All Questions
6,548 questions
0
votes
0
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85
views
A naive looking question about Gelfand-Kirillov dimension
Let $A$ and $B$ two affine algebras, $A$ a subalgebra of $B$. If we have a left $A$-module $M$ we can extend the scalars: $B \otimes_A M$. I will denote the resulting $B$-module by $N$
How are $\...
- 3,318
-1
votes
1
answer
209
views
Every abelian group can be embedded into a ring [closed]
Let $(G,0,+)$ be an abelian group. Does there always exist a ring with unity $(R,0,1,+,\cdot)$ and an injective homomorphism of groups $ \psi:(G,0,+)\rightarrow (R,0,+)$?
Is this hard to prove, or are ...
- 143
1
vote
0
answers
118
views
Cyclic homology with coefficients in a bimodule
I've recently been trying to understand Hochschild and cyclic co/homology better, and I've noticed that while it's common to define the Hochschild homology $\mathrm{HH}_{\bullet}(A;M)$ of an $R$-...
- 11.8k
2
votes
0
answers
148
views
Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
- 24.9k
1
vote
0
answers
64
views
Reference request for non-banded Toeplitz matrix
I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix.
I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
- 11
1
vote
1
answer
67
views
Low rank matrices which preserve maximizers of quadratic form
Suppose $x, y \in \{0,1\}^d$ are binary vectors. For a matrix $M$ consider the quadratic form,
\begin{align}
x^T M y + (\mathbb{1} - x)^T M (\mathbb{1} - y)
\end{align}
Does there exist an $M$ such ...
- 297
6
votes
1
answer
273
views
Let $f$, $g$ be two complex polynomials satisfying $f(x-1,y)g(x,y)=f(x,y)g(x-1,y-1)$, what we can say about $f$ and $g$?
Let $f$, $g$ be two non-zero polynomials in $\Bbb C[x,y]$ satisfying the identity
$$
f(x-1,y)g(x,y)=f(x,y)g(x-1,y-1).
$$
What we can say about $f$ and $g$?
In particular, if $f$, $g$ are two non-zero ...
1
vote
0
answers
77
views
$M^2=0$ defines a Koszul algebra ? What if $M$ is Manin's endomorphism's of Koszul $A$ ? (Here $M^2=\sum_k M_{ik}M_{kj}$ - resembles $d^2=0$).)
Consider a matrix $M$ which elements $M_{ij}$ are generators of some algebra $K$,
impose new relations: $M^2=0$ and get a new algebra $K_{2}$.
Question 1: Is it true that $K_2$ is Koszul algebra when ...
- 24.9k
1
vote
0
answers
60
views
Bounding the length in a module of evaluated skew polynomials
Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \...
- 223
1
vote
0
answers
45
views
Rank of Hadamard product of column-wise polynomial evaluations and row-wise exponential evaluations
Consider the Hadamard product $A \odot B$ between two special matrices $A,B \in \mathbb{R}^{n \times m}$. The columns of $A$ are evaluations of polynomials, while the rows of $B$ are evaluations of ...
- 21
1
vote
0
answers
162
views
Integral points on "complex exponential surface" in $\mathbb{C}^3$
I encountered the following object in $\mathbb{C}^3$ defined for $m\in\mathbb{N}$ by
$$A_m:=\lbrace (z_1,z_2,z_3)\in\mathbb{C}^3|(2^{2z_3}m-1)2^{2z_1+z_2+1}+3^{z_2-1}(2^{2z_1}-2^2-3^{z_3+1}m)=0\rbrace$...
- 91
0
votes
0
answers
36
views
Conjugate gradient-like algorithm with multiple search directions
I am solving an $n*n$ system $Ax=b$ in CUDA where $A$ is a sparse matrix. Currently I am solving it using the conjugate gradient algorithm.
I have noticed that $Ax$ where $x$ is $n*1$ has roughly the ...
- 71
11
votes
0
answers
436
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
2
votes
1
answer
345
views
What's the explicit value of this determinant
Let $n\ge2$ be a positive integer, and let $b_1,\cdots,b_n, c_1,\cdots, c_n$ be variables.
Recently, I met the following determinant:
$$\det A=\left|\begin{array}{cccc}
1 & b_1+c_1 & b_1^2+c_1^...
- 87
1
vote
0
answers
189
views
The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 173
3
votes
0
answers
40
views
Filtering a pre-Koszul algebra to get a homogeneous associated graded algebra
In Priddy's paper "Koszul resolutions", on p. 42 he defines an algebra $A$ to be pre-Koszul if it can be written as a quotient of a free algebra $F = F\langle x_i \rangle$ with generators $\{...
- 4,254
0
votes
0
answers
34
views
Support of a function acting on an algebra?
