User trew - MathOverflow

Classification of Hopf algebra with exactly two 1-dimensional modules

2013-03-19T20:56:35Z

Is there a classification of indecomposable non-semisimple finite dimensional Hopf algebras with exactly two 1 dimensional modules? If not, is there one when all simple modules are 1 dimensional and there are only two simple modules? If there is no full answer, I'm also interested in some examples of such Hopf algebras. By the way: is there a formula for the number of simple modules involving the structure of the group of grouplike elements like in the case of a group algebra? Thank you for answers. edit: Im also interested for non-semisimple indecomposable k-algebras whose basic algebra is a hopf algebra(in general or more special with only 2 simple modules).maybe there is some kind of criteria?

Answer by trew for An injective smooth function with injective differential must have a continuous inverse?

2013-03-19T21:04:17Z

Yes this is true since your map is open by the local diffeomorphism theorem. We can find for every $ x \in U$ an open $U_x $ such that f is a diffeomorphism from $U_x$ to $f(U_x)$.(i think we need that f is in $C^{1}$ not just differentiable)

heller functor of finite order

2012-02-06T16:31:40Z

Hi, let A be a finite dimensional selfinjective algebra. Assume mod A is periodic.That is for every finite dimensional module with no projective summand we have: $ \Omega^{n} (M) =M $ for some n ,where $ \Omega $ is the Hellerfunctor(giving the kernel of a projective cover of M ). When is it true that $ \Omega $ is of finite order,that is $ \Omega^{n} $ is isomorphic as a functor to the identity functor of the stable category of modules ?(Assume there is a common biggest period for all module if necassary)

Thanks for help

A question about the existence of a specific extension of a character.

2011-07-06T18:21:41Z

general situation:
Let $ N \leq G $ be a subgroup,and let $ \chi \in Irr(G) $ be an irreducible character of G such that $\chi_N $ is not irreducible( i dont think that this is really needed) and let $ \psi \in Irr(N) $ be an irreducible constituent of $\chi_N$. Assume there is an subgroup H of G with $ N \leq H \leq G $ to which $ \psi $ is extendible. Then there is a character $ \mu \in Irr(H) $ with $ \mu_N =\psi$ and $[\chi_H , \mu] \neq 0.$ NOTE: the problem here is to find such an extension with $[\chi_H , \mu] \neq 0.$ Is this true in general?
I have the following specific situation: Let J be a nilpotent finitedimensional algebra over a finite field. Then G=1+J(called finite algebra group) is a p-group,$N=1+J^{2}$ is a normal subgroup. In the paper "On characters and commutators of finite algebra groups" written by Halasi,he writes:
Lemma 3.1: "Let G=1+J be a finite Algebra group and $\chi \in $ Irr(G).Then the following properties are equivalent:
1.There exists a proper algebra group H and $\varphi \in Irr(H)$ such that $\varphi^{G}=\chi$.
2.$\chi_{1+J^{2}}$ is not irreducible.
Proof:...
Assume now that $\chi_{1+J^{2}}$ is not irreducible and let $\psi \in Irr(1+J^{2})$ be a constituent of $\chi_{1+J^{2}}$.Let H be a maximal algebra subgroup such that $\psi $ is extendible to H.Then H $ \neq $G.We choose a $\varphi \in Irr(H) $such that $\varphi$ is an extension of $\psi$ and $\varphi$ is a constituent of $\chi_H$...."
I wonder why there is such a $\varphi$ .Is it true in the general situation or what properties of 1+J and $1+J^{2}$ or H(being maximal) are used here? Thanks for helping

Modular Representation Theory of the generalized Quaternion group

2012-01-16T22:46:21Z

Hi, here (http://www.math.ku.dk/english/research/conferences/group.actions2011/problem.session.seattle96-maybe.pdf/ Conjuncture 3.6) i read that the indecomposables of the generalized Quaternion group,which has tame representation type,are still not classified. Is this still the case and if yes what do we know about the representation theory of those groups? Thanks for the help

What do we know about periodic modules in p-groups?

2011-12-15T13:40:07Z

Hi,

a module in KG,where G is a p-group and K a field of characteristic p, is called periodic if $ \Omega^{n} M = M $, for a natural n. In general the full subcategory of periodic modules seems to have also wild representation type( http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ijm/1255989005 ). I wonder if there are still some interesting results about periodic modules. So I search for a kind of up-to-date survey paper listing such results. some questions are:

In which dimensions can a module of period n occur?(results like in this paper: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ijm/1256048241 where it is proven that a power of p divides the dimension)

Which periods can occur in a given group?

Is there any interesting relation of the subcategory of periodic modules and the pure group structure?

