Vasily Gorbounov, “Generalized electrical Lie algebras and invariants”

[00:00:10]I period.

[00:00:27]Wait.

[00:00:39]>> Yeah.

[00:00:42]Okay. This is my pleasure to introduce from high school speaking about the electricians.

[00:00:50]Please.

[00:00:52]Uh thank you very much for uh making me a part of this conference. Um I've been talking for a while about uh electrical le algebraas. It's a joint project with with Ardi Vinstein and later Azat Benodinov joined us with with Arcadia.

[00:01:12]We talk about it maybe for eight years now. But maybe for the first time today I will I will offer you a good reason why why electrical algebbras can be useful apart from just being a curious mathematical object. We will see I'll try to uh to do that. Uh so um um the name electrical sounds exotic for uh an audience in in the department of mathematics. So indeed uh this le algebbras at least a very important example of them uh appeared first in the work of uh Thomas llam and pilaskki um uh when they were studying electrical networks and I'll I'll say a few words what does it mean for a mathematician to study an electrical network uh and then uh after that a student of Thomas uh uh Sue u um looked at the examples of electrical algebbras uh of other type and uh uh for a long time we didn't know if this algebraas appeared anywhere else in the literature but uh uh recently we've uh learned that indeed uh this construction is not entirely new it appeared for example in the work of Veranov and Katarina and many other uh authors uh when they were studying categorification and representation theory. But uh I guess the most uh the most interesting for us recent discovery was that uh electrical le algebbras uh in in the disguised appeared in the work of uh um Gante Zinwinsky on um models and representation theory and uh uh um and that connection uh uh looks to us very interesting. And I'll try to explain a little bit about this uh connection today. So the work of Gilfand Zinsky is from 1984 but in fact uh uh the it was built on the on the work of other people. So it actually goes even further in history.

[00:03:35]Um so before I give the definition oops of uh uh the precise mathematical definition I'll explain uh where directic comes from. So in the theory of networks, a typical problem is uh you have a network and by a network I mean a graph with an extra data and in in in in that uh uh part of network theory an extra data is a collection of vertices in a graph which I announced to be boundary vertices and using this boundary vertices I can make a measurement in the electrical networks the measurement is obvious. You can measure the electrical resistance between particular uh particular u vertices in your graph if you if the edges of the graph are resistors. So you get a matrix uh uh of the size equal to the number of boundary vertices and then a typical problem is to what extent this matrix characterize the network. In other words, given a metric, this matrix can you recover the architecture of the network apart from the boundary vertices which are given to you? It's where you made your measurement. It's sort of tmography inverse problem and it this problem has a beautiful solution. Of course, you don't expect to recover the original network. You recover a class of the networks which are equivalent to each other. And that is uh uh a part of the network theory which was developed by Alexander Pnikov in early 2000. Uh and his idea was well so as I said the network can be quite complicated but it's sort of from electrical theory we know for example that you can take a simply network and and the measurement you make on the boundary vertices would be the same. The uh obvious example is uh parallel connection of resistors.

[00:05:40]You can replace by one resistor if you change the uh uh the the resistance of each of the edge by the formula like that one.

