Title: Cohomology of semi-direct product Lie algebras
Abstract: This is joint work with Dietrich Burde (University of Vienna, Austria). Intrigued by computations of Richardson, our goal is to compute the adjoint cohomology spaces of Lie algebras which are the semi-direct product of a simple Lie algebra s and an s-module. We present some theorems and conjectures in these cohomologies.
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According to myself whoops do you see my screen I see the screen very well yeah okay so shall I start introducing you I I think it’s time right yes you can please to my cell phone so yeah welcome everyone it’s a pleasure to have frish vaman with us
Today um giving a talk yeah freedle is an expert in chology and many other things but today’s talk will be about chology yeah about a joint chology of semidirect products of Le albas please immediately go ahead fre thank you very much Y and thanks the organizers for the
Opportunity to to speak in this seminar and I must apologize to those who heard already part of the talk in Lisbon last year but um okay I will speak about some new developments in the end okay my talk is joint work with di bord from the University of Vienna in
Austria and our starting point is um a conjecture of tamas p and so first of all we will talk about this uh I will talk about this conjecture unfortunately we are not ready yet to to prove it but from these considerations about the conjecture we
Um uh finally came up with a program to compute uh the chology of uh lee algebras which are semidirect products of uh semi simple or even simple Le algebra and uh module yeah and uh so this is what I um understand by Computing chology via representation Theory because the representation theory
Of the module comes into um these chology computations okay and so the the points one to four are the subject of our article already published article uh together with dri Border in 2023 and the the fifth point is about new computations which will hopefully somehow someday become another
Art okay let me start with pwi’s conjecture so um it is a conjecture about uh at least one lecture of the conjecture is um about finite dimensional complex Lee algebras and the conjecture states that um a finite Dimension complex Le algebra is semi simple if and only if the H1
With trivial coefficients is zero and all the hn with ad joint coefficients are zero so it is a um chological characterization of semi simple Le algebras unfortunately it’s conjectural for the moment and so in the remark okay um in the remark below I I um explain to you um what these uh
Chological conditions mean H1 with trivial coefficients being zero means that g is equal to its commutator subalgebra h0 with ad joint coefficient zero means that the center zero and H1 means that all derivations are inner and H2 that uh G is algebraically rigid this implies geometrically rigid but is not
Equivalent to it and so uh P’s Concor was originally stated in terms of lipnet homology but it boils down to this statement in ordinary chival irber chology okay and just because I’m a chology guy I I have two slides where I um recall um chival irber chology of a
Lee algebra to you so when you have a Le algebra G and a a g module M then you then you have a space of code chains these are the the the linear maps from the exterior products of G with values in the module and uh on these spaces of code
Chains you have a cound operator which uh is defined as uh these uh sum of sums yeah there’s one sum um with the the module action uh coming in and one sum with a with a bracket yeah so it’s depending uh on the bracket structure of the Lee algebra and on the
Module structure uh of M and so this Co boundary operator satisfies d^2 equals to zero and thus the the image of D is contained in the kernel of the next D and the chology spaces are then the the chival irber Lee algebra chology spaces of G with values in
M and so to come back to the uh to the these interpretations for the P conjecture the the uh I have these examples here the first um uh co- boundary operator going from C1 to C2 is ju with trivial values is given just given by the bracket and so
Uh the H1 is just linear forms on the Lee algebra which vanish on the bracket and so H1 being to equal to zero means that g is equal to its commutator subalgebra and so the zero SC boundary operator goes from yeah with ad joint coefficients uh goes from G to to one
Cod chains and uh is just given by the action and here for Adon Co efficients the actions the ction is given by the bracket thus the h0 is just the center and uh yeah uh for for the uh First Co boundary operator with ad joint coefficients one can easily see that the
H1 is derivations divided by Inner derivations yeah so outer derivations if you want to and uh what what is very important for us for the um for for the SQL is that we we need uh to know what it means for Le algebra to be unimodular this
Means that the traces of all these adjoint operators at x given by the bracketing by X are zero for all X in in G and this is for example the case if H1 of g c is equal to zero because in this case um G is equal to the commutator
Subalgebra and so this little X here can be written as a sum of commutators yeah but add is a morphism of Lee algebra so this makes it the trace of a sum of commutators of add yeah and and the trace of um a commutator is zero thus um in case H1 is
Zero the Le algebra is unimodal okay now uh what do we know about P’s conjecture um rather little things but it is connected to another subject which which was very important in the 1990s um it was um um um I forgot his name um famous djon mathematician
