Questions tagged [group-algebras]

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Hochschild cohomology of a group algebra

Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
Qwert Otto's user avatar
1 vote
2 answers
174 views

Prove that the ideal of $\mathbb{C}G$ generated by a family of elements $\lbrace p_i\rbrace_{i=1}^n$ is equal to $\mathbb{C}G$

Given a finite abelian group $G$ consider the group algebra $\mathbb{C}G$ and a set $\mathcal{P}=\lbrace p_i\rbrace_{i=1}^n$ of elements of $\mathbb{C}G$. Define $I$ to be the ideal of $\mathbb{C}G$ ...
Marcos's user avatar
  • 577
3 votes
1 answer
306 views

A generalisation of induced representations

Let $G$ be a finite group, and $H\subseteq G$ a subgroup. Let $F$ be a field. Let $W$ be a finite-dimensional $F[H]$-module. Let $T$ be a left transversal of $H$ in $G$. Then we can define: $W^G=\sum_{...
semisimpleton's user avatar
1 vote
1 answer
122 views

Example of a group algebra with commutative Jacobson radical

I am searching a simple example of a finite group $G$, so that the Jacobson radical $J(FG)$, of group algebra $FG$ is commutative, where $F$ is a finite field. I know example for that if $G$ is any ...
neelkanth's user avatar
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4 votes
0 answers
96 views

A weaker version of a theorem of P. Hall on noetherianity of $G$-modules

Recall that a group $G$ is polycyclic if it has a finite series of subgroups $G=G_0 \rhd G_1 \rhd \cdots \rhd G_l =1$ for which each factor $G_{i-1}/G_i$ is finite cyclic or infinite cyclic. A group ...
M.Ramana's user avatar
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7 votes
0 answers
97 views

Optimizing computations with nilpotents in a group algebra

Of course, I have a very concrete problem at hand, which has been vexing me for about a year now. But let me start with a question that has a better chance of having been answered. Let $G$ be a ...
darij grinberg's user avatar
0 votes
0 answers
63 views

Identities for operators in group algebras

Let C[G] be a group algebra for (typically) infinite noncommutative group G. fix f,g -- functions $f,g : C[G]\times C[G] \to C[G]$. Let us consider the family of operators on $C[G]$ such that for the ...
Andronick Arutyunov's user avatar
0 votes
2 answers
129 views

Examples of isomorphic non-equivalent twisted group algebras

Let $F$ be a field, $G$ be a finite group and $\alpha \in Z^2(G, F^*)$ . The twisted group algebra $F^{\alpha}G$ is a $F$-algebra with $F$ vector basis, $\{\bar g : g \in G \},$ and multiplication ...
Eloon_Mask_P's user avatar
4 votes
0 answers
96 views

Convolution algebra of a simplicial set

Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their ...
Josh Lackman's user avatar
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4 votes
1 answer
761 views

When is a group algebra Koszul?

Let $KG$ be a group algebra of a finite group $G$ such that the characteristic of $K$ divides the group order. Question: When is a block of a group algebra (or even the whole group aglebra) a Koszul ...
Mare's user avatar
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1 vote
0 answers
68 views

The influence of the derived subgroup of the unit group of a group algebra

Let $FG$ be a group algebra in which $K$ is a field and $G$ is a group. Suppose that every element in the derived subgroup $\mathcal{U}(FG)'$ of the unit group $\mathcal{U}(FG)$ of $FG$ is a ...
Tran Nam Son's user avatar
2 votes
0 answers
239 views

Existence of $\sqrt{2}$ in a finite group algebra over $\mathbb{Q}$

I cannot find a finite group $G$ such that $\exists x\in \mathbb{Q}[G]$ with $x^2=2e$, where $\mathbb{Q}[G]$ is the group algebra of $G$ over $\mathbb{Q}$. I also could not prove it does not exist. ...
Hugo MTV's user avatar
  • 143
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1 answer
425 views

Is any finite-dimensional algebra a sub-algebra of a finite-group algebra?

For $A$ a finite-dimensional algebra over a field $K$ Does there exist a finite group $G$, such that $A$ is a sub-algebra of $K[G]$ ? Where $K[G]$ denotes the group-algebra of $G$ over $K$. In case ...
Hugo MTV's user avatar
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2 votes
0 answers
124 views

invariant decomposition of $\mathbb{C}[S_k^n]^{S_k}$

Denote $S_k^n = \underbrace{S_k \times \dots \times S_k}_{n \text{ times}}$ and let $S_k$ act on $S_k^n$ conjugate diagonal, so that $$ \pi (\sigma_1, \dots, \sigma_n)\pi^{-1} := (\pi \sigma_1 \pi^{-1}...
Felix Huber's user avatar
1 vote
1 answer
128 views

Nilpotent elements of index $2$ in group algebra $FA_4$

Let $A_4 = K_4 \rtimes C_3$ be alternating group on $4$ symbols and $F$ be finite field containing $4$ elements. By definitions of group algebra and augmentation ideal, there exist a natural map $$\...
HIMANSHU's user avatar
  • 381
6 votes
0 answers
150 views

Subalgebra of group algebra generated by idempotents

Let $G$ be a finite group, and let $A$ and $B$ be two abelian subgroups of $G$. Let $K$ be a number field such that all characters of $A$ and of $B$ take values in $K$. Let $\mathcal{O}_K$ be the ring ...
Ehud Meir's user avatar
  • 4,969
2 votes
0 answers
65 views