[00:05:59]And same goes for inline connected resistors. You can always get rid of the midpoint. So in in theory there are many networks which would have the same measurement. So the idea is to construct the minimal network which reproduces your measurement and Alexander Pnikov provided a beautiful theory. So with minimal uh representative of the class of equivalent networks for a given uh measurement can be viewed as an orbit of an action of a group. So you choose a set of generators in a group for example JN or SLN uh a highest vector and apply uh this generators one by one to this highest vector. and he explained in a clear way how this orbit can be translated into the operation of gluing uh out of uh essentially trivial graph like this. So I I uh uh you should view this graph as an empty I mean each lollipop is actually representing one one uh boundary point no edges but edges would be the bridges between this lollipops. So he explained how to assemble any network with a given given measurement matrix via this operation and that construction was applied by Thomas Lum uh to electrical networks when the measurement as I said is the electrical resistance between the boundary vertices and uh uh with an effort of several people Thomas of course uh made the major contribution ution but uh other people contributed it uh to it as well. It was realized that the group which should reconstruct for you the electrical network given a uh uh uh resistance matrix is actually simplectic group but it's not a standard simple group. If you view it as a subgroup on uh in in SLN uh each such group would be preserving a form. Uh uh would be preserving a particular simplectic form. Uh he gave a description of this group in a different language. He said okay this is a group which has a set of generators and the number of the generators is equal to the number of uh positive simple roots and the relation between these generators are like this. So if your generators are far apart, if the indexes are far apart then the generators compute uh commute and if the indexes are neighbor of each other then uh uh you get a commutation relation like this and it's clear that this commutation relation is a deformation of the commutation relation for for the nil potent le uh so I'm talking about lean notic group. So this commutation relation if you put zero in the right hand side would be exactly the commutation relation for the nil potently algebra uh in um in SLN. Uh so the electrical group he discovered is a deformationation of that but accidentally it happened to be at the same time the group uh the simplectic group uh uh uh which preserves a particular um um simplectic form. Uh however in his work there was no matrix representation for this group and uh and uh the study of uh this that type of groups was done uh by by le theoretical methods by purely theoretical methods. You assume generators and relations and you you you write long commutators uh as a basis uh etc etc. So the point was maybe we didn't state it like that in the beginning but the point was to find the nice um um matrix representation of such a group and uh oh okay and uh and the leading idea was uh the guiding idea was like that. So an electrical transformation is transformations as I said there are transformations which don't change the resistance uh the uh effective resistance between boundary points like these two but there is one more uh this one was discovered in 19 1899 by by Edwin Canelli an electrical engineer and it's a beautiful transformation so you you you you have a network and inside of your network you have a piece which looks like a star. So this point in the middle is supposed not to be the boundary point. And you can perform the surgery. You can cut out this triangle uh this star and glue in a delta or a triangle. And uh in order not to change the resistance at the boundary points, you need to perform a rational transformation on the on the original uh uh resistances in in this in this network. So the original resistances denoted by R1 R2 R3 and the new resistances are denoted by R A R B and I RC and they they are connected by this uh transformations which are people called start triangular or delta triangular transformation.

[00:11:36]uh so that is a naturally piece of theory of electrical network and we wanted to use it uh and we wanted to use it in the following ways. So there are there are uh you know from the theory of integral models uh people like uh our matrices and our matrices are matrices which are say start with 2x2 matrix you can make it acting on a tensor product of any number of uh vector spaces two uh two dimensional vector spaces uh by an operator figj. Say f was the original operator then ti j is the operator which acts on the indices i and j is this 2x2 matrix and as identity on the rest of them of of them and and and and there is a generalization of young bster equation more complicated equation which is called uh the local young bster equation. So it requires for the following. It it calls for the following. It requires a operator fee uh which acts on a set of variables xi um uh and the following identity is is holding. So the left hand side and the right hand side are exactly like in the young buckster equation. Uh operator f12 is followed by the operator f13 and followed by the operator f23. But uh the identity occurs only if the variables on on the right hand side and the left hand side are connected by some rational transformation. And we naturally wanted to take this transformation to be the start angle transformation here except that you see it's it's completely symmetric and homogeneous. And in order to satisfy this equation, one at people knew it for a long time. One need to uh take the inhomogeneous form of of uh uh start triangular transformation. It takes the following form and then one finds a matrix V uh which would fit data and the matrix V uh people as I said knew it for a long time is given here at the top line. So it's a it's a 2x2 matrix which would act on on the tensor product of a large of large number of vector spaces of dimension two with indexes i and j and then this so this is a n matrix it satisfies this uh local young bster equation and and there is there are no le algevers inside yet and then uh one does what people in in integraable system do they you can multiply this r matrix By the permutation you get what is called R check matrix and then you observe that the R check matrix is nothing else but the exponent of an important matrix R R - R - R and that was our starting point. Ardi actually asked look here's an important matrix uh what le algebra guys like that would the guys like that would generate?