Um uh asking the question whether there are Lee algebras which are not semi simple and which have H1 with trivial coefficients equal to zero h0 with Edon coefficient zero and H1 with Edon coefficient zero uh flat MOSI flat oh I’m sorry Moshe plate asked this questions question and um there were
Answers given by angelopulos in 1988 and Ben s benadi in 1996 and uh they constructed examples which are not semi simple but which satisfy these conditions and for the to up to today uh to my knowledge Ben aadi’s example is the smallest one and
Uh in this line of thought um there is a recent paper very interesting paper by Garcia pulo and Salgado who show that if you suppose that the Ley subalgebra is simple not only semi simple but simple then uh benard’s example is indeed the smallest example yeah so there is still
Work some work to do in in order to make it um uh available to to all these sympathetic Lee algebras but um uh garia poo and Salgado came very close to it already Okay now let me come back to our contribution dri and mine contribution to uh to an approach to the P
Conjecture first of all let us um uh observe something which is very uh well known for sympathetic Lee algebras um for sympathetic Le algebras when you have an a billion radical then you must be semis simp yeah and so there’s a standard method of proof of it
Um the when you have the Ley DEC composition of of a Lee algebra of a finite dimensional complex Le algebra then uh this means that g is the semidirect product of this Ley subalgebra s and the radical yeah and and so the semidirect product bracket is
Of this kind here and so there’s a possibility to to define a derivation on G by putting it equal to zero on the Ley subalgebra and equal to the identity on the radic and then uh it is uh very uh easy computation to show that it is indeed a
Derivation and this is what I did here but if the Le algebra is sympathetic it means that all derivations are inner so this derivation must be an inner derivation an inner derivation meaning a deration of this form at X yeah but G is unimodular yeah
So it has zero Trace zero Trace but it is the identity on the radical so the radical must be zero yeah so the um observe that this reasoning does not work when you have U the bracket of u and v here in the in the second term of the semidirect
Product bracket yeah if you have the bracket of u and v in the second term you would would have it uh in all three factors here and so two times in the in the factors below and only one time in the factor upstairs here yeah so it needs really uh sympathetically with an
A billion radic okay and so with DED we thought about this proof and um I came came up with um with an with a variant of this proof which shows that when you have a sympathetic Lee algebra so with uh solvable radical n if the central extension of this Neil potent Le algebra
N is split then G must be semi simple and the proof is exactly the same meaning um write down what it means to be um so first the semidirect product in the L decomposition and then the the Neil potent part you can decompose in terms of the central extension yeah so
You have three components um and the last component is in the the center of N and then Define a derivation just by putting it equal to the identity on the center of n yeah and because because the central extension is split you can take as a cycle the zero
Co cycle and this makes it possible that this um this uh map D which is the projection on on the center is a derivation yeah when you have a non-trivial CO cycle it would not be a derivation but thanks to the fact that the you can take the co cycle to be zero
It will be a derivation and then the the rest is exactly the same the derivation must be inner um but the the Le algebra is unimodular so um the center must be zero but N is a Neil potent Lee algebra it must have a non-al center thus uh G must
Be sism yeah so this is kind of our contribution our very modest contribution to to an approach uh towards the p conject and let let me just say some words why I think that this can be an approach to to solve the P conjecture um the the P con is about Le
Algebras which have lots of chology spaces which vanish and here we we have a proposition which which tells you um one of these chology spaces is not Vanishing and it is this uh um this H2 of n with with uh value with the H2 of n n over ZN with value in ZN
Yeah and so there you have a non-al chology class and um I think I I didn’t uh have the time to to work this out in detail but I think that you can um uh um trans transgress this two class of n over Z of
N with values in ZN to a class A three class of G with values in G yeah this this is the challenge yeah and if you can do so then you solve the the P conject oh okay the it’s still hypothetical but this is what I would propose as an approach to the
P okay uh you see see we um kind of uh uh failed to to show the P conjecture together with Di and so we uh we did some other things on the positive side and so uh during this project we came um across for example Richardson’s paper where Richardson shows that there
Exists a rigid Lee algebra with non-trivial H2 with adjin coefficients and in this paper he he computes um uh yeah uh chology of uh semidirect products of SL2 with values in some modules yeah and so we we ask ourselves can we have a conceptual approach to to this kind of
Computations so this is um what we call then Computing chology with representation Theory or by representation Theory so in the last part of the talk I will only talk about Le algebras which are the semidirect product of a finite dimensional complex uh semi simple Lee algebra x s
