This video features a lecture by Bernhard Keller from the Université Paris Cité, France. It was presented at the “Eighth Isfahan Seminar on Representation of Algebras” on December 13, 2023.
Seminar Page: http://portal.math.ipm.ir//Events//ISRA-VIII.htm
Structures in particular a in particular a infinity algebras and their morphisms and as you perhaps know they generalize DG algebras and their morphisms as introduced Yesterday by Professor Chen and the plan is as follows there there will be three parts a first part on a Infinity algebras first a bit of
Motivation then the definition and some examples obtained from H Shield Prology then in the second part I will uh State uh kades V’s theorem on minimal models and I will introduce morphisms of a infinity algebras and also a infinity categories and functors and then in the third uh section I will reformulate the definitions of a infinity algebras and their morphisms using the bar
Construction I will look at a infin modules and the derived category and I think time will not allow to look at Twisted objects yes sorry I have to delete this point so let us start with a bit of motivation um a Infinity algebras appear in several subjects maybe the most
Uh the oldest occurrence of a Infinity algebras apart from topology is a simplec geometry where they appear in mirror symmetry namely the the so-called a Bas side of mirror symmetry is given by fukaya categories of simplec varieties and fukaya categories are a Infinity categories if you want to learn
More about this you can look for example at James Pascal’s lectures at the house of winter School in 2020 you can find it on YouTube then uh another important occurrence of a Infinity structures is the link between DG categories and infinity categories where here by an Infinity category I mean a quy category
In the sense of dral lri and many others uh there is the so-called DG nerve functor um which to each DG category Associates an a infinity and infinity category um so which is just a simplish set with certain properties and to to Define this Infinity Factor one this DD nerve Factor
One looks at a Infinity functor categories then um a third uh occurrence a Infinity functor categories are used to describe certain homotopy categories which will be important in several Chens lectures tomorrow namely U he will need to consider the homotopy category of homotopy short exact sequences in in a DG category for
Example in an exact DG category and then finally let me not omit um motivations which come from classical representation Theory let me uh state two problems uh which appear in classical representation the Theory and where a Infinity structures are helpful for this let me fix a field K and let’s
Look at an associative unital K algebra it can be finite or infinite dimensional and a complex of right modules over this algebra then uh we often form the homology of such a complex just given by these sub quotients of the component um these contain some information about
The complex of a modules but not all information and well if we just look at them as Vector spaces we can ask or if we yeah if we look at them as a modules we can ask what additional information is needed to construct to reconstruct the complex up to Isom up to quasi
Isomorphism from its homology yes so given its homology what is the additional information needed to reconstruct the complex itself up to quism and here the answer is well this uh the homology of the complex carries a unique minimal a Infinity model module structure over the given algebra a and
This a Infinity model structure encodes precisely the information which we need to reconstruct the complex from its homology now another problem from classical representation theory is the study of modules with a filtration by a given sub quotients uh belonging to a given family of modules so we take B and
Algebra and we take NB modules and we look at the category of B modules having a finite filtration with sub quoti among vmis so in other words in other words we look at the closure under iterated extensions of the set formed by these n modules and it is clear that this
Uh category of iterated extensions is strongly related to the Y algebra of the sum of the M but uh it is not so clear how we can reconstruct it from the UN algebra and and in general we cannot however um this y algebra carries a canonical minimal a infinity algebra structure
Which does encode the additional information we need so we we can endow this un algebra with higher multiplications so that it becomes an a infinity algebra and this a infinity algebra contains all the information to uh reconstruct the category of filtered modules and let me remark that this
Problem is closely related to the rep presentation theory of boxes in the sense of the KF School in particular there’s work by oeno from 97 in 2005 and uh this uh this idea was successfully applied the idea of representations of boxes and their link to a Infinity theory was successfully applied by
Stefan kikan kmer and of sienko in 201 13 to the study of Quasi Hy de algebras and their exact Boral subalgebras in particular the problem of uniqueness of such exact moral sub algebras okay so I I hope at least one of these motivations will have motivated uh all everyone in the
