 My talk will be essentially about some homology calculations, but it involves a lot of combinatorics in the sense that first of all, these complexes we consider, or they are small subcomplexes, they at the same time carry some least structure, which is, gradually, algebras, and the least structure is kind of very nicely defined, can be described combinatorially. And of course, the other point where combinatorics involved is our proof, so in the proof of the fact that homologies are sitting in one place in certain complexes, we use, at some places of the proof, we use some arguments from non-commutative Grimler-based theory for ideals kind of defined by differential in these complexes. So this is not just using problems Grimler-based computations for homologists, it is just elements of the proof where Grimler-based theory gets involved. So in this complexes we calculate, they appear from so-called pre-Kalabiya structures, which in turn appeared in open strings and this structure was also present in many other places in algebraic geometry and in selective geometry and in connection with mirror symmetry conjecture. But I will not talk about this, I will just give definition of pre-Kalabiya algebras as they appear initially and translate it to the definition of higher cyclic-Hochschild complex. Then I describe this small sub-complex which allows us to calculate essentially homologists of this higher cyclic-Hochschild complex and show what is combinatorial description of Leigh-Brekitt on this small sub-complex. And then kind of, of course, I will probably do not have time to show whole proof of the fact that this higher-Hochschild complex have homologies, it is pure, so it have homologies only on one place. But I will give you a structure of the proof and show maybe some elements of the proof which Grimler-based is involved. And as a result of this purity proof, for this digital algebra of the complex, the higher-Hochschild complex, we have a result of formality of this complex, so deformation theory kind of described by this. But this is a direct consequence of the fact that homologies are pure, you can argue more or less standard using this ill-infinity structure which is obtained by homotopy transfer on homology. And the fact that homology sitting in one place do not give you too much freedom, so it should be formal as a result. I probably will not talk about this, but the fact is that purity of the complex is very good property which allow you to have formality and all consequence for deformation, so on and so forth. So now I will start with definition of plecalabia structure as it was given by Konsevich Vlasopoulos and Zeidel some time ago, so this is kind of initial definition which is defined as an infinity structure on a direct sum of algebra and it's dual shifted by one minus d. So this is when I talk about deep plecalabial algebra, but I will not actually pay too much attention to shifts in this talk. So then this infinity structure on a plus a star should be cyclic invariant with respect to natural evaluation pairing on a plus a star, which means that if you take your form here it is nth multiplication in your infinity structure and permute your arguments, so pull alpha one to the nth, then the form is preserved. Modulo sine which is minus one in power degree of alpha multiplied by sum of degrees of all elements through which you pull this alpha, so sum of all these degrees. Here I use convention, shifting convention. And form itself is just evaluation form, so if you have elements from a plus a star, where a is from a and f is from a star, then form is just evaluating of f on b and j on a with appropriate sign. And third requirement in the definition is that a itself is an infinity subalgebra in a plus a star in this infinity algebra structure which you have. It can be defined for a associative algebra but also for a infinity, it doesn't matter. So, and then the notion can be reformulated in terms of this high hawkship complex and this reformulation this allows many, it is kind of useful in many respects, so it allows applications of homological tools and it's also allow extension of this notion by translating everything in this terms of complex work categorically so you can extend the application of the structure. And one point also important that here you suppose when you take a star it is kind of better to suppose that a is final dimensional or at least have final dimensional graded components because otherwise a star can give you sub problems. It's not the same as a and so on and so forth in the infinite dimensional case but in formulations in terms of high hawkship complex this requirement, this finding this requirement is not needed. So now this complex have many gradings so I give you a slice of this complex of degree n capital and component of degree n in this slice so and this amounts to home from n capital copies of powers of a which are mapped to n's power of a with certain additional structure which I explain in few minutes so this complex actually comes from dualization of power of bark complex so you utilize it by home as by models over a in power n from n's copy of bark complex to this model a n with a following by model structure on it so this is kind of important point in all this formalization that we do this n copies of a by the following n by model structure so we multiply element of a n not as you would do naturally just multiply each term from the right and from the left by corresponding term but we twist second the element by which you multiply from the right so you twist this b1 bn and multiply a1 x y to b2 and so on and at the last term you will get bn so the left right multiple is twisted this is this cyclic structure on n's power of a so then this differential on bark complex when you consider this as a home software k give you differential on this complex and this element so of the complex kind of understood can be pictured as so to speak multi operations where you have for each group of entries li you have one