With an S module finite dimensional s module which is looked at as an abilon Le algebra and so in order to compute the chology of of this kind of Lee algebra we we use the hor Shield s formula which tells us um that the chology uh the the degree N chology is a
Direct sum of tensor products of degree K and degree L chology such that K plus L is equal to n and it is the K chology of of S of the semi simple Le algebra s with trivial coefficients and the L chology of this trivial ailan Le
Algebra V with values in G and then you have to take the S invariance of it so uh so everything boils down to to compute this second tensor factor factor here and for this we use the long exact sequence in chology of the ailan Lee algebra
V with values in V with values which is a sub module of G and the quotient module G over V and then you can observe that the subm module V is a trivial V module because it’s anilan Le algebra and but the the quotient module G over V
Is also a trivial V module yeah but G is not in general G no not not only in general G is not a trivial V module unless V is trivial model okay and so for for the trivial Le algebras these chology spaces boil down to uh c-chain spaces yeah it’s just the
The linear Maps the N chology of V with values and V is just linear maps of the exterior product of of V the nth exterior product of V with values in v and then when you take the the S invariance of this then you come to S
Equivariant Maps yeah and here the the the representation Theory comes into game yeah you have to you have to evaluate these uh spaces here and therefore you have to know uh the the exterior products of your module as an S module and then uh have to know
Whether there is V or G over V meaning as an S module um isomorphic to S yeah whether s or V are in this in the decomposition into irreducibles of this n exterior product of V yeah this is kind of the challenge and this is what I mean by Computing chology with representation
Here okay uh yes so as we are using a long exact sequence it is all always easy to work at one end of the long exact sequence or on the other end yeah so on on the low dimensional end we can get uh propositions like this uh when
You have a simple s module then the H1 is equal has Dimension One and uh yeah from this um from this reasoning which I already um explained to you uh that you you looking at these home spaces between exterior products of V with values in s on the one hand and
With values in v on the other hand yeah and so from this reasoning you can have criteria whether the the the cas chology space is non zero and so one Criterion is here um if the K minus first exterior product of V does not contain a factor
Which is isomorphic to an ideal of s and uh if the case exterior power of V does not contain a factor isomorphic to V then the HK is uh non-rival okay but um so let me uh go to the other end of the long exact sequence
Meaning to not to low Dimension but to very high dimensions and there we have um uh the following proposition if G is n dimensional now uh still um still a semidirect product Lee algebra but we can even have um Neil potent radical it’s it’s not necessarily a billion but Neil potent is
Also uh okay with us then again under representation theoretical Criterion U if the S module n over the um derived algebra of n does not contain the trivial s module then hn is zero so let me show how how this works we always start with the hor Shield s
Formula so uh G is n dimensional so the the the uh the Neil potent radical is M dimensional and so s is n minus M dimensional and so the the the um extremal chology space hn which is of the dimension of the of the Lee algebra um in this direct sum from the
Hor Shield s formula there’s only one combination which can give something nonzero and this is h n minus M of s and uh hm of n with values in G and the s invariance of it yeah so the proof is not over um continues here so what we need
To compute is we need to compute hm of n with values in G and the S invariance we use the long exact sequence the long exact sequence stops here and um we know that uh hm of n with value in with trivial values is is one dimensional
Um and so when we now have a trivial module yeah the G Over N is a trivial n module uh then uh it is just uh S as an S module yeah so the S and variance are zero and this means this La VAR last space in the long exact sequence is zero
Now let us compute the the the space before the space we want to to know and this is this hm of n with values in n and the S invariance of it and for this we use P Duality yeah and this is a very nice
LMA uh the LMA is when you have a un modular Le algebra then The Duality is here as equivariant yeah this is a LMA I can suggest to you and so then we obtained by by panare Duality uh just this space uh appearing in the in the um in the claim and over
The the derived um subalgebra of N and so this s module does not contain the trivia module by assumption therefore it is zero the S invariance are zero and uh thus these two spaces are zero and therefore this space in the long exact sequence must be zero and this then
Gives that the hn of Jus values in N is zero okay this is one uh one uh example of how we use here representation Theory to compute chology um maybe questions up to this point okay then um in some cases we can compute the whole chology and so this is kind of our
It’s not a big theorem but it’s our main theorem of of the paper with D um uh when we we we are back now to to these uh semidirect product Le algebras so a semisimple Le algebra here is even a simple Le algebra slnc and a module and as a module we
Take the natural uh module of Dimension n yeah so for example SL2 then it’s uh vectors with two components um and in this case we can compute the whole chology and it’s just given by the chology of sln with with trivial coefficients but which is with a shift in