[00:14:40]I I took you through this uh um uh uh slides because next slide I would give you general definition and and and and you see that uh and you see that this definition if I didn't explain to you where it came from wouldn't make much sense. It would look ad hoc. So I hope my my intention is clear. I wanted to motivate this definition. So if I look at the previous slide uh then uh um uh all the parameters of deformationations which are which will appear in the next slide I equal to one. So again the observation was that you can actually deform this guy and that's the way you deform it. So uh if you start with a so now the setup you're given a km algebra given by a set of um simple roots and the carton matrix and uh for each neotent uh generator ei you introduce the new neotent generator by by this formula. So ui is ei plus a i where a i is a parameter uh times the commutator of ei and fi and minus a 2 um a i 2 fi. So the the the question was what le algebra do this guy generate? Uh by the way the parameters ai are not supposed to be necessarily non zero. Some of them can be zero. you can only partially deform your ne potent generator into into negative roots. So some of them can be zero and some some some some are not zero. So if you let us look at the simplest example SL SL SL2A.

[00:16:26]So the block a typical block the generator U in this case would look like that. then you immediately recognize that this generator is is actually conjugated to ei to to to the generator E. So locally this deformation is conjugate to the standard nil potent elements. But if you take commutators of them if you start forming long commutator if consider if you consider the holy algebra generated by them it's not going to be conjugate to u to to the nil potent it's it's an algebra of its own right. It looks like because because the conjugation you use is is not commuting. uh so you cannot um uh conjugate the long commutator using using the conjugation of of the factors.

[00:17:17]Uh so it's a new algebra. It's a uh many parameters family uh deformation of the nil potent le algebra in your kmoody algebra and and the question is what is it? So that was our question and u u we didn't realize immediately that it was the same algebra lumb was talking about Thomas Lamb was talking about uh until we did some calculations.

[00:17:43]So the first calculation of course we did we checked how do the ser relation would look like for the generators UI and we discovered that it's exactly the deformation of uh of the uh ser relation for for the importantly algebra. So the parameter which pops up is the element of the carton matrix a iig ji i time a j. So if all a i are equal to zero you recover back your n potential algebra but if they are not then it's a one it's a many parameters deformation well we've uh uh we've proved that it's a flat deformation dimension is uh is uh is not dropping um and uh and uh and therefore we we saw that this is a maybe new way of looking at the calculation of Thomas lam and pav pilki uh we provided a matrix well first of all we provided a new definition of the electrical algebra and uh uh together with a matrix representation for it and and Thomas and P confirmed that they didn't know this so it's it's indeed was was some sort of new observation so first we recovered the theorem um well first we we did another observation that in fact together with this definition of of the electrical algebra. You can get more canonical uh uh description of the generator's UI if you use as a conjugate this element uh GA which is the product uh is it written? Yeah. uh GA which is the product of the exponent of the elements fi at least in the case SLN then uh uh your your formula for the generators transform into sort of somewhat simply a formula so the new generator UI is going to be ei + b i minus one given by this formula times fi minus one so these two are commuting and therefore you still get an important But this observation is interesting. So um you get a new description of the electrical le algebra and it looks uh if I get it on the pre on the next slide it looks very much like much well studied deformation of the important algebra which leads you to uh the le algebra soom in this case your deformation is ei plus b i the same fi it's not any longer important it's semi simple but it's also a deformation of the algebra of the nil potent le algebra. So you have two distinct deformations of the nil potent algebra. One in in the case of uh in the case of sln one is s o and another is sp. So the orthogonal and the simplectically algebra somehow come as a as a uh deformation of the same root of the need potently algebra and it's a it's an an interesting connection uh in many ways uh however uh maybe I don't have anything substantial to add to it right now.