Audience so now let me come to the definition of a infinity algebra the the notion originally comes from topology from the work of starf as he he was interested in group likee spaces spaces endowed with a multiplication which is well defined up to homotopy and associative up to
Homotopy um for example uh loop spaces spaces of Loops in a topological space and and in order to study these he invented first a Infinity spaces and then a Infinity algebras a Infinity algebras appear when we look at cellular complexes associated with a Infinity spaces here you see a picture of starf
This must be from the 60s I suppose when he made this invention and uh well he is now 87 and still often goes to lectures in particular online lectures here you can see him in 2020 okay now uh to give you the formal definition let me fix the field
K um some some things can also be developed over a commutative ground field but uh since I don’t have much time let me just stick with the case of the field and let me introduce a bit of notation um I will denote by GK the category of graded Z graded Vector
Spaces just a moment problems with my Pointer yes okay now it works the category of Z graded spaces and in this category the morphisms which we consider are homogeneous morphisms but of arbitrary degree okay and this this category is endowed with a tensor product given in the in the usual way the nth component
Of the tensor product is the sum over p + qals n of VP tensor WQ this is the definition of the tensor product at the level of objects and then we also have a tensor product defined at the level of morphisms namely the tensor product of two morphisms f and
G whose degree is denoted by the absolute value we Define the tensor product of the two morphisms by by this formula when we evaluate the tensor product of morphisms then as you can see we have to uh switch the position of G and V yes on the left and on the right
The position of G and V is Switched and then we have to use the coal sign rule whenever we switch two symbols of degree p and Q we have to insert a sign minus one to the P * q and so this is uh very important as
So so in the SQL I will often write tensor products of of morphisms and additional signs appear when you apply these tensor products to tensors and uh well fortunately this allows to get rid of of of these additional signs so we get a few signs less with uh thanks to this
Rule okay then we also have in this category we have an internal home so home is again a graded Vector space and the nth uh component of this graded Vector space is given by the homogeneous morphisms of degree n from V to W now starting from this category of
Graded uh Vector spaces we form the category of complexes as usual a complex is a graded Vector space together with a an endomorphism inside this category endomorphism of degree plus one and whose square vanishes and we have morphisms morphisms in this category are always of degree zero and commute with
The differential and we we the category inherits a tensor product and here you see a first instance of the use of the Cal sign rule yes we we the the differential of the tensor product just appears like this without any signs the signs uh appear when we apply these formulas to
Elements and again we have an internal home and the differential now is the uh graded commutator with the differentials of V and W and we have the shift or suspension factor which shifts degrees by + one and changes the sign of the differential and finally we have the homotopy
Category which has the same objects as the category of complexes and where morphisms are given by uh homotopy classes of morphisms of complexes so we go modul the this Subspace where H runs through all homogeneous morphisms of degree minus one from the to W so this is just h0o of
The morphism complex and more generally if we look at hn of the morphism complexes we get morphisms in the homotopy category from V to the N suspension of w okay and now with these notations I can State the definition of an a infinity algebra it’s a zed graded Vector
Space endowed with graded Maps so homogeneous Ken Maps MN from the nth tensor power in the sense which I have just defined of a to a they are defined for n greater equal to 1 and they are homogeneous of degree 2 minus n and satisfy a an infinite
System of quadratic equations namely uh the first one is that M1 composed with M1 vanishes so in other words M1 is the differential of a complex so a indow with M1 is a complex then the second one can be written like this if we compose M2 with
M1 then that’s equal to M2 composed with this sum and well here you recognize the uh differential of the tensor product of two copies of a so this rule just says that M2 is a morphism of complexes from the second T power of a to a and is aism of complexes and well
Equivalently uh one can also say that M1 is a graded derivation of degree + one with respect to M2 okay then the third uh quadratic equality um links M2 and M3 uh we can write it like this so here on the left hand side you recognize the
Associator for M2 if we evaluate this on three elements say of degree zero to simplify matters then we get uh this difference yes which is the associator for M2 and on the right hand side well as you can see here we we take M3 and we take its commutator with the
Differential yes this is the differential of the Third tensor power this is the differential of a and so and this is the graded commutator between M3 and these differentials so what we see here on the right hand side is the boundary of M3 and the morphism