Dimension so let me explain the proof and once again the the proof will not stop on this page but will go on on one other page um so first of all uh just uh by comparison of Dimensions we can never have that the ex the case exterior
Product of V is equal or isomorphic to S yeah because uh these binomial coefficients can never be n^2 minus one uh and therefore in uh uh therefore the the the factor in the long exacted sequence which corresponds to h k of V comma V over G
Yeah V over G is as an S module isomorphic to S this factor is always zero yeah and so uh we have that the space which is interesting to us the HK of V with values and G and the S invariance of it is just this home space
Yeah of s equivalent maps from La Lambda K of V with values in V okay and so on the next slide I will explain uh what this space gives oops so it is nonzero only for k equal to one this is what I claim and this is once again a very um
Nice little Lama uh the Lama um which you probably know is that uh you have an isomorphism of Lambda K with the homomorphisms of Lambda n minus K comma Lambda n yeah just by um the exterior product you take a K uh yeah a kfold exterior product and
An N minus K fold is z product and you you uh uh multiply them together to have an N Fold exterior product yeah um so the Lambda n of v as a v is n dimensional here is just C and the Lee algebra acts via the trace and thus trivially by
Unimodularity yeah and so uh this is then just the trivial module here and uh so we have here an isomorph for isomorphism of Lambda K of V with the Dual of Lambda n minus K of V yeah and so one can ask when is uh this
Um the natural module V of Dimension n isomorphic to its dual module and this happens only for n equal to two only for SL2 the two-dimensional module is isomorphic to its dual and for all the other simple Le algebras sln yeah for great n this is not the case and
Therefore um the only space surviving here is the space for k equal to one and this makes the the dimension shift here in the in the result okay um some remarks about it we we can with the same kind of method we we can even more easily compute the chology with trivial
Coefficients um and uh one can also uh compute liet chology using the the methods uh of P which P showed to us in in homology and which uh we have um translated into chology together with yurk um and and then you can see for example that this five-dimensional Lee algebra SL2 C with
Um um with a semidirect product in in the two-dimensional module uh is rigid as a Le and as a light AR so can see small things like that and um now I come to the last part of my talk so these are results which we computed more recently together with
Di and uh so Di has um always this notation which uh which is maybe not so standard for for him the the the module VN is n dimensional so n is not the highest weight but the dimension of the model yeah um and so uh with with the same kind of methods
We we can show uh propositions like this when you have SL2 and uh uh a direct sum of uh the module VN kfold direct sum of the module VN then you have a um an H1 which is K squ dimensional and uh when you have a direct sum of VN and a
VM with n uh and M different distinct integers then you you have Dimension to so we did computations of this this kind yeah but um so for in some sense we are we are still at the beginning yeah we we are kind of searching for for patterns in this chology yeah we we
Would like to have a theorem which uh which tells us all the chology spaces under these and the those conditions yeah but um so at the moment we we are kind of testing and for for testing SL2 is kind of the the best uh lee algebra we can
Work with the representation theory of sl3 is still very well known but um the things you can say about a general simple Le algebra are maybe not enough to to compute these uh chology spaces yeah so um so what we can one of of these patterns is for
Example we can show that when you take an even dimensional module yeah you take SL2 semic product with an even dimensional irreducible module uh then the H2 is always zero all these Le algebras are rigid yeah and so uh in order to show this you need the
The decomposition of Lambda 2 of this module into irreducible factors and all the irreducible components are odd dimensional and therefore the the spaces which always come into must be zero and this makes the chologist zero yeah so for example um one one thing we we don’t know yet is what
Happens if we take the odd dimensional modules yeah um all already Richardson knew that then we we do have a non-trivial H2 and this is a non-trial H2 in his theorem yeah rigidly algebra with a nonrival H2 uh but uh we would like to to compute it explicitly for all odd dimensional
Modules and so for the moment we we we have some ideas but we we don’t know the the general answer yet and so at the last slide I would like to show you that in in some cases uh we can uh have the whole chology yeah we can compute the whole
Chology um we already saw uh one instance of um of that uh for the natural module uh and sln we can compute the whole chology yeah and here is another instant um for SL2 and the threedimensional module we can also compute the whole chology by the same methods and so
Um but for the moment we have computed only the chromologic spaces so one one can do many more things here for example one can ask oneself um what is the bracket yeah on on these chology spaces you have a a Le bracket H1 in fact is Lee