Quick: for a measurable function $f$ its support on Euclidean space is clearly just the subset where $f$ does not vanish.
Now, let’s have $f$ acting on an finite Lie algebra, f.e. $\mathfrak{gl}$ as $...
0
votes
0
answers
71
views
When is a submodule trivial?
I am a beginner concerning module theory, but I need it for my PhD. Sorry for obvious questions therefore.
Given a left $C(G)$-module $(V, \tilde{\rho})$ where $C(G)$ denotes the group algebra over a ...
0
votes
0
answers
46
views
submodules in a direct sum of semisimple modules without common simple factors
Let $A$ be an associative (unital) algebra. Let $M_1,\cdots, M_r$ be pairwise non-isomorphic simple $A$-modules and let $V=\bigoplus^r_{i=1}V_i$, where
$$
V_i=M_{i,1}\oplus \cdots\oplus M_{i,n_i}\...
- 620
0
votes
0
answers
61
views
Combinatorial counting question related to count (anti)commuting N-tuples of matrices (more generally $(X_1,...X_n): F(X_i,X_j)=0$ - only one F)
Consider some finite set $S$ (can be matrices over $F_p$), consider some symmetric relation $F(s1,s2)$ which values are True or False (for example - matrices (anti)commutate or not).
Question 1: can ...
- 24.9k
0
votes
0
answers
64
views
Proof of a folkloric result about PI-algebras [duplicate]
I am not not an specialist in PI-algebras, but I can say I have a rather good understanding on the subject.
It is, of course, interesting to discover if an algebra $A$ is a PI-algebra. But it is also ...
- 3,318
0
votes
0
answers
66
views
Random elliptical potential lemma
Elliptical Potential Lemma: Let $V_0 \in \mathbb{R}^{d \times d}$ be positive definite and $a_1,a_2,...,a_n \in \mathbb{R}^{d}$ be a sequence of vectors with $||a_t ||_2 \leq L < \infty$ for all $t ...
- 61
1
vote
1
answer
294
views
Nearest Kronecker product to sum of Kronecker products
I am interested in efficiently finding the closest Kronecker decomposition to the sum of $k$ Kronecker products:
$$\min_{A,B} || A \otimes B - \sum_{i=1}^k A_i \otimes B_i ||_F$$
where $A,A_i$ are $p \...
- 111
5
votes
2
answers
397
views
Ring with vanishing $K_0$
Suppose we have a ring $R$ such that the Grothendieck group $K_{0}(R)=0$.
Question 1: Does it follow that there exists two positive natural numbers $n\neq m$ such that
$R^{m}$ is isomorphic to $ R^{n}$...
- 855
0
votes
0
answers
106
views
Is there a counterpart to "a group acting on a space" to "a ring acting on a space", especially if the space is a (Lie) algebra?
I have read something that comes close to this is a module, but as I have understood, a module requires my space to be an abelian group, which would not be the case for a Lie algebra.
So, if I have ...
0
votes
0
answers
124
views
Do unitary adjoint representations on $\mathfrak{sl}$ form a ring?
I am not too deep into abstract algebra, but I need it badly for my PhD. Therefore, I would be happy for some help here!
I try to give some sense into:
Let $\mathrm{Ad}_{g*}M= U^\dagger M U$ be a ...
0
votes
1
answer
127
views
update rule for the inverse after a rank-1 update plus scaled identity
Is there an update rule for $$\left(\tilde{X}^T\tilde{X}+\alpha\cdot I\right)^{-1}$$ with $\tilde{X}=[X\;\; a]$ as a function of $A\triangleq (X^TX)^{-1}$, $X$ and $a$?
I know that when $\alpha=0$ we ...
- 503
3
votes
2
answers
451
views
Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture
Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
- 24.9k
3
votes
1
answer
472
views
Tips for how I can proceed with my Lie theoretical problem?
$\DeclareMathOperator\SL{SL}$I am looking at a map from a Lie group into a Lie algebra $\phi$:
$$\phi: \SL(n)\rightarrow \mathfrak{sl}_n$$
$$ P \rightarrow U_1^\dagger P U_1 + U_2^\dagger P U_2.$$
$P$ ...
7
votes
1
answer
238
views
Hadamard product decomposition with lower rank matrices
Given integers $k$ and $l$ and a matrix $A$ of rank $kl$, can we always find a matrix $B$ of rank $k$ and a matrix $C$ of rank $l$, such that $A$ is the Hadamard product of $B$ and $C$, namely $A=B \...
- 71
1
vote
1
answer
363
views
Proj construction and nilpotent homogenous elements in graded ring
Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined ...
- 5,998
0
votes
0
answers
33
views
determinantal ideal of sum of Galois conjugate matrices
Given $n$ matrices $A_i \in \mathbb{Z}^{m\times m}$. I am interested in the ideal $I_d(A)$ generated by the $d\times d$-minors of $A = \sum_{i=1}^n x_iA_i \in \mathbb{Z}[x_1, \dots , x_n]$.