output from this model so you have cyclic set of operations with r i outputs inputs and number of this operation is n capital and they are kind of situated cyclically on the circle so and some of all inputs is this small n which is a give you writing of your slides on the complex so yeah of course for n equal n capital equal to one you have the complex to a as here a is only in one copy you do not have any special cyclic structure on it so you have just usual home which is hohschild complex usual hohschild complex so now we do it will be proven on this complex because namely we consider now that complex was high hohschild and this is high cyclic hohschild complex so we consider not all elements of home but only some of them so if you consider this operation is this real of element of the operations then there is natural action of group that n on this multi operation namely you can turn this wheel such that two outputs coincide and some sign to this turning this is how that n acts on on homes in previous in hohschild complex in high hohschild complex and when we consider only operations which are invariant under this turning then we get elements of high cyclic hohschild complex so now i i always operated with the slices so now let's take kind of bigger picture so n is not restricted we take any n so number of inputs can be arbitrary but number is outputs of outputs fits then we have a slice of complex and then we can take arbitrary number of outputs as well so we have infinite again some or product of the sub complexes this give you whole high hohschild complex and the least structure on this high hohschild complex is defined quite naturally again in this interpretation of elements of such wheels so you have a graded prelease structure on such wheels defined by inserting of one one output of one operation to all inputs of another operation with appropriate sign this is graded prelease algebra and then if you take associated the algebra a b minus b a this sign you get a graded the algebra structure on this high hohschild high hohschild complex and high cyclic hohschild complex and both so then what will be in terms of high hohschild complete definition of the collabial structure it is element of high cyclic hohschild complex shifted which is a solution for our carton equation in this v algebra so now let us consider a small sub complex in this high hohschild complex which we will use to calculate to show that high hohschild complex is so also it will be least sub algebra not only sub algebra but least structure on this least sub algebra is described combinatorially why is this highly non-commutative wheels or so this small sub complex appeared appear from a well-known quotient of the bar complex which is this small resolution here is the kernel of multiplication in algebra a and this is a kind of starting point for defining this non-commutative wheels so then we define this sub complex zeta over algebra a is by model home over a in power n from power of this resolution to the same model as I described before and take again invariance under that action this is our small complex zeta or n slice capital n slice of this complex inside high hohschild complex and this quasi-azomorphic sub complex in case algebra is smooth because if in a sense that kernel of multiplication in this of this algebra is projectile model then it is quasi-azomorphic sub complex so it can be used for homology calculations so now we need to choose some basis in in this complex zeta and this basis on initial complex which consists of the excess which is this difference so number of excess is equal to number of generators a number on number of the excess is equal to the number of excess operators of algebra a and this of from here we denote the xistar let's say at this point to describe the basis in zeta n we also pass from arbitrary smooth algebra to let's say for now free algebra but it is also can be done for pass algebra of quivers most of them so I will tell later more precisely now in the form of a as free algebra on n generators x i x in r generators x i x r so basis will consist of following monomials which are again written in a circle monomials with labels so you can have labels uh delta uh crisp number of deltas is the same as number of excess as generators of free algebra and there could be some size other type of labels in your word here it is one but it could be any number of labels side and in between there are just from monomials from free algebra you want you to you and whatever are monomials from free algebra so these are our non commutative which form basis in small subcomplex zeta and now we want to embed precisely to fix some embedding of this complex zeta to higher hawkshade complex for that we should specify how element of the basis of x i produce you produce for your operation element of higher hawkshade complex which is this multi multi operation uh and this we do in the following way uh so we define uh let's say in this picture uh we take xy delta build which contains three deltas and two size and this word will define for us uh map from a cube to a in power five in the following way so you have three inputs three let's say monomials from free algebra and uh we glue them to deltas with uh by by axis in these words we glue them to deltas with the same index in some way first so we found x we found delta with the same index and glue them here then how and and do the same with all three nodes then how we read outputs of the separation we suppose that there is some positive orientation on the plane so all arches are here related clockwise so outputs will be the following starting from nearest xy we read part of initial xy delta monomial and then the part of input word which give you this word as a outcome as a first outcome this is this word here then here you read this word and this word till the nearest xy this will you out will give you output uh second output here the next output and here if if there is no xy you read word from output then then monomial