[00:21:02]Um um so as I said in the case of uh um uh SL we recover the result of Thomas lum for any set of parameters which are not zero. We recover we we prove that the electrical algebra is uh is isomorphic to the simplectic algebra. We found the um the particular simplectic form which depends on the parameters which it preserves.

[00:21:31]uh however if some of the parameters are not equal to zero then you get new algebbras and roughly uh that is not a theorem but it's uh it's probably not far from a theorem so what is important is whether your ai is zero or not so you can think of a iis for the purpose of classifying the classes of isomorphisms depending on the parameter size you can think of i's being the sequence of zeros and ones and then uh uh the number of isomeorphism classes uh for a given n is roughly the Fibonacci number which corresponds to n. So it's it's the very first observation I I made that uh uh when you have a single parameter a single uh neoton generated generator deformed then you can can conjugate it back to the neotently algebra. So it says that if you have only one parameter nothing interesting has happened. If you have two parameters which are far from each other, they would be commuting. So nothing interesting is happening. So what is important is a block consecutive block of nonzero parameters.

[00:22:42]If they appear that you get a new isomeorphism class and this isomeorphism class this isomeorphism classes relate the neotently algebra and the electrical algebra SPM.

[00:22:56]Okay. Um next calculation was is also important for this talk. Uh we've uh calculated the electrical le algebra for SP itself. So we picked up the uh Chevalier generators for SP uh um and deformed them according to our formula and then actually it's a result of quite interesting calculation. uh what you get is a direct sum two commuting sub algebbras. First is isomorphic to uh uh the electrical snn but with the parameters b1 bn minus2 and the other one is isomeorphic to the electrical sn + one with the full set of parameters you started with. So it's a it's a it's a it's an interesting calculation and it would be relevant for us later. Uh also you can uh uh take the fine algem algebra and and uh and consider the deformation. In this case we prove that that uh the electrical algebra electrical algebra would be a sub algebra of uh of a fine SL. It's pro so it's by construction is the sub algebra. Its properties are are not very well studied at the moment but it appears to be quite an interesting le algebra infinite dimensional certainly.

[00:24:26]Um an obvious observation which which we made next was that our le algebra electrical le algebra does not intersect the opposite barrel. I mean the intersection is zero and therefore uh you can talk about uh a generalized if you wish a gen a general form a generalized form of the u gaus decomposition. So the usual Gaus decomposition says that every matrix in JLN splits into a product of U a low bar and and and upper uni potent if the um uh principal miners of your matrix are not zero. Well the same theorem holds here with an interesting exception with an important exception. every matrix in JN splits in into a product of a barl and and and a copy of electrical le group. Now I I I I maybe run ahead of me. So I define the electrical le algebra and by electrical league group I mean the league group generated by the exponents of of of of the elements of electrical le algebra and and uh uh uh uh an example of such uh such decomposition I put here. So this this elements would be important for us later. They would uh be connected to the invariance of the electrical algebra.

[00:25:52]But I I've I've done this calculation.

[00:25:55]It's quite cute. So you you you split this and and right I didn't say so in the classical Gaus decomposition you require the principal minors not to be zero uh in order for decomposition to exist. And here another condition appeared. Look, we've got uh uh H1, which is a uh u H1, H2, H3, which are the diagonal elements of your barl. And and and in order for the decomposition to exist, you require H1 and H2 H1 not to be equal to zero uh and H2 not to be equal to zero. Otherwise, you are in trouble. But what are H1 and H2? Well, you see that it's it's a polomial which starts with a principal minor and after that you get terms which deform the principal minor. So the statement is outside of the divisor defined by the principle deformed principal minor such decomposition exists. So if your classical decomposition calls for one set of devices which outside of which such a decomposition exists here you get a uh a new set of deformed set of devices which provides such a decomposition.

