# Questions tagged [group-algebras]

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### 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

**0**answers

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

**0**answers

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

**1**answer

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

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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

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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

**1**answer

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

**3**answers

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

**0**answers

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**

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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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

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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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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**

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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

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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**

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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

**2**answers

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

**2**answers

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

**2**answers

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

**3**answers

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

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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

**2**answers

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

**2**answers

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

**3**answers

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

**3**answers

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 ...