Complex from the third T power of a to a and so in other words this means that this means in particular that this associator is zero up to a homotopy given by M3 yes and so M2 is a morphism of complexes which is um associative up to given a given
Homotopy homotopy given by MC okay and well the next uh here is also the next uh equation in this series um yes so here on on the left hand side you can see a a higher commutator if you like yes you you form the commutator between M3 and M sorry associator higher
Associator associator between M3 and M2 yes we have M3 and we multiply two of the arguments together in all possible ways and here on the left we take M2 and we multiply three of the arguments together in all possible ways and we throw in suitable signs and this is the
Higher associator and it should also be homotopic to zero it should be the boundary of a given homotopy M4 and the general uh equation in this the in this system infinite system of equations is is this one uh so on the left hand side we have a a higher
Associator and on the right hand side we have the boundary of MN boundary in the morphism complex and here a higher associator yes we we have in total we have n arguments we um we multiply some of them together and then we multiply the result we throw in suitable signs that
Gives us the higher associator and it should be homotopic to Zero by the homotopy given by m MN so in pictures um here we have our n arguments A1 up to MN we multiply some of them together by Ms and we get a new string of arguments and we multiply these together
All of them are multiplied together we throw in suitable signs and take the sum of all possibilities of doing this and that should give zero so here I have moved this ter to to the left in in this formula as zero on the right so I move
This differential also to the left and well yeah this notation um with the little segments uh has the advantage of giving uh few things to draw uh if you if you’re ready to draw a little a little more uh Strokes then then you can
Draw this in trees yes so we view the M as operations with uh n inputs and one output so we have our n arguments we multiply some of them together using Ms and then we multiply we leave the others alone and we multiply the the result of Ms together
With the others using muu and again uh we have suitable signs and the sum of all possibilities gives us zero okay and now uh some consequences of the definitions uh well so first of all the algebra a with M2 is not an associative algebra in general but the homology of a
Becomes an associative graded algebra for the multiplication induced by M2 then secondly well if if our algebra is concentrated in theq Zero by chance um then it is an associative algebra for M2 because the MNS vanish simply for degree reason yes the in in degree in degree one it is of
Degree plus one so that takes us out of degree zero in degree two it is the multiplication M2 and degree three it is of degree minus one which again takes us out of degree zero Etc and so all the mn’s except M2 vanish so that we get an associative algebra
And also uh if it’s if it’s not necessarily concentrated degree zero but all the MNS for n greater equal 3 vanish then the the aums say precisely that a endowed with M1 and M2 should be an associative differential graded algebra in the sense of Professor Jen’s lectures yesterday and conversely when whenever
We have a DG algebra uh it can be considered as an a infinity algebra with Vanishing MN for n greater equal to 3 and uh here notice that I have not spoken about Unity I have not mentioned units uh there are various possibilities of introducing units we can introduce
Strict units or units or homological units um I I will say a little bit about this later but it’s it’s something which should not uh worry you it is uh It is Well understood what goes on with with units okay so let me give uh the first examples um of possibly non-trivial a
Infinity structures more General than where where we have non vening higher multiplications um for this let’s take B an ordinary associative algebra concentrated in degree zero let’s fix an integer greater or equal to n and what we will do is uh we will deform this algebra we will consider
First order deformations of this algebra given by H Shi Co chains respectively Co Cycles so um okay so we have fixed our integer now we choose an indeterminant Epsilon of degree 2 minus n whose square should vanish yes we just take a a graded version of the Dual numbers of
K and we fix linear map with n inputs and one output in B so in other words a hook shield code chain and hook Shield n code chain and now we consider the algebra a whose underlying Vector space is just the tensor product of B with this algebra of
Dual numbers graded in an onal way and we endow it with Vanishing M1 with the M2 which is the natural multiplication of the tens of product then we Define ml to vanish for all L different from 2 and n and we Define MN uh using this composition yes so yes remember a is
This tens of product so to Define MN we take this tens of product and we project onto the N tens power of B we we set Epsilon equal to Zer that gives us M from a tensor n to B tensor n then for B tensor n we apply the Cod