[00:27:08]Okay. So now uh a representation theory.

[00:27:12]So what we've learned with ari uh so far is the following. So the the uniotent le algebra and unipotent group have in each representation have a distinct set of invariance one in each representation and every irreducible representation. Um uh u so since the electrical le algebra is a deformation of the uh uh near potent uh le algebra the guess the conjecture was that uh the invariance of the electrical algebra must be the deformation uh deformationations of the um of the uh invariance of the uniotent le algebra and indeed it so we first stated the conjecture Okay. So for any complex le algebra uh uh G uh and every simple finite finite dimensional G module with the highest weight lambda there is a set of uh coefficients uh uh and a vector uh v lambda a which sits uh in a vector v lambda beta which sits in the beta weight component of this uh uh this module uh uh the new deformed invariance invariant would be just a linear composition of linear combination of this and uh I will illustrate it by example. So take SL take SL SL4 take the uh vector representation of it then the invariant would be the basis vector V1 but uh for the electrical algebra it deforms like this. Likewise for the second uh uh exterior power uh that would be the uh um u invariant of the SL4 itself. But in order to produce the invariant for the electrical SL4 you need to do you need to uh consider this linear combination and of course this vectors are exactly the weighted components of this module and the precise coefficients are calculated there. So by now so here's another example if say you I mean for all class of minuscular or semi- minuscular or quasim minuscular uh you can write explicitly with formulas uh for the uh ad a joint and uh and u and uh for a joint representation you also can write with vectors. So by now it's a theorem.

[00:29:54]So this conjecture is a theorem for all classical groups uh namely SL ln s spn uh and the minuscule and quasi minuscule uh uh representations we have an explicit description of this invariance we have a general theorem which I will maybe which I will mention later but I mean in these cases you can actually do this calculation it's a pleasant calculation you start with invariant with an invariant and start formulate it and obtain a precise formula. Okay. So the next step is this is what uh is a new this is what uh is a new part of this talk comparing to other talks. We we recently learned about it. Verano pointed out to us that probably our work is uh is related to the work of Gilfontin Zilivinsky who studied the representation models for classical groups. So by representation model I mean given a uh simple maybe reductive group uh it's a representation of this group which splits into a direct sum of all irreducibles each with multiplicity one of course one such uh representation is is known you can uh for jian for example You can consider the uni potent no it's general construction but you can consider the uni potent and the functions on it then peter ve theorem would tell you that this is exactly the model for representation but gant and zilinsky somehow were not happy with this uh uni potent subgroup and they uh using uh um various results obtained from them uh before them uh produce the following table. So the top line of this table are the classical groups GLN or N and S py and the uh m not m0 is a subgroup uh uh in each of them. So what is the uh JLN with the um with the um u uh half integer index and what is SP with um odd index they explain it's just a subgroup of the respective group which which uh preserves a given uh vector or a given linear function uh a functional on on a representation um uh And uh uh the claim they made is that if you consider the functions on the appropriate space then that would be a model. Meaning algebraic functions regular functions on this modulus on this homogeneous space splits into a sum of u all reducibles with multiplicity one. And if you read their paper, they uh they they really uh uh are excited about this this result because the way you count uh um the simple summons in this uh uh um uh model representation is by finding what they call the spherical vector an invariant vector with respect to m. So the claim is every irreducible of these groups uh for m node which I listed below has a unique spherical vector an invariant with respect to this group. So if you take talk talk about JN then uh uh there is where and and spn is a subgroup where spn is embedded in the obvious way you embed sp into jian and that into j l n + one.

[00:34:10]Um right uh so if you take if there is a unique spherical ve vector in every irreducible representation of this group respect to sp and they found it amazing because from the examples it's clear that this uh spherical vector has nothing to do with the usual highest ve weight vectors which define this representation. It's a totally different vector.