Wedderburn decomposition of wreath product of cyclic p-groups

Let $G$ be wreath product of cyclic group of prime order $p$ by itself, i.e. $G=C_p \wr C_p$, where the action of $C_p$ is taken as cyclic permutation on generators of first $p$ cyclic groups. Can we ...
HIMANSHU's user avatar
  • 381
1 vote
0 answers
146 views

Wedderburn decomposition of semisimple group algebras

Let $G$ be a finite $p$-group. What can we say about the Wedderburn decomposition of the group algebra $FG$? Here $F$ is a finite field of characteristic co-prime to $p$. Can we say something in the ...
HIMANSHU's user avatar
  • 381
6 votes
2 answers
352 views

Is there a countable discrete infinite group $G$ over which the group algebra $\mathbb{C} G$ is semisimple?

I am seeking for an Artin $k$-algebra (especially for group algebra) which is infinite-dimensional over some field $k$. It's known that any complex group algebra has trivial Jacobson radical. So I ...
Master Gang's user avatar
9 votes
1 answer
520 views

Units of group algebra of dihedral group

Question: Can we fully describe the group of units (=invertible elements) $(KG)^\times$ of the group algebra $KG$ for $K=\mathbf{F}_2$, $G=D_\infty=\langle s,t|s^2=t^2=1\rangle$, the infinite ...
YCor's user avatar
  • 60.1k
26 votes
4 answers
2k views

Units in the group ring over fours group after Gardam

Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$. This conjecture asserted that if $\Gamma$ is a ...
Ville Salo's user avatar
  • 6,337
5 votes
0 answers
117 views

Is the group Hopf algebra left and right adjoint?

Suppose that $G$ is a group and $k$ is a field. Then it is well known that the group ring (group algebra) functor $k[\bullet]$ is left adjoint to the group of units functor, the latter of which ...
Mark.Neuhaus's user avatar
  • 2,004
0 votes
1 answer
301 views

Group algebras and group automorphisms

Say, we have a countable ICC group $G$, a Hilbert space $H$ with a basis indexed by the group elements, the group algebra generated by the left regular representation of $G$ on this Hilbert space, and ...
Chilperic's user avatar
  • 111
12 votes
0 answers
510 views

Does $\mathrm{Ext}^1(M,M) \neq 0$ imply $\mathrm{Ext}^2(M,M) \neq 0$?

$\DeclareMathOperator{\Ext}{\operatorname{Ext}}$The first question is about group algebras: Question 1: Let $A=kG$ be a group algebra (with $G$ finite) and let $M$ be an indecomposable $A$-module. ...
Mare's user avatar
  • 25.7k
2 votes
1 answer
318 views

Name and properties of this combination of group algebra and semidirect product?

Given a field $k$, a group $G$, and a homomorphism $\phi : G \to \mathrm {Aut} (k)$, we can define a ring $\widehat {k [G]}_\phi$ as follows: As an abelian group it is isomorphic to the group algebra $...
Itai Bar-Natan's user avatar
3 votes
2 answers
369 views

Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different

In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
worldreporter's user avatar
5 votes
0 answers
472 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 $...
David Handelman's user avatar
2 votes
0 answers
120 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 modules ...
Mare's user avatar
  • 25.7k
3 votes
0 answers
59 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\...
Alireza Abdollahi's user avatar
6 votes
1 answer
345 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 $G$ ...
Alireza Abdollahi's user avatar
5 votes
0 answers
266 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+...
Meisam Soleimani Malekan's user avatar
1 vote
0 answers
191 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 ...
Josh Lackman's user avatar
  • 1,188
14 votes
1 answer
557 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. ...
Benjamin Steinberg's user avatar
13 votes
3 answers
608 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 ...
JeremyR's user avatar
  • 380
8 votes
0 answers
330 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 ...
Andrei Jaikin's user avatar
11 votes
0 answers
330 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 ...
Seva's user avatar
  • 22.8k
0 votes
1 answer
313 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 ...
Jack's user avatar
  • 397
3 votes
0 answers
185 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 ...
Jonathan Rayner's user avatar
0 votes
0 answers
430 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 ...
Jack's user avatar
  • 397
9 votes
1 answer
869 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 ...
Nguyen lan Lee's user avatar
10 votes
3 answers
1k 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....
Konstantinos Kanakoglou's user avatar
3 votes
1 answer
177 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 \...
Mahmood Al's user avatar
9 votes
1 answer
341 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 ...
M.fouladi's user avatar
  • 389
1 vote
1 answer
367 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 ...
Liam_V's user avatar
  • 11
11 votes
1 answer
841 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?
Peter Kropholler's user avatar
16 votes
1 answer
331 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-...
Dave Witte Morris's user avatar
3 votes
1 answer
277 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 ...
groupalgebra's user avatar
3 votes
1 answer
213 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 \...
Hans's user avatar
  • 2,863
28 votes
1 answer
2k views

Does GL_n(Z) have a noetherian group ring?

Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL_n(Z)$ been considered in the literature? Motivation: a recent trend has been to study "...
Steven Sam's user avatar
3 votes
0 answers
283 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-...
Ali Taghavi's user avatar