Questions tagged [group-algebras]
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5
votes
0answers
146 views
Slightly noncommutative Nakayama's lemma?
Nakayama's lemma asserts the following. If $R $ is a commutative ring with an element $s$, and $M$ is a finitely generated module such that $sM = M$, then there exists $r \in R$ such that $rM =0$ and $...
2
votes
0answers
57 views
Uniserial modules for group algebras
Recall that a module is uniserial in case it has a unique composition series.
Let $G$ be a finite group and $kG$ its group algebra, that we assume is not semi-simple.
Questions:
Can uniserial ...
3
votes
0answers
50 views
Zero divisors with support size 3 in complex group algebras of residually finite groups
Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\beta$ is a non-zero element of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ such that $1\...
6
votes
1answer
193 views
Zero divisors in complex group algebras of residually finite groups
Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ ...
5
votes
0answers
151 views
Are these element in a group algebra of a torsion-free group zero divisors?
Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)?
$$1+x+y,\quad 4+x+x^{-1}+y+...
1
vote
0answers
130 views
Idempotents in Group Algebras
What is known about idempotents in Lie group algebras (such as on the classical Lie groups)? Specifically the self-adjoint ones. Is there anything interesting to say? I haven't been able to find much ...
11
votes
1answer
162 views
Group rings such that every (countably generated) module has a maximal submodule
Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma.
I am interested in the following question, with two variants.
...
13
votes
3answers
394 views
Is this sum of cycles invertible in $\mathbb QS_n$?
I am interested the following element of the group algebra $\mathbb{Q}S_n$:
\begin{align}
\phi_n=2e+(1\ 2)+(1\ 2\ 3)+\dotsb+(1\ldots n)
\end{align}
where $e$ is the identity permutation. My question ...
7
votes
0answers
118 views
Torsion in a tensor product over a group ring
Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra.
Is it true ...
11
votes
0answers
308 views
“Small” zero divisors in $\mathbb C[\mathbb Z/p\mathbb Z]$
If $p$ is a prime, and $a,b$ are non-zero elements of the group algebra $\mathbb C[\mathbb Z/p\mathbb Z]$ satisfying $a\ast b=0$, then $$|{\rm supp}\ a|+|{\rm supp}\ b|\ge p+2.$$ This is easy to prove ...
0
votes
1answer
178 views
Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
3
votes
0answers
124 views
Orthogonal basis for decomposition of induced representation of derangements
Background
Let $V_n$ be the $\mathbb{C}$-module spanned by the set of derangements (permutations with no fixed points) inside the group ring of $S_n$. We make $V_n$ into a $\mathbb{C}S_n$-module ...
0
votes
0answers
140 views
Intersection of two subspaces of a Hilbert space
Background:
Let $D$ be a Klein Four group and consider free product $Z/2Z\star D=<a,b,c,d|a^{2}=b^{2}=c^{2}=d^{2}=bcd=1>$. Now we consider group algebra generated by $Z/2Z\star D$ with inner ...
8
votes
1answer
424 views
Kaplansky conjecture (consequences)
The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements.
Questions
Do we assume that $F$ has any ...
0
votes
0answers
60 views
Unit group of $\mathbb{F}_2[D_6\times C_\infty^n]$
Let $\mathbb{F}_2$ denote the finite field on two elements, $D_6$ the dihedral group of order $6$ and $C_\infty^n$ the direct product of $n$ copies of the infinite cyclic group $C_\infty$. I have a ...
7
votes
2answers
450 views
About the classification of commutative and of cocommutative, fin. dim. Hopf algebras
I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
3
votes
1answer
165 views
Elements in the group von Neumann algebra which are not summable
Let $G$ be a discrete group. I am wondering if there is a recipe which can be applied to find elements in the group von Neumann algebra which are not absolutely summable, i.e. $T \in VN(G)$ while $T \...
8
votes
1answer
249 views
Real rank zero of group $C^*$-algebras
The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...
1
vote
1answer
236 views
Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group
I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$.
Ultimately, I'm interested in finding a ...
11
votes
1answer
579 views
When is the integral group ring Noetherian?
The integral group ring of a polycyclic-by-finite group was shown to be Noetherian by Philip Hall. Are there any other known examples?