Chain C to these n arguments and multiply the result by Epsilon to end up here in b t k Epsilon and that is of course a Subspace inside a and this composition is by definition MN okay and then if you like as an exercise you can show that this
Construction of a together with the MLS yields an a infinity algebra if and only if our H Shield n Co chain is in fact a hor Shield n Co cycle it’s differential it’s H Shield differential vanishes yes here I have written down the H Shield differential and well you will yes you
This already looks very similar to the conditions uh which Define an a infinity algebra and in fact we we have this equivalence and so this this means if you like that H Shield n Co Cycles uh always determine first order deformations of a given algebra in the in the realm of a Infinity
Algebras okay yes so far for the definition and the first examples now uh let me come to minimal models minimal models will yield a lot more examples so again let me take for B an orinary associative unital algebra and M some B module as we have seen in uh
Problem two uh problem two or problem one yeah well actually in both problems we have SE such a module and the claim is that the yeda algebra of this B module has a canonical a Infinity structure whose M1 whose differential M1 vanishes and whose M2 is the UN product on this x
Algebra okay now to to get this structure well we we proceed as we would proceed to in order to compute the X algebra together with the Y product so we choose a projective resolution of M in other words a aasi isomorphism between a complex uh right bounded complex with
Projective components and the complex M concentrated in degree zero and we look at the endomorphism DG algebra of this complex projective moduls the and morphism DG algebra introduced by Professor Chen yesterday so it’s uh its n’s component consists of the uh graded endomorphisms of P of
Degree n so this this is also in uh in Conformity with the notations which I introduced at the beginning the multiplication in this DG endomorphism algebra is simply given by the composition of graded endomorphisms and well this is a differential created algebra and thus an infinity algebra
And of course as you as you know the homology of this differential graded algebra gives us the UN algebra this is one way of computing the UN algebra with its product uh induced by composition of gr endomorphisms of the projective resolution and now well there there is a
General theorem which tells us that if we have some a infinity algebra a then on its homology we also have a canonical a infinity algebra structure with Vanishing differential this is kad’s theorem from 82 we take an a infinity algebra we look at its homology for for M1 for the differential
M1 and then the claim is that this has an a infinity algebra structure such that M1 vanishes yes it’s the natural M1 on homology the natural M1 on homology is the vanishing differential M2 is induced by the M2 of a so in this example it would be induced by the composition of graded
Endomorphisms so here it is the product and so this is the first uh condition which we have and then the second condition is that we should have a quasi isomorphism of a Infinity algebras between the homology of a in doubt with this a infinity algebra structure and a
Itself and this uh quasi isomorphism should become the identity in homology so it should lift the identity of homology if we take homology on both sides here yes on the left we have the zero differential so homology is H itself on the right hand side we have H
Star a and fi should induce the identity between these two copies of hstar a okay um I have not yet defined what quasi isomorphism of a infinity algebra is but I will do it in a in a few minutes and yes so I said I I mentioned canonical structure it is actually not
So canonical this a infinity algebra structure it is unique up to a non-unique isomorphic of a infinity algebra so if we have we have if we have two such structures on H star a they are related by an isomorphism of a Infinity algebras now an isomorphism of a
Infinity algebras is not just a linear isomorphism which commutes with all the structure it is something uh much more complicated it has higher components um so this uh equivalence relation is is a highly non-trivial equivalence relation and in particular the the misis for this structure are are
Not unique as linear maps not at all yes although the the collection of all these Maps is unique up to nonunique isomorphism of infinity algebras okay so as I said the these non-al Notions of a infinity morphism and a Infinity classy isomorphism will be defined in a minute
In a few minutes and let me introduce a bit of terminology we say that an a infinity algebra is minimal if its differential vanishes yes so so what we construct here is a minimal infinity algebra structure a Infinity quasi isomorphic to the given a infinity structure on
A and in this situation we call H star a with its a infinite structure a minimal model for a so in in small examples one can compute this completely explicitly suppose that we are given an algebra b by uh quiver with relations so we take this A4 quiver and we require this
Relation um for M we take the sum of the four simple modules associated with the vertices and then it is not hard to compute that the UN algebra in this case can be described by by this R yes we we rever the the arrows in degree zero give rise to