[00:34:37]It's not and then uh when we saw this uh list we realized that uh this is actually our theorem except for the middle case. So the electrical le algebra and therefore electrical lee group for jn is exactly spn minus one but it now depends on many parameters. So there are many of them.

[00:35:01]uh and the same is for SP the uh electrical le subgroup inside of sp is the product of the appropriate uh electrical uh factors. So again we have a whole family that guy is still we still cannot I mean that subgroup we still cannot fit into our picture. So instead we offer as a substitute or as an addition to it an electrical so and and and and now the theorem about the invariance I stated before says that each of this electrical subgroups provides for you a model for any values of the parameter because they deform the highest weight vectors of the uh uh um reducible representations. Moreover, now we can there is nothing special about this particular copy of for for example SP because all SPS are conjugate inside of SL. So in in particular we can make this SP conjugate to any of our electrical SL and therefore the deformed invariant deformed highest vector for this copy of electrical SL uh would map into the spherical vector. So the spherical vector is just a conjugate. It's just a image under the matrix which provides the conjugation between the deformed uh uh um deformed uh highest weight vector and the spherical vector. And I want to emphasize that it I mean I'm not an expert on representation theory but it it it does look interesting before you take the deformation you have just just the unitary uh uh uni potent sub and of course it is not at all uh uh isomorphic to SP right so there is no way to uh relate directly if you don't know this family of uh deformation formations. There's no way to directly relate the highest weight vectors and spherical vectors. And then you use the deformation of this guy. I mean, they're still not isomeorphic. Just one is a deformation of another, but the invarians are deforming and that's enough for you to count the irreducible representation.

[00:37:27]So we find it quite an interesting um usage of this. So maybe this is the first time the electrical lee group Lee algebra happened to be useful. that it provides at least to my mind some sort of conceptual explanation for for for the gale fund zillinsky model and uh uh in the rest of the time I will mention um uh some calculations we have done with Misha Shapi and and and Arcadi uh u uh uh in Aradi Bernstein again um about the cluster algebra structure um um uh related to the electrical uh uh group. So here's the here's the setup.

[00:38:24]So Misha explained that there are many types of electrical uh many types of cluster important cluster algebbras. One of the important cluster algebra are the invariant functions on uh uh SLN with respect to the uni potent subgroup of SLN. So it's uh it's uh it's a set of invariance uh of of that type when you take G. So you study functions on SLN over it's the same as the invariant functions uh um and Zinsky and Famine uh found a remarkable structure cluster algebra structure on on this space I have here listed all the clusters. So the variables pig J here are the flag miners of the uh of a matrix. Every minor is a function on a group. So by flag minor I mean as a minor whose uh columns follow 1 2 3 etc. But the rows can are indicated by the subindex. So this is a 3x3 matrix whose rows are one to four and columns are one to three. So this is a full list of clusters. Uh uh so each vertex represent a cluster and the variables around it are forming this cluster. So this is a seed I think in the terminology of cluster algebbras.

[00:39:56]Now the the edges tell you what cluster transformations you have between these variables. So most of them are plucations except to make the mutation graph complete and by what these three ends are supposed to be glued into one point.

[00:40:15]So it's a graph on a sphere. Uh in except that to make this picture complete uh Zillinsky had to invent this new guy which is called omega. It's also a function but it's a it's it's a it's a polomial in in the flag miners. uh uh so without it you wouldn't have such a nice graph of mutation. It's not the same uh quiver uh uh or uh which Misha was talking about. It's an honest graph of mutations. So each cluster is a vertex and out of this vertex there are three possible mutations to that cluster that cluster that cluster etc. Um so now uh uh let us uh set for us a modification of this problem. If before we consider the invariance of the function uh invariant functions on SLN with respect to the unipotent group. Now we have a deformation of the unipotent group the electrical group. And let us replace the uh unipotent group by the electrical group. So we would be studying the set of invariant functions of the type like this.

[00:41:38]And this invariance as I said I I already had them on the blackboard. This invariance are nothing else but uh but the deformed flag miners.