11
votes
0answers
179 views
Must nonunit in group algebra of free group generate proper two-sided ideal?
Let $F$ be a free group and $k$ be a field. If $x$ is an element of
the group algebra $k[F]$ that is not a unit (equivalently, that is not
a nonzero scalar multiple of an element of $F$), must the 2-...
3
votes
1answer
168 views
Units in a finite semisimple group algebra
Let $G$ be a finite group and $k$ a finite field, with the characteristic of $k$ not dividing the order of $G$. Then $kG$ is a finite semisimple group algebra with the interesting property that an ...
3
votes
1answer
133 views
Intersection of Maximal Left Ideals with Finite Dimensional Quotient
Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I \...
2
votes
0answers
198 views
(Non trivial) coidempotents(Co-$K$-theory)
I was interested to know about coalgebraic version of "Idempotents".
So I seached the web and I found the following interesting post :
https://math.stackexchange.com/questions/689322/co-idempotents-...
1
vote
1answer
223 views
$Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$
I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say
$$\mathbb{Z}^2\rtimes_n\mathbb{Z}=\...
9
votes
2answers
530 views
Are the reduced group Von Neumann algebra/ Group $C^{\ast}$ algebra functorial in the case of LCH groups
Let $G$ be a LCH group and $\mu$ be its left Haar measure. Call $\lambda_G : G \to U(L_2(G,\mu))$ the left regular representation. We can define the reduced $C^{\ast}$ algebra and reduced Von Neumann ...
11
votes
0answers
386 views
Group rings isomorphic over F_p, but not over Z_p ?
Suppose given a prime $p$.
Question: Do there exist finite groups $G$ and $H$ such that ${\bf F}_p G$ is isomorphic to ${\bf F}_p H$, but such that ${\bf Z}_p G$ is not isomorphic to ${\bf Z}_p H$ ?
...
5
votes
0answers
1k views
Is the radical of a homogeneous ideal homogeneous?
Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this generality,...
0
votes
0answers
131 views
Are all of compact support functions of $A(G)$ in its abstract Segal algebras?
Let $G$ be a locally compact group. We know that if $G$ is abelian and $\cal F$ implies the Fourier transform, for every Segal algebra of $G$ say $S^1(G)$, ${\cal F}S^1(G)$ is an abstract Segal ...
15
votes
2answers
901 views
Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?
Due to the first (and very helpful) answer I received, I've reformulated the question a little: $G$ and $H$ are now assumed to be $p$-groups.
Let $p$ be a prime, and let $\mathbb{F}_p$ be the field ...
2
votes
2answers
170 views
vanishing of certain product in group algebra
Let $G$ be a finite abelian group. When $\prod_{g\in G\setminus 1} (1-g)$ vanishes in (say, complex) group algebra of $G$?
It is easy to see that for cyclic group $G$ such product does not vanish, ...
6
votes
2answers
2k views
The functoriality of group C* algebra structure
Let $G$ and $H$ be discrete groups and $f:G \rightarrow H$ be any homomorphism of these groups. I have three questions about it:
1) How to prove the functoriality of the construction of universal $C^*...
23
votes
3answers
3k views
Splitting the determinant polynomial into linear factors - a Dedekind problem
Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial
$\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq i\...
0
votes
0answers
368 views
Amenability of an “almost Hamiltonian” group
Here is another interesting question that I can't answer on my own.
Let $G$ be a countable, discrete group such that for any subgroup $H$ of $G$ and any element $s$ of $G$ we have $[H : sHt]$ is ...
4
votes
2answers
493 views
Reference for von Neumann algebras coming from a group algebra twisted by a 2-cocycle?
I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce.
Does someone know some good references (article, book)? It would be very helpful for me.
To be more precise, ...
5
votes
2answers
743 views
Universal characterization and explicit description of elements of the group von Neumann algebra and the crossed product
Group von Neumann algebras and crossed products for a locally compact group G
can be constructed in many different ways.
For example, one can take the von Neumann algebra generated
by certain ...
12
votes
3answers
2k views
The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$
Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
15
votes
3answers
2k views
Is the group von Neumann algebra construction functorial?
Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set ...