itself and then another output will give you another input will give you next output so this way you get get five outputs so number of outputs is the number of deltas in your word plus number of size and number of inputs are number of deltas and then you do this gluing in all possible ways and take linear combination of those this define for your operation uh so this word is input here these two green words are green outputs here this word is input here this word is output and so on and so forth so you get element of a higher hochschild complex which is a map from power three but sometimes it's zero power of a sometimes it's first power of a two uh at number of outputs is five so to a in power five so this is realization of uh non-commutative c delta monomial as a operation so it is kind of uh highly non-commutative monomial because you see you you multiply what what this operation does you multiply this monomial not like usual linear monomial you can multiply from the right and from the left but you can multiply this from a certain number of sites which correspond to the number of deltas in this monomial and then you get output which is also not a plain not two words or not plain word but it is kind of tuple of mononials uh I will give some more examples uh let me show you how it works so now we embedded this small sub complex in a particular way to our high hochschild complex and uh not that in this embedding um we get only as elements of small subcomplex we get only those homes where powers of a in this partition of power n in this partition in the incoming element is can be only zero over as you can see here so this is another characterization of subcomplex so let me to make it a little clearer let me give one example uh of uh what can be obtained uh with help of this non-commutative words elements of small subcomplex so let's uh explain in this language uh this double Poisson bracket which is known as a structure which allow non-commutative kind of historically already known as a structure which allow non-commutative Poisson bracket and representation spaces on algebra if you have this map from a square to a square on algebra which satisfy this double madness identity double E code and symmetry so let us obtain this map from a square to a square from uh c delta third from non-computative third and for that we need to take word which contain two deltas and no size then you will have two inputs and they give you two outputs so this will be more from a square to square and you can see that uh with this definition you will automatically if if map obtained from c delta third then this map automatically will satisfy limits identity let me say right away that symmetry identity will come from the fact that we take uh the two invariants and uh jacobi identity will be additional condition which come from our cartel equation uh but uh why uh it will be satisfy live double limits identity so double limits identity says that if you in input plug some product of two elements then what will happen you kind of should multiply one of one part of this element to this element of tensor product here and multiplication and and the other way and multiplication here on on the this is element of a square the structure of a b module on this a square uh let's consider outer structure so it means that if you multiply product of a b by c from the left you multiply left component and if you multiply from the right you multiply right component from the right so if you have this notation uh then live limits can be uh seen exactly from our interpretation of this operation given by c delta third as operation from the higher hawkshift complex i just described so indeed when you let you word here is product of two words you one you two then when you plug it here in all possible ways you can plan plug it in such a way that division border between this words you one and you two is kind of higher than point where you plug it or lower than point where you plug it so in the first case you see that result of your uh operation is uh that you just perform operation on v and you two and this glues this you one to the beginning of the tensor product so that shows from the left you multiply the left component of the tensor product and here you get from this sum of other type of elements in the sum you get another sum on here when you two just multiply from the right by the result of break it between these two so this is a kind of simplest example how this uh pre-collabial structures in form of high hawkshift complex or sub-complex produce non-computative or some structure or analogous things for bigger number of inputs so now what is the me uh break it on uh this non-computative monomials let's take two monomials and compose them as we should interpret them as a operation as a multi-operation and compose them as multi-operations should be composed so as I said before this composition of insertions of all outputs to inputs the operation of composition of elements of high hawkshift complex we just described before so what this operation will mean for elements of sub-complex of high of sub-complex data uh so uh if we uh want to take a bracket of these two c-delta words then uh actually surprisingly enough in a way it uh break it of these two words as a obtained as a through the composition of elements of high hawkshift complex of the svils actually will give you element which again expressed expressed via c-delta monomials more precisely how uh we do the following so if we want to take bracket of let's say monomial a and monomial b we glue any x from b to any delta from a is the same index uh cut up on this uh place x and delta disappear we open it up and we get new c-delta monomial we do this for all x and set deltas sum up and do insertion in another way for ba and sum up again so this will this linear combination of new c-delta monomials obtained in such a way will express uh use the same operation on the level of high hawkshift complex as operation expressed by this uh combination of this