Thank you

edit: Another question: Can we give an example of a periodic module in an arbitrary KG?Maybe there is a canonical construction.

edit2: after reading parts of benson im a bit confused.for example in the introduction he says compelextity 1 is equivalent being periodic.But he says something else in a later theorem. Is the following correct?: M has complextity 1 iff

$M_E $ has maximal complextity 1 for an elementar abelian subgroup E of G iff

M is a direct sum of indecomposable periodcis and projectives iff

in the minimal projective resolution the terms have bounded dimension.

Dimension of the radical of the tensor product

2011-12-15T13:26:50Z

Hi, are there any useful formulas for $dim_{k}(rad(M \otimes_{k} N)) $(for example relating it do $dim_{k} M $ and $dim_{k} N $ and $dim_{k} rad(M)$and $dim_{k} radN $) where M and N are finite-dimensional KG modules(Characteristic p) and G is a finite group with p divides the order(or some interesting specialisation like a p-group if you want) Thank you!

Answer by trew for Ways to prove the fundamental theorem of algebra

2011-09-09T20:05:22Z

There seems to be a new elementary proof using only bolzano-weierstraß and an inequality: http://de.arxiv.org/PS_cache/arxiv/pdf/1109/1109.1459v1.pdf

Irreducible representations of the unitriangular group

2011-06-19T12:43:07Z

Hi, I wonder how much is known about the irreducible representations of the nxn unitriangular group over a finite field with q elements.
I know that all characterdegrees are a power of q and all degrees which occur are known.But what is known about the irreducible representations or the complete charactertable at least for small values of n?For example is the charactertable for n=3 known? Thanks for helping

Answer by trew for Irreducible representations of the unitriangular group

2011-08-14T12:21:29Z

Here is what I found out about the characters when n=4.I dont know if thats interesting and how to get the actuall irreducible characters then.Maybe someone has an idea: There are $ q^{3} $ linear and from http://fourier.math.uoc.gr/~marial/uni1.published.pdf there are $q^{3}-q $ characters of degree q and $q(q-1)$ characters of degree $q^{2}$. Now lets look at the characters of $G/Z(G)=1+J/J^{3}$ ,where Z(G) is the center of the group and J are the lower triangular matrices with zeros on the diagonal.This is again an algebragroup,so we have: $q^{5}=q^{3}+aq^{2}+bq^{4}$,where a is the number of the degree q characters of G/Z(G) and b the number of degree $q^{2}$ characters. Assume $ \phi$ is a degree $q^{2}$ character in G/Z(G),then $ [ \phi_{Z(G/Z(G))} , \psi] \neq 0$ ,for a linear character $\psi $ of Z(G/Z(G)).But then since $\psi$ is G/Z(G) invariant as a character of the center and using Clifford: $\phi_{Z(G/Z(G))}=q^{2} \psi$ and then $ \psi^{G/Z(G)} = q^{2} \phi +...$ which is not possible because of $ \psi^{G/Z(G)}(1)=q^{3} < q^{4} =q^{2} \phi(1)$. So we have b=0 and a=q^{3}-q from $q^{5}=q^{3}+aq^{2}$.So all the degree q characters of G are also the degree q characters of G/Z(G). Let now $\chi$ be a character of degree q of G/Z(G) then we can choose a linear character $\psi$ of Z(G/Z(G)) with $[\chi_{Z(G/Z(G))} ,\psi] \neq 0 $ and again as above: $ \psi^{G/Z(G)} = q^{2} \chi +...$ Since $(1_{Z(G/Z(G))})^{G/Z(G)}$ has all linear characters as constituents $\psi^{G/Z(G)}$ can only have degree q irreducible constituents.So for all(there are $q^{2}-1$ such) the nontrivial linear characters $\psi_k $ of Z(G/Z(G)),we have : $ (\psi_k)^{G/Z(G)} = q \sum\limits_{i=1}^{q} {\chi_i}$. Similary one can show that for all nontrivial linear characters $\vartheta_k $ of Z(G) one has: $(\vartheta_k)^{G}=q \sum\limits_{i=1}^{q} {\phi_i} $,where $\phi_i$ are degree-$q^{2}$ characters of G.

Answer by trew for A question about the existence of a specific extension of a character.

2011-07-06T23:24:24Z

Hi, I just saw a theorem in a paper of Isaacs which could be useful. Assume additional to the general case(with same notation)that N is normal in H and H/N is abelian. Then the lemma:

"Let N be a normal subgroup of H and H/N abelian.Let $\vartheta \in Irr(H) $and $\psi \in Irr(N)$ and $[\psi,\vartheta_N] \neq 0$.Then every Z $\in Irr(H)$ with $[Z_N,\psi] \neq 0$ has the form $Z=\lambda \vartheta$ for a linear $\lambda \in Lin(H/N) $ "

tells us that every irreducible constituent of $ \psi^{H} $ has the form $\lambda \vartheta$,where $\lambda \in Lin(H/N)$ because if Z is an irreducible constituent of $ \psi^{H} $ then $[Z,\psi^{H}]=[Z_N,\psi]$.We can then write $ \psi^{H} $ as a sum of all different $\lambda \vartheta$ with multiplicity one,because $[\psi^{H} , \lambda \vartheta]=[\psi, (\lambda \vartheta)_N]=[\psi,\psi]=1$.