[00:41:51]Here they are. So you you you you had uh um so this is a flag minor x11. It doesn't change if you act on it by the uh uniotent upper triangular. But if if you want to keep it fixed uh under the electrical upper under the electrical deformation you need to add this guys.

[00:42:15]So that is a new invariant or new invariant function with respect to uh uh the electrical group. Likewise this guy this is the um uh flag minor and in order to make out of it uh an invariant with respect to the electrical group you consider this linear combination etc. So we replace each flag minor on this picture by the deformed flag minor. And then we ask ourselves what changes what what stays the same and what changes does the structure of of uh uh uh clusters uh uh stay the same or or it explodes and there is nothing good etc. and and also the maybe a consideration. So I I I I had it on this slide. A remarkable result by Misha, Misha and Alec uh was uh the use of person brackets which uh Misha uh described before. uh so this cluster algebra structure produces a pluson structure whose brackets on the cluster variables from the same cluster are lo canonical and that gives you a description of a cluster. So here I gave you a picture and I told you is the clusters and Misha would tell you well if you take three functions if you take two functions and you want to know whether they belong to the same cluster you just calculate the uh post bracket between them if it's lo canonical they are in the same cluster if not then not um so um that is a a sort of information we start with and now let us see what happens when we replace uh the uniotent algebra uh uni potent group by its electrical uh deformation.

[00:44:12]The remarkable fact is well everything is in place but sort of in a different way. So all the all the uh cluster transformations um all the pla relations get deformed. Here's a typical example of a deformed plucy relation. So if I replace pi pi j by its electrical deformation then the pukar relation which before did not have this term would gain an extra term but this extra term is again a flag minor.

[00:44:55]So your cluster transformations get deformed depending on the on parameters and uh and uh uh you gain an you just gain an extra factor an extra summon. So what happens to po bracket? It's also an interesting exercise. We figured it out completely in this case of uh SL4. All the all the brackets of course stay closed and that is not a surprise because the electrical le algebra is co ideal and therefore the appropriate group has the property that the homogeneous space over it would would inherit the poson structure from the group. So the bracket is there but it's not any longer it's not any longer lock canonical. does keep the long canonical part but gain another uh um summon and the meaning of this we are trying to uh understand right now um we have uh we have a tactic so instead of introducing whole set of nonzero deformation parameters we can do it one by one we can just deform one uh uh nepotent generator and although as an algebra as a E algebra as I told you before it's going to be the same it's not going to be the same as a pos uh manifold if you factor with subgroup out and and well I guess I would I would I would be responsible for that maybe the guess is if uh Gman Shapi and wine theorem described for you each cluster in terms of poson bracket namely functions belong to the cluster. If the bracket is lo canonical then the electrical deformation starts telling you something about the edges in this graph. So if that deformation term tells you that there is an edge and I can actually show you what kind of age.

[00:47:04]So if you if you look at this cluster then if you take the posson bracket of P1 and P3 you will get you you I'm sorry if if you take the poson bracket of the right P1 and P3 you you end up here.

[00:47:19]It's a different cluster. So it tells you that there is an edge from this vertx to this vertx. So that that is only conjectural. So I wouldn't be taking your time explaining our conjectures not very well cooked conjectures yet. But the feeling is that electrical electrical the electrical deformation produces a structure where it appears like the clusters are gone although everything is closed under the bracket and under the cluster transformation. But it just tells you something more about the mutation graph of the original cluster structure. or maybe it's a beginning of uh um surfacing of a new structure which we don't know the name for yet. I think I it would be a good time to place to stop. Thank you very much.

[00:48:12][applause] Questions?

[00:48:19]I can translate a question from Maxim was chairman in the morning but he wasn't attend your talk he asks if there are electrical algebra possibly there are magnetic >> yeah yeah uh uh well I mean >> this can be interpreted I believe he was joking >> well uh so um of course it's a matter of uh u opinion But in our opinion uh um in our opinion the deformation which I had on the blackboard is a candidate for what you can call the magnetic algebra namely uh so there as I said there are two distinct uh distinct uh deformationations of an important algebra.