The matrix ...
- 61
2
votes
0
answers
85
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
3
votes
1
answer
343
views
Reference request: about “SNF” (Smith Normal Form)
I've read about some studies on the Paley I Construction. Among them I found the following notations ( See this page: https://documents.uow.edu.au/~jennie/matrices/32P02.html ).
$$SNF:1,2^a,4^{b},8^{b}...
- 41
3
votes
1
answer
426
views
Is Malcev completion an embedding?
The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by
$$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$
the group-like part of the completed (by the augmentation ...
- 985
2
votes
1
answer
184
views
Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q) $, where $P_i$ - do not depend on $N$?
Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating ...
- 24.9k
1
vote
0
answers
138
views
Questions on integer matrix multiplication
Question 1:
Given two integer matrices $A$ and $B$, and let $C$ be $AB$.
$C$ can be very big in pratice, so what is the fastest way to compute the statistical data of $C$?
For example,
$$A=\begin{...
- 696
0
votes
0
answers
103
views
Matrix of the minimal projective presentation of a $\tau$-rigid module
Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the ...
- 839
4
votes
0
answers
99
views
If matrices $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup \mathbb{T}$, does $\text{rank}([A,B])=\text{rank}([A^*,B])$?
Here $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$ denotes the unit circle in the complex numbers.
This holds, if we have $\sigma(A)\subseteq \mathbb{R}$ or $\sigma(A)\subseteq \mathbb{T}$ (independent of $B$...
- 151
0
votes
1
answer
54
views
Orthogonality in Hilbert algebras and congruence
Consider a finite-dimensional Hilbert space $V$ (say, over $\mathbb{C}$) and a finite-dimensional Hilbert algebra $A$ (i.e., Hilbert space with a compatible associative unitary algebra structure). ...
- 327
4
votes
0
answers
116
views
When is the intersection of cosets of a conjugacy class $0$-dimensional?
Let $G = \mathrm{SL}_n$ (say); let $K$ be a field. Let $g$ be a regular semisimple element of $G(K)$, and $\mathrm{Cl}_g$ its conjugacy class, considered as an algebraic variety. Then $\mathrm{Cl}_g$ ...
- 20.2k
4
votes
0
answers
118
views
Adjoining new factors for primes in UFDs
It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
- 18.7k
2
votes
0
answers
72
views
Gradient descent over the set of complex symmetric matrices
In the course of my research (somewhat related to compressive sensing), I am trying to determine a complex, symmetric matrix $L$ (i.e. $L = L^T$) through the following optimization formulation:
$$ \...
- 21
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
2
votes
1
answer
158
views
How to decompose a given polynomial by ideal generators
Given a finite set of polynomials $f_1, f_2,..., f_n$ of variables $x_1,...,x_m$, generating the ideal $I$, suppose that we have one more polynomial $g\in I$.
What is the algorythm for decomposing $g$ ...
- 728
6
votes
3
answers
434
views
What is known about finite dimensional modules over the nilCoxeter algebra?
Recall that the nilCoxeter algebra $\mathcal{N}_W$ for a Coxeter group $W$ is given by the $\mathbf{k}$-basis $x_w$ for each $w\in W$ and multiplication $x_ux_v=x_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$ ...
- 1,066
0
votes
0
answers
37
views
Largest root of the Adjacency matrix of two graphs (comparison)
Let $G$ and $H$ be two graphs whose spectral radius (largest eigenvalue) of the adjacency matrix is the largest root of the following polynomial:
$$P_G(x) = x^6-x^5-(2a-n+5)x^4+(2a-n+1)x^3+2(5a-3n+5)x^...
- 199
1
vote
1
answer
115
views
Equivalent definition of Spin group in terms of automorphisms
Let $\mathrm{Cl}(\mathbb{R}^n)$ denote the (real) Clifford algebra on $\mathbb{R}^n$ with respect to the Euclidean inner product. Let $\mathrm{Cl}^0({\mathbb{R}^n})$ denote the even part of $\mathrm{...
- 1,903
2
votes
0
answers
135
views
Tensor product of finite extensions of $\mathbb{Q}_p$
Consider the tensor product of finite extensions of a field $F$ of characteristic zero. (I am interested in the case $F=\mathbb{Q}_p$.)
$(1)$ If $M$ is a finite Galois extension of $F$ with Galois ...
- 622
1
vote
0
answers
148
views
Vanishing (infinite) tensor products
Since the advent of free probabilities and QFT, infinite tensor products of $R$-associative algebras with units has become more familiar to the working mathematician.
Starting from the (permuting) ...
- 3,738