We have $[\chi_H , \psi^{H}]=[\chi,\psi^{G}]=[\chi_{N},\psi] \neq 0$ Choose an irreducible constituent of $ \psi^{H} $,called $\phi$ with $[\phi,\chi_H] \neq 0.$ On the one hand we have: $(\psi^{H})_{N} = |H:(N)| \psi $ and on the other (since $[\phi,\psi^{H}]=[\phi|{N} , \psi]$):

$(\psi^{H})_{N} = a \phi +etc$ Comparing these 2 gives: $\phi|{N}=c \psi$ for natural a and c and etc as a combination of other caracters. we have to show that c=1 to finish the proof. $c=[\phi|{N} , \psi]=[\phi,\psi^{H}]$,but we showed above that $\psi^{H}$ has only irreducible constituents with multiplicity one. I am thankful for proofreading this or giving hint to avoid the additional assumptations.

p-groups realisable as 1+J,where J is a nilpotent finite F-Algebra

2011-06-17T22:23:09Z

There is a functor F: {finite nilpotent Algebra over a finite field F} -> {finite p-groups} sending J to 1+J.(You can think of J as the Jacobsonradical of 1F+J) and sending f:A->B to F(f):1+A -> 1+B with F(f)(1+a)=1+f(b). I want to know which classes of p-groups are realisable as 1+J and maybe if you see some interesting properties of the functor F. For example if J^2 = 0,then G=1+J is elementary abelian and if J^3=0 then G=1+J is special. Such realisations could be interesting since,we have for example in general G' $\leq $ 1+J^2 and a series 1+J $\leq$ 1+J^2 ... $\leq$ 1+J^k =0 and there it is known that the characters of 1+J are induced from linear characters of a subgroup 1+A where A is a subalgebra of J.

Answer by trew for Good books in Modular Representation Theory

2011-02-19T15:58:20Z

Hi, a very elementary written book is Local Representation Theory by Alperin.The second book on finite groups by Huppert has also a big part about modular representation theory.(you should read the first book too,with an long introduction to representation theory in the semisimple case) a more advanced book is that of feit and a recent (2010) book is "Representations of Groups: A Computational Approach" by Lux and Pahlings.It has many nice examples with gap and the sporadic groups.You should also take a look for the books of curtis and reiner.The one from 1962 is easy to read and might be the best introduction.

Applications of the Theorem of Gelfand-Naimark

2011-01-16T20:12:24Z

Hi,
I am interested in the correspondence of algebraic results about C(X) (the space of continuous functions $X\to {\mathbb C}$(complex numbers) or $X\to {\mathbb R}$(real numbers) and topological properties of X,for example results like this .(Does by the way someone know what's the "deep result by bkouche" mentioned in the paper?)

Can you by the way use this result to prove interesting theorems with this translation
(like: -a manifold(or even a CW-complex) is paracompact
-the theorem of Tietze, etc?)

There are many such correspondences which are obtained by using Gelfand-Naimark but I couldn't find literature where you can find full details with all needed definitions and proofs.(I couldn't even find a proof of the categorical Gelfand-Naimark theorem in the nonunital case,only some sketches.) Does such literature exist for a beginner in this topic? The book "Basic Noncommutative Geometry" written by Khalkhali is a good source but omits most details (see page 16 for a little list).

So I would be glad if you can recommend to me a good book/link, or write a nice result here if it's not too complicated.

Example: X is connected iff C(X) has no idempotents because direct sums of subalgebras correspond to disjoint union of closed subspaces,since C is an equivalence.

Proving interesting theorems about S_n using its character table.

2010-12-01T21:56:50Z

Hi, i wonder if there are interesting proofs about $S_n$ (group theoretic or not) using its character table. Using the Murnaghan-Nakayama rule you can for example prove that for $n>4$ $A_n$ is the only normal subgroup of $S_n$ because there are no nonlinear characters $x$ and $g$(not 1) in $S_n$ with $ x(g)=x(1)$, since $x(1)>x(g) $ . Do you know any other nontrivial theorems about $S_n$ with a proof using its charactertable ?

