Methods in Noncommutative Algebraic Geometry:
As it turns out, an arbitrary scheme $X$ also appears ...
1
vote
1answer
37 views
Is Artinian algebra with finite injective dimension of itself is self-injective？
Let $A$ be an Artinian $R$-algebra .(i.e.there is a ring homomorphism $\alpha:R\rightarrow A$,where $R$ is a commutative Artinian ring,the image of $\alpha$ lies in the center of $A$ and $A$ is ...
0
votes
0answers
48 views
Jacobson radical of the $\mathbb{k}$-algebra $\mathbb{k}[x]/\langle{p(x)}\rangle$ where $\mathbb{k}$ is a field
Let $\mathbb{k}$ be a field and $p(x)\in\mathbb{k}[x]$. I am trying to prove that the Jacobson radical (the intersection of all maximal ideals of $A$, denoted by $\mathrm{rad}(A)$) of the $\mathbb{k}$-...
3
votes
0answers
50 views
Lang's proof of the uniqueness of multiplicities for semisimple modules
In Algebra by Serge Lang (XVII, Proposition 1.2, p. 643-644) it is shown that for a simple $R$-module $E$ and $n, m \geq 0$ with $E^{\oplus n} \cong E^{\oplus m}$ it follows that $n = m$.
Lang argues ...
4
votes
0answers
34 views
Natural transformations from coinvariants to invariants
Consider a group $G$ and a ring $R$. Write $R[G]\overset{\varepsilon}{\longrightarrow}R$ for the counit map. Write $(-)_G\dashv \varepsilon ^\ast \dashv (-)^G$ for the adjoint triple on modules ...
12
votes
1answer
268 views
The mysterious isomorphism between coinduction and induction
Let $G$ be a finite group.
This answer mentions an explicit yet mysteriously preferable isomorphism from the coinduction functor to the induction functor.
This instructive answer observes the ...
0
votes
0answers
33 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
25 views
How do I prove that every fractional ideal of an order in a division algebra is a full lattice?
Let O be an R-order for some Dedekind domain R, let F be the field of fractions of R and D be a division algebra over F. A fractional left ideal of O is an R-lattice I in D such that OI in I (I ...
1
vote
0answers
40 views
Are all quaternion algebras over the rationals skew fields?
If I understand correctly, any quaternion algebra over the rationals is a noncommutative associative division algebra. I am currently working with implementations of quaternion algebras in MAGMA and ...
3
votes
0answers
18 views
Does Pontryagin duality extend to class two nilpotent or maybe even metabelian locally compact groups?
A metabelian group $G$ is determined up to isomorphism by "abelian data" $(A,M,\alpha)$ -- an abelian group $A:=G/[G,G]$, an $A$-module $M:=[G,G]$, and a cocycle $\alpha:A\times A\to M$ giving the ...
2
votes
1answer
63 views
group-like elements of a Hopf algebra and the group algebra
Suppose we are given a $n$-dimensional Hopf algebra $H$ over field $\mathbb K$. I want to prove that
$H$ contains n distinct group-like elements if and only if there exists a group $G$ such ...
3
votes
2answers
72 views
Maximal ideals in a group ring when the ring is a field
Is there any simple characterization of the maximal ideals in a group ring $R[G]$ when $R$ is a field, perhaps in terms of maximal subgroups of $G$?
0
votes
2answers
32 views
Field where $e^2 + f^2 = -1$ implies existance of matrix $A$ with $A^2 = -E_2$?
Suppose we have a field $K$ with a solution for $e^2 + f^2 = -1$. Is there a matrix $A \in K^{2\times 2}$ with $$ A^2 = \begin{pmatrix}-1 & 0 \\ 0 & -1 \end{pmatrix}? $$
0
votes
0answers
29 views
Solving linear systems for integer values in MAGMA
Say we are given a quaternion algebra D over a number field F as well as a maximal $\mathcal{O}_F$-order $\Delta$ $\subseteq$ D and say we have a $\mathbb{Z}$-basis $\omega_1, . . . , \omega_n$ for $\...