[00:49:15]one uh is near potent another is semi-simple and that is a candidate for uh could be a candidate for be called magnetic and in fact from the work of uh Anton and Boris Bkov and Dimat life and Galashian Pilowski we know that so there is a there is a a chain of electrical theories and and the electricity is one of the members. The other members are easing model percolation percolation and and and ports models. So Galashian and Pilaski studied the easing model from sort of same point of view. They naturally arrived to to this algebra.

[00:50:05]Uh so maybe maybe that would be magnetic algebra. We try to marry them actually.

[00:50:10]We try to >> I commend I have another suggestion uh for this magnetic algebra possibly because everything I believe started from your consideration of electrical networks and uh all the resistant resistors that are the real numbers and uh you can add some more involved elements namely you can add capacitors you can add inductances so you get something and and here the resistance will depend on the frequency so you have some function an analog of of this uh of this network and all the Kiroff theory works perfectly well for for this alternating current and RCL change and you can add also mutual inductances mutual capacitors etc and uh actually a lot of is known about them and and something parallel theory can be based on on these uh alternative current chains. For for instance, it is known that uh the resistance which is is known is as impedance. It is a rational function of frequency. If you have a finite complexity chain and it's it cannot be arbitrary actually this isn't soal you have lots functionality it has certain properties analytic properties and the inverse problem that you mentioned already it was also solved roughly 100 years ago that is if you have a herloic function how to reconstruct uh a chain with exactly this index And the the solution is not unique game. There are several several suggestions. For instance, there you can decompose it into fractions and on based on the element of this continent fraction you get the element of this RCL chain. So I think that this can be some development in this direction.

[00:52:24]>> Thank you very much. Well, indeed. Thank you for bringing it to uh to the talk. I think with Anton we kind of trying to think about it, but yes, it's good that other people think so too. It's great.

[00:52:36]So, maybe we are on the right track.

[00:52:38]Thank you very much.

[00:52:39]>> Okay, more questions.

[00:52:42]So, I have a small one maybe a bit well strange. I mean, if you start from the very beginning, so you have a really big physical physical electric network.

[00:52:52]really physical but big one some factory. So can you use some uh cluster geometry stuff to solve some optimization problems for that?

[00:53:03]>> By optimization problem what do you mean exactly?

[00:53:06]>> How to how to construct this network >> the minimal representative?

[00:53:13]>> Well >> yeah I certainly we can do the following. You start with an arbitrary network. You want you you single out boundary vertices. You want to keep the resistance at them fixed. You can use these two guys to reduce the number of edges to what you expect it to be not greater than n * n minus one /2 where n is the number of boundary vertices. But then the remarkable fact which which is is begging to uh be interpreted in in the cluster algebra language and I believe it is done that this triangular transformation does not change the number of edges and that takes one minimal graph and produces another and all these minimal graphs are forming the clusters and and I think that certainly can be made precise in the examples uh But maybe Anton knows uh better about the current state of that. Yes, it's uh it is it's all about I mean people from the very beginning suspected that the cluster alge structure must be there because you have all the ingredients.

[00:54:22]You have a uh quantity you want to preserve. You have the minimal graph and this minimal graph is not unique. there are rational transformation on the on the [snorts] on the weights of this graph which turn one graph into another graph and that so yeah >> so can can I also if you consider RCL chains this alternative current there are actually lots of optimization problems for the synthesis of of these chains that is you have to to put very specific uh values of capacitors etc inductance to to get a chain with with optimal some certain optimal properties for inance there are lots of optimization problems for for electrical filters etc. So and this is actually is used in electrical industry maybe very interesting any more questions thanks a LOT FOR INVITING >> [applause] >> AND WE HAVE A citman's break.