Algebra and this Lee algebra H1 acts on the other uh HK yeah and um so for example uh here it it must be the trivial Le trivial one-dimensional Lee algebra this Le algebra G has a non-trivial infinitism deformation yeah a unique uh infinitism de nonzero infinitism deformation and this
Infinitism deformation because of H3 being Z zero um extends to a formal deformation of of the Lee algebra but uh so this these are some things you can say uh just by in inspecting uh these uh chology spaces but um there’s also an action of this onedimensional Lee algebra on H2 on H4
And on H5 and in order to compute this what one would need to to uh take real code chains and and compute with with the elements and so this we did not do yet and uh actually this is kind of the last Lee algebra where we can compute the whole
Chology by our methods yeah when we try this for the seven dimensional Le algebra SL2 C with values in V4 yeah uh did we try this no maybe we we tried only the odd dimensional one so take the eight dimensional Le algebra SL2 semidirect with
V5 then I come to to some long exact sequences which I cannot where I cannot specify all the terms and uh I I can some say something about the low dimensional terms about the high dimensional terms but in the middle there’s some there are some chology spaces which I which I cannot
Compute and so one one should ask yeah are there different methods to to compute uh the chology and one method with we which we did not try yet but which looks uh also promising is um the method by gr graduation yeah gradation the the Lee algebra G is a
Graded Lee algebra SL2 has an element H which acts by bracket on on the whole Le algebra and um you you have a basis by I vectors yeah and so in this situation um the chology uh is given by the subc complex of uh uh code chains of total degree zero this
Is a theorem by Dimitri fuks theorem 1.5.2 in his book and so um maybe this can also serve to compute the chology and go into uh these uh Corners where we we did were not be able to to conclude yet okay this is everything I wanted to say thank you for your
Attention thank you very much for this very interesting and Clear Talk are there any questions or comments I don’t see everybody it’s very hard for me to do this now I think you just go ahead you unmute yourself and go ahead because I don’t see everybody on my screen so I’m
Sorry no questions no comments and I have to ask a question right go ahead yeah no no I can you go one slide back please I think it’s one slide back because no yeah so for the events you did it right so so it’s really only the I I just want
To make sure so so the next case is V5 as you said correctly yeah yeah yeah okay so we can this is only about the age2 yeah ah it’s not for all okay no no no not for all not for all okay oh I thought yeah okay okay so even V4 would
Be interesting for all yeah okay yeah yeah yeah and did you try this if I may ask or you Um I I did so many computations I I don’t remember actually no no I it’s not a fair question but you said something and then you said V5 all of a sudden so I just wanted to yeah clarify yes I I think for V4 I did not
Try the whole kology yeah should I should try that yeah you’re I mean that would be in also in the light of this proposition yeah it would be interesting yeah what is this largest one that you can do all and not just H2 right that would be the next
Exact exactly yeah okay yeah but I if I remember correct yeah yeah if I remember correctly it was V3 and not V4 okay okay h and for V3 but for V3 apparently it’s not true that H2 vanishes right so for the OD case yeah yeah okay so you cannot expect an anal
So it’s more comp obviously more complicated for the odd Dimension right as you mentioned of course yeah yes and the H2 is nonzero this is the nonzero H2 which Richardson used yeah yeah you said it also yeah yeah I don’t remember this paper at the moment but you mentioned it
Yeah okay yeah it’s a very nice paper I know but I don’t remember the details at the moment yeah no no yeah I have a small remark I’m just back from teaching sorry I’m so late yeah but to the question about SL2 and V4 I can answer
It because in low dimensions of course there’s a distinction how to compute by by Theory and long exact hology sequence or just by computer and of course this is still computable just by computer so we know the answer for SL2 and V4 or I know it at least because you just can
Ask the computer which will not be possible in general of course yeah but for low Dimension then by low I mean something like Dimension at most 20 uh yeah maybe only 15 I don’t know some depends on your computer power uh you can do it directly but but do you
Do you speak about H2 or about the whole the whole you can compute them all okay yeah but very soon and is there a pattern somehow I mean especially for the odd ones do do you see a pattern there if I may ask this we would like to see a pattern
Yeah so no uh the problem problem is for for a pattern you need really a bit more information and that’s uh not really feasible only for H2 I see a pattern but not for the old chology okay and and if I may ask what is the pattern for H2 I’m
Sorry the pattern is very easy for the even dimensional irreducible modules in the thing it’s is always zero that we know yes and for the ones the pattern is that the result is always one Dimension okay it’s called that was my question exactly yeah so it’s stays one
Yeah yeah but we don’t have conure this is a conjecture yeah but but the the I mean what what you did compute with the computer confirms this conjecture that was the question that was the question of course right okay thank you yeah are there any other questions or comments
If this is not the case let us thank freedi again thank you yeah very nice fish thank you that was my comment yeah