Extensions in between the simple elements of degree plus one yes I I look at right modules and then we have this relation which gives rise to an element in X2 from X2 of S4 by S1 and well you you can compute that the composition of any two consecutive extensions vanishes and that
M3 of these three extens x1’s is uh non- trial and equal to to e in in one uh one choice of minimal a Infinity structure yes um okay well so when one can construct these minimal models inductively or one can construct them using so-called homological perturbation Theory I don’t
Have time to give you the details but you can find them in the appendix to the notes at the end of the notes there’s an appendix which uh explains homological perturbation Theory applied to this problem it’s a very nice there’s a nice algorithm um due to mer kulov and explained for example by
Conage okay and then uh well of course it is uh nice if the higher MNS can be chosen to be to vanish yes we say that the in a infinity algebra a is formal if we can find a minimal model where the higher MNS vanish yes uh so even the vanishing of
The higher MNS is not uh something uh independent of the choice of minimal models there can be minimal models with Vanishing higher MNS and and a Infinity isomorphic minimal models with non Vanishing higher MNS but if we can find a minimal model where they vanish then we say that the infinity algebra is
Formal and that’s of course a desirable property and sometimes uh for a graded algebra uh one can show that all all a Infinity algebras a with homology B are formal and in this case the this graded algebra is called intrinsically formal yes so so and a graded algebra is
Intrinsically formal if we cannot uh endow it with uh higher structure where necessarily some MNS are are non zero okay and there’s a Criterion for intrinsic formality due to Cle and Thomas from 2001 new um in terms of H Shield corology if these by graded field homology groups
Vanish for a graded algebra B then this is sufficient for the algebra to be intrinsically formal okay so this is of course very nice when we have an intrinsically formal algebra then in order to know that a is formal we just have to look at its uh homology as a as a graded
Algebra okay and uh well in practice one also encounters nonformal algebras in many situations um and it’s a sufficient condition for non formality of DG algebra is uh that there are nonvanishing Massy products what is a messy product um let us take three homogeneous
Elements U V and W yes we can view them if you like as m from the free model of rank one to itself and then we suppose that UV is homotopic to Zero by an element little a and that v w is also homotopic to zero it’s the co- boundary of an elemental
Little V and so if we look at the composition of the three of them then in homology this composition vanishes for two reasons the first reason being the existence of a and the second reason being the existence of B and then the uh difference essentially
Of these two reasons becomes a a CO a CO cycle and so defines a class in homology which of course depends on the choices which we made on the choice of a and on the choice of B and on the choice of U V and W in their homology classes
So so this element this difference lives in a certain quotient um uh but this element inside this quotient always equals the element of M3 applied to UV to the classes UV and W and this for any choice of minimal model yes so we we have something which
Is independent of the choice of minimal model and therefore if uh if the algebra a is uh if if this is if this does not vanish then uh M3 will not vanish for any choice of minimal model and so the algebra is not going to be
Form okay and now yes as promised let me come to the definition of morphisms of a Infinity alas as we have already used them in Kadesh relas in the statement of Kadesh relas theorem okay let’s take a and b to a Infinity algebras then a morphism of a Infinity
Algebras is a family of graded Maps so family defined for n greater equal to 1 so the of degree 1 minus n so in particular for for for n equals to one we just get an ordinary map of degree zero between these two gred spaces and this map F1 of degree 0
Should commute with M1 so it should be a morphism of complexes and then if we compose fub1 with M2 we should find the same thing as M2 composed with fub1 * F1 yes which means that fub1 is a morphism of algebra up to some correction term and this correction term is the boundary
Of F2 in the morphism complex and so F1 commutes with multiplication up to a given homotopy homotopy given by this F2 of degree minus one okay and then more generally we have this uh rule for the higher terms um yes so it is it is actually not so complicated except perhaps for for
The signs um if we if we view it again in in a picture so on on the left hand side what do we do we we have our n arguments we multiply some of them together we get a new string of arguments and we apply the
The the morphism f u to this string of arguments and this should be the same thing as if we apply the FI in to all the arguments and then we multiply the new string of arguments together using M okay yes so this is the notion of
Morphism so again you can see it is it is a non-trivial notion somehow the the main part is the F1 which is of degree zero and which will induce an algebra morphism in homology and uh the higher structure is encoded in the f2s F3 Etc which uh measure the