new c-delta words uh so here there is a picture which explained it in a way so it it shows that when you consider operations obtained defined by by this c-delta monomials and whenever you take an element which comes from outside of monomial itself some let's say x i which is sitting in the output and do with this what you would do for real operations in in the difference between composition of these two operations all such uh monomials all such insertions of outside elements will cancel with some element from uh composition on the other way so only actually the insertions which I showed before insertions of elements of the c-delta monomials itself to each other only they give result for the bracket so this is how how how combinatorially one can see that this bracket is closed on the sub-complex zeta so now let me say something about uh homology calculations themselves so we have this small sub-complex zeta um and differential which comes from differential in high hawkshift complex of course we have both version without taking the invariance so what we will prove we will prove that for free algebra on number of variables bigger than or equal than two so it is not true for polynomial algebra variable or for pass algebras of query with at least two vertices this complex is four so homologies are sitting in one place and so differential in this complex which comes from high hawkshift complex looks like this and so we uh elements of of them in itself is such a word from non-commutative algebra on variables xi on variables deltas in the same quantity as xi and xi's and on such a word our differential is just substitution of arbitrary xi apart from the one which is sitting on the first place so when u1 is empty then it will be something a little bit different a little twisted but if xi is sitting inside the word it is just gets substituted by this expression so yeah some of commutators of axis and deltas and if xi is sitting in the first place if there is no this ui or u1 here then this xi is substituted by also kind of by commutator by the delta and axis but this x gets uh into the end of the world world world so you kind of get this expression after substitution of first xi and then the rest other other xi's are as before substituted by delta and you take some of this expression here yeah so as you see this differential substitute xi's by delta and axis so we have co-homological grading by degree of delta on this complex and homological grading on degree of co-homological grading by degree of xi which gets smaller when you apply differential and by degree of delta it is homological grading so you can get new deltas and we will consider in this proof the slice of the complex which have fixed degree m by xi and delta so what we essentially prove is that all homologies of the complex are sitting here in degree zero in this co-homological grading by xi and so our kind of yeah first main kind of basis of induction let's say in in general the proof is by m by by we go we go by these slices here and so basis of induction is m equal to 2 for m equal to 1 there is homology sitting here but if you put a arrow from k from the field here then this homology will disappear so this is kind of trivial step and here we set all our considerations and I should say that in the whole proof the most kind of subtle place where we should ensure that homologies are vanishing is this place the closest to the place where homology sitting which is of course not surprising so here we will need more careful considerations which will which using this Grubner basis arguments mainly for degree of xi is equal to 1 so again about the structure of how we proceed first step we do is a reduction to the complex with different differential namely differential is straightened so to speak so we consider another differential which does not act differently on xi whether they are first in the world but it substitutes just all xi by delta and we take some by all the substitutions and first step which is not difficult we prove that actually we reduce the effect for our complex this will be our main result which we want to prove for our curly differential that if or more precisely yeah okay I should thank you words also about so we should prove result for the small complex with the curly differential it will follow that for the complex where we take invariance and the ten the ten action it is also true because this the ten action commutes with differential so if you turn this wheel differential you can differentiate after or before it doesn't matter and then a purity of the whole higher whole set complex will follow from purity of this small sub complex because it's quasi-zomorphic as I already said so now the main work will be done in the complex with straight differential and reduction to the complex from the complex which we need with this twisted differential to the complex with straight and differential is not difficult but here already we have some appearances of grubner basis considerations namely for degree of element xi degree of element bigger than two so not in this difficult place so to speak but down in in this down the picture we have easy to see that we can reduce our differential to straight and differential and okay maybe I do not have time to say how but it is really not difficult and when we go to this place this red red line in the complex grinding then we already get a bit more subtle consideration where we will work modulo the so we get expression of differential modulo the ideal generated by our delta which defined our differential so so here we need to do something in the in this ideal which is quite small one so to speak so it is close to three polynomials but still considerations for the grubner basis in this ideal allow us to proceed and again to get rid of this first xi in the expression and move side to the left model of the image by this if if element was from the kernel the model of the image remove xi and by this getting closer to showing that image coincides with