Questions tagged [group-algebras]

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-1 votes
1 answer
215 views

tensorproduct, p-adic groupring [closed]

Suppose there is a cyclic group $G$ and a prime $p$. Why can one write $$ \mathbb{Z}_p[G] \cong \mathbb{Z}_p \otimes _\mathbb{Z} \mathbb{Z}[G]$$ Is this some theorem which has a name? Thanks for ...
9 votes
1 answer
355 views

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,...
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 ...
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 $\mathbb{C}G$ ...
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 ...
3 votes
1 answer
307 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_{...
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 ...
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 ...
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 ...
0 votes
2 answers
130 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 ...
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 ...
4 votes
1 answer
762 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 ...
16 votes
1 answer
332 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-...
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 ...
2 votes
0 answers
240 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. ...
5 votes
1 answer
427 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 ...
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}...
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 $$\...
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 ...
14 votes
1 answer
558 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. ...
2 votes
0 answers
66 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 ...
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 ...
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 ...
19 votes
1 answer
628 views

Group rings isomorphic over $\mathbf{F}_p$, but not over $\mathbf{Z}_p$?

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 ${\bf Z}_p H$ ? ...
9 votes
1 answer
524 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 ...
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. ...
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 "...
3 votes
2 answers
370 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 $\...
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 ...
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 ...
3 votes
0 answers
284 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-...
14 votes
2 answers
546 views

$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $\ell^1(G)$ is a flat (right) $\mathbb{Z}G$-module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by-finite group. Denote the ...
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....
2 votes
1 answer
320 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 $...
13 votes
3 answers
609 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 ...
5 votes
0 answers
473 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
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 ...
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\...
6 votes
1 answer
346 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
0 answers
268 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
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 ...
8 votes
0 answers
331 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
0 answers
332 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
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 ...
3 votes
0 answers
187 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
433 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 ...
9 votes
1 answer
870 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 ...
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 \...
0 votes
0 answers
372 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 ...
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^*...