Deviation of the suc successive fi to verify a commutation with multiplication okay and then uh we say that such a morphism is a quasy isomorphism f F1 is a quasy isomorphism of the Associated complexes we say that the morphism is strict if all the higher FIS
Vanish um so this is the this would be the naive notion of morphism it’s something a linear map which commutes with all the data and then we have the identity morphism which is the strict morphism whose F1 component is the identity and we have a natural formula for
Composition and we will see for example so now in Kadesh Will’s theorem we have seen that um each DG algebra is a Infinity qu isovic to a minimal a infinity algebra and somehow conversely we can one can also Al show that each a infinity algebra is a
Infinity quasa aric to a DG algebra if you like a a kind of this will be a quite a big DG algebra yes so on the one hand for a given a infinity algebra we have a minimal model with Vanishing differential and the smallest possible components yes or the components are
Equal to the homology so they are the smallest possible in the quasia aism class that’s the minimal model and then if you like we have a kind of maximal model where we have where we will have very large components um but we will only have M1 and M2 no no higher
Operations we will have a true differential graded algebra and yes so in particular this this also shows that if you like from from the point of view of qua isomorphisms um a Infinity the class of a Infinity algebras is not essentially bigger than the class of DG algebras yes
We can um up to quasi isomorphism they they are they are the same um but we will see that a infinity algebras have a lot of nice properties which which DG albas do not have and similarly for for their models and uh let me let me stop
Here for the first part of the lectures thank you very much thank you ber well no talk uh are there any question comments there is a question in chat is there a question in the chat Uh you mentioned that am2 is not an associative algebra in in general yes but if we view am1 as a complex in the homotopy category and equipped with the multiplication map M2 is it true that am1 is indeed an algebra object in the hot category yes yes absolutely that is
True it’s an associative alra object in the homotopy category yes yes yes up to homotopy it M2 is associative and the homotopy is given by M3 absolutely yes and uh yes of course uh one can also ask whether given an algebra in the homotopy category we can find an a
Infinity algebra which it comes from and I don’t have a counter example of my head but I’m sure that this is not the case I any question this and I also one question and uh which properties of associative algebra is be transferred to the a infinity I
Means if if we have if we have an associ algebra then we we constract the a infinity algebra I mean if it possible such that it’s homology just this associ algebra um sorry I did not uh understand everything the the the the sound on my
Side is not so good um so we take you we we take a graded algebra right and we want to and and assume that there exist the a infinity algebra such that the homology of this Infinity is equal this alra right and which properties of this algebra can be transferred to the a
Infinity I mean it possible to use this a infity algebra for studying some properties of this algebra uh I see so you would I see you so would so one could say that your question is uh given an a infinity algebra uh if we know its
Homology what can we infer about the a infinity algebra itself could one reformulate the question like that uh yes but I mean I want to study this algebra not Infinity but by using the infinity ah as you you want to constru to study this algebra not the infinity algebra Um yes um Um yes that’s uh yes that’s not so obvious I mean in if you like uh we can uh we can view the the a infinity algebra we can the infinity algebra structure can be viewed as a deformation of its of its homology and so uh the question amounts
To asking uh if we know a deformation of our algebra what can we infer about the algebra itself so in other words the the algebra itself it’s a degeneration of of its homology and uh sorry of of of the infinity algebra and uh so the question is related to the question of which
Properties are preserved under degeneration yes and uh yes and I yeah I don’t know uh I don’t know a general any general uh results uh on this question it will very much depend on the particular situation I think thank you yeah and then there is another question
By Edmund just to reconfirm you said that there are examples of algebra object here yes yes I think so yes yes yes yes I’m sure there are such examples yes yes yes yes yeah examples of algebras in the homotopy category which do not come from a infinity algebra yes next there is question
This yeah this is this is again related to a problem from from from deformation theory yes there are there are uh there are usually there are nth order deformations which cannot be uh extended to higher order deformations and and this yeah this is and this this is related to to the question yes
So are there any more question okay so thanks than you Berard again and we will start I think