kernel or in case of straight and differential we just show if we should not go always through moving this xi but we just show that differential is actually the same we can get rid of this first xi model of the image so differentials curly one and straight one will be the same so reduction does not use any induction by by by place of xi so i i wanted to show this reduction more precisely but i think i do not have time anymore anyway i and then after we reduce we we proceed with the main proof for the straight and differential which uses some facts about what are generators of ccg model for the ideal generated by again this delta here so we again work in the quotient of free algebra by ideal generated by delta and then it was mainly for free algebra but then we can model some changes substituted by pass algebras of clear because this all theory of ccg is sent in in and grebner basis are developed for by edgreen for arbitrary algebra with multiplicative basis pass algebras in particular case of this there are only a few subtle places so here for example we consider three polynomials on two types of variables which is the free product of two free algebras and in case of quiver it will be free product of quiver plus algebra times free algebra on deltas and this should be presented somehow that is a quotient of another quiver algebra which is possible to do and then we can proceed in this quotient of free algebra and it works again so yeah thank you i think i should stop here already over time just a minute so it's okay so thank you for your quite interesting talk questions comments yes i have a small question you just mentioned the free product of these two free algebras in this spirit of kourosh i i yeah maybe maybe i mean that you will not have relations between these two sets of generators okay okay so if you multiply two free algebras you get free algebras on the okay i understand your partitions the alphabet in two okay yeah i also have a question so then you consider this poison structure and then you just have two circles and then you just multiply then you just take some break and then it's becomes one circle i just remember then and we maybe some discuss uh uh hoph algebra on pozitroids uh with jarar and kristoff then it was also similar because pozitroids they are characterized as uh necklaces say that this is just some permutation which is on the circle and then we have to multiple just such circle permutation and then we just else just have found some point where we have to glue then we just make ball and then it becomes one common circle and so the question is any some relation to some uh uh some algebra which says you have abstract but but if we just consider some concrete algebra for example function on grass mania and on something like this if we just do that so what we will get uh so have you in other in in other ways so if have you some examples with uh non-obstruct but but then you just take some algebra a for example functions on some manifold and then did your construction yeah for example this double person bracket is an example but it is kind of um not exactly the least the separation which is considered the important structures so for algebras uh i'm not sure i know uh some examples we namely this least yes this would be interesting to know and as for for this uh positrons you said that you glue this in a particular place or you glue it by all possible places oh unfortunately uh i forgot many years ago i only just uh that i see this picture with drink or i say that's somewhere i saw so many probably many instances there it would be interesting i don't remember how how how we need to investigate can you give me some references of where i i thought i have to find maybe some notes uh i don't i don't think that we have paper but we just there are some notes i will try to find and there is in cluster algebra i think there is although this is uh so this is a poison uh structures so maybe also double i think there is a walk by uh chapiera and white stain uh so they consider the different brackets and then they i don't remember do they double these brackets but maybe also could be some geometric implementation of your theory okay i i may have a kind of general question uh you mentioned and at the beginning uh a formality result or uh and uh and there's a called poisson structure coming into the the picture of course these are double person brackets so it's quite different uh does it have a meaning to speak about quantization here um or not at all actually what what this uh that what this formalities supposed to say that we actually kind of we have a description of this l infinity structure on on the digital algebra so you you actually have all you can can know about deformation factor as a result so this is what it says about deformations uh and and this is not only actually double poisson brackets it's kind of poly poisson yeah so this is kind of one of the points because about yeah but it also says about double that the kind of aptochomotopy we know the picture we know is it there is only how how this digital algebra is the infinity looks like okay may i ask a question uh there is about the small complex so is it uh yeah maybe you are right i i say it later that everything is so it's uh homotopy it's uh quasi is amorphic to the initial one for smooth algebras for algebras the kernel of multiplication is projective module over algebra for projective bimod then this uh two resolutions and you take homes and they are they have the same homologist so this is called as a morphism on on this complexes normally whenever you have smooth algebras sit again and they are under i understood that they are uh but more more than that they are sometimes homotopy is amorphic ah yeah this is here i'm not sure we we use kind of argument for homotopy as morphine to get formality but it goes through the cell infinity structures on homology this uh homotopy transfer structure and homotopy is machine of them i'm not sure if you think about this okay thank you