Examples of nice families of irreducible polynomials over Z

2010-11-09T17:41:36Z

Hi, i search for irreducible polynomials over Z which have variable coefficients you can "choose". Since I found nearly nothing in books or the internet i hope you can help me. Here 3 examples: Let g a polynomial over Z with degree smaller then n/2 ,then: $g* (\prod_{i=1}^n (x-a_i)) -1 $ is irreducible if the a_i are all distinct. Here you can choose n the coefficients of g and the a_n so its a nice example. Another one,I found, is from Furtwängler : $x^4 (\prod_{i=1}^{n-4} (x-b_i)) -(-1)^n *(2x+4) $ where the b_i are strictly increasing.Can you generalize this example ?I think it should work also for some integers other than 2 and 4. Here is another nice example : http://mathoverflow.net/questions/18094/polynomial-with-the-primes-as-coefficients-irreducible

I try to find examples where its easy to control zeros modulo p of the some irreducible polynomials and its derivation for another problem.

A result about Characters of F-algebra groups

2010-09-28T17:16:21Z

Let G be an F-algebra group(G=1+J , where J is the jacobson radical of a finite dimensional F-algebra ,where F is a field of prime characteristic) In a paper of Isaacs ("Characters of groups associated with finite algebras" from 1995) there is a claim of Gutkin with a wrong proof.It says: Let x be an irreducible character of G ,then $ x=a^G $ ,where a is a linear character of some subgroup H<=G of the form : H=1+U where U is multiplicativly closed F-subspace of J. This result would be a generalisation of the result of Isaacs paper,but its unclear to me if it has been proven until today.Does someone know if the result is true and can recommand me a paper/link for more information?

Minimal conditions for the exponential law for compact-open topologies.

2010-09-25T18:37:00Z

What are the minimal conditions on three topological spaces $X,Y$ and $Z$ such that with the compact-open topology the map

$${(X^Y)}^Z \to X^{Y \times Z}$$

given by taking adjoints is a homeomorpism. The map sends $f: Z \to X^Y$ to $g:Y \times Z \to X$ by the relation $g(y,z)=f(z)(y)$.

This result is known for $Z$ Hausdorff and $Y$ locally compact. I'm interested in a proposition of the form the adjoint construction is a homeomorphism of mapping spaces if and only if some statement regarding the spaces $X$, $Y$ and $Z$. It would also be interesting to see some counterexamples,for example for Z not Hausdorff,etc.

Comment by trew

2013-03-26T18:37:16Z

sorry, i forgot to add that im looking for nonsemisimple and indecomposable algebras of the above type.

Comment by trew

2013-03-19T21:30:58Z

no im interested in any field.but if you have some examples over C you can post them too of course

Comment by trew

2013-03-19T21:17:04Z

en.wikipedia.org/wiki/Inverse_function_theorem the fact that the dimension must be the same follows from you assumption that Df(x):R^n -> R^m is invertible(so you can use linear algebra to see that n=m)

Comment by trew

2013-03-19T21:12:52Z

I think its called Inverse function theorem in english

Comment by trew

2012-02-06T17:49:01Z

thats right i guess.thanks

Comment by trew

2011-12-17T14:20:21Z

thank you for your edit!But I think it has period 1 for p=2.

Comment by trew

2011-08-14T17:45:56Z

I dont know what Gabriel thought but here is an answer why the term is used from: amazon.de/… page 42 says:"...There are two main reasons for using the term quiver rather than graph:the first one is that the former has become generally accepted by specialists;the second is that the latter is used in so many different contexts and even senses( a graph can be oriented or not,with or without multiple arrows or loops) that it may lead,for our purposes at east to certain ambiguities.

Comment by trew

2011-06-19T19:24:32Z

thanks,i know that result.it would be interesting to know what has been done for n=4,before i try ;) I know for example that all characterdegrees with their multiplicity is known.can someone by the way say what exactly "wild" means?

Comment by trew

2011-06-18T15:54:22Z

thank you.do you know a paper which contains most of the known results about those groups?by the way: if someone has an interesting example of a p-group not realisable as 1+J,I would be glad if you post it.

Comment by trew

2011-02-19T16:08:08Z

Ok,that sounds strange so i changed that.I dont know how to explain.Maybe just try to read it.

Comment by trew

2011-02-08T20:58:33Z

I think its an interessesting question wheter there is an exact solution

Comment by trew

2011-01-17T08:57:42Z

sorry,i forgot about your article ;)

Comment by trew

2011-01-16T21:54:43Z

Thank you!I know this from the book of pedersen.Its a very nice application and there is even a one-to-one corresponende of the compactifications of a locally compact space and essential extensions of $C_0 (X) $ described in the book of Khalkhali.

Comment by trew

2011-01-16T20:36:54Z

Thank you very much!

Comment by trew

2010-12-01T22:22:23Z

thank you!does someone know if you need to use that result to prove Murnaghan-Nakayama,which also gives you that the character table entries are integers?I think so.