Questions tagged [group-algebras]
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64
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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 $\mathbb{C}G$ ...
- 397
3
votes
1
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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_{...
- 817
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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 ...
- 141
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92
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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 ...
- 1,160
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63
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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 ...
- 32.2k
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63
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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 ...
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2
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103
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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 ...
- 103
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93
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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 ...
- 1,073
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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 ...
- 25k
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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 ...
- 141
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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. ...
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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 ...
- 79
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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
$$
\pi (\sigma_1, \dots, \sigma_n)\pi^{-1} := (\pi \sigma_1 \pi^{-1}...
- 219
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Nilpotent elements of index $2$ in group algebra $FA_4$
Let $A_4 = K_4 \rtimes C_3$ be alternating group on $4$ symbols and $F$ be finite field containing $4$ elements. By definitions of group algebra and augmentation ideal, there exist a natural map $$\...
- 381
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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 ...
- 4,959
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63
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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 ...
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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 ...
- 381
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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 ...
- 765
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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 ...
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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 ...
- 6,162
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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 ...
- 1,984
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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 ...
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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. ...
- 25k
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1
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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 $...
- 1,075
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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 $\...
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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 $...
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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 ...
- 25k
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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\...
- 3,698
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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 $G$ ...
- 3,698
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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 $\mathbb CG$)?
$$1+x+y,\quad 4+x+x^{-1}+y+...
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Idempotents in Group Algebras
What is known about idempotents in Lie group algebras (such as on the classical Lie groups)? Specifically the self-adjoint ones. Is there anything interesting to say? I haven't been able to find much ...
- 1,073
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Group rings such that every (countably generated) module has a maximal submodule
Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma.
I am interested in the following question, with two variants.
...
- 36.6k
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Is this sum of cycles invertible in $\mathbb QS_n$?
I am interested the following element of the group algebra $\mathbb{Q}S_n$:
\begin{align}
\phi_n=2e+(1\ 2)+(1\ 2\ 3)+\dotsb+(1\ldots n)
\end{align}
where $e$ is the identity permutation. My question ...
- 380
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Torsion in a tensor product over a group ring
Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra.
Is it true ...
- 1,378
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"Small" zero divisors in $\mathbb C[\mathbb Z/p\mathbb Z]$
If $p$ is a prime, and $a,b$ are non-zero elements of the group algebra $\mathbb C[\mathbb Z/p\mathbb Z]$ satisfying $a\ast b=0$, then $$|{\rm supp}\ a|+|{\rm supp}\ b|\ge p+2.$$ This is easy to prove ...
- 22.4k
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Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$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 ...
- 387
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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 ...
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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 ...
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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 ...
- 611
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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....
- 7,608
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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 \...
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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 ...
- 379
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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
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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?
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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-...
- 3,881
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votes
1
answer
270
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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 ...
- 31
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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 \...
- 2,761
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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 "...
- 9,962
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(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-...
- 295
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$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}=\...
- 1,498