Cohomomorphisms and Operads (contd.)

0:00 What is a symmetry object? Is it just a group of automorphisms? My answer is no. The deformation is not just the classical deformation with cumulative parameters. And no, an automorphism object or an endomorphism object is not just the usual classical story. The answer is this, we should in all cases try to construct some kind of universal object which are non-commutative spaces of the same kind as we started with. So it's non-commutative algebra, then non-commutative algebra. And then to treat this new non-commutative spaces as spaces of deformations or spaces of symmetries or something like that. And my second remark is that at least at some elementary basic level, this prescription is so robust that you can even work with it, not only with this more general story. So this is just the continuation of the step-by-step construction of the internal So I'm just repeating step-by-step these statements, replacing the associative algebra objects by their empiric analogs. And I'm restating here the moral I've already formulated. we should consider the non-convulsive space of morphisms of A to B linear generators as represented by this new operand called morphism. So from the pure, if you are a pure categorical theorist, you will say that I have enriched the category of operands by introducing new objects of morphisms which are objects of the same category and by introducing new composition laws which are represented by in this basic category so it's a very well known categorical constructions which here

2:30 reappears in a rather cumbersome setting but it's very useful so in particular it produces a whole lot of kind of hope for us it should be considered as a kind of group the generalization of a group into this context. And simultaneously, it produces a whole lot of representations of this hot conference. Everything is in the same construction. So in particular, analysis of quantum groups and things like that are now this much reproduced. So application one, as I said, is this construction But there is a second duplication which has no analog for associative algebras. Namely, there is an old question in algebra, what is the deformation space of something? For example, associative algebra. And we can be naive and say, okay, let's choose the system of generators. And that's like the right multiplication law in terms of the system generating indefinite parameters. This indefinite parameters then should satisfy several algebraic relations because you want your multiplication to be, for example, associated or or something like that. And then you say, OK, this system of equations defines my information. But again, as I said, you are not obliged to treat these coefficients as elements of your ground field or the ground field. You are not even obliged to treat them as elements of an associative reading. So you can do that and you get an interesting story. In the case of operat, you should treat them as elements of a new operat. So, in the classical language, if I have an operat, like a very classical operat, like associated with B or whatever, in the space, then the structure of an algebra on the space of the given type can be defined as just one operatic of morphisms from A to the operatic endomorphisms of B. It's not just algebraic, it's operatic because you are considering

5:00 And now, we have a beautiful, open new possibility. Instead of classical set of maps or scheme of maps from A to operatic endomorphism, we can consider the internal homomorphism of that. And thus, we get exactly a solution to the problem I have stated at the beginning. I want the deformation space to be something non-community and now it's non-community I get an operatic deformation space for example even on the structure of the structure of associative algebra so it's a new object which deserves Also, I would like to note that the notion of laybook graphs should open the way you study C-star generalizations about graphs in the book. It should be done. I think the time is right. We should speak not only about C-star algorithms, but C-star algorithms. Okay, and now just a few words about how concretely look this The first story became known again when the time when quantum groups became very popular And at that time, I suggested to groups related to this story, but on the category of quadratic associative algebra. So we are containing associative algebras, portions of tensor algebra model of quadratic relations. And in this category, it's very beautiful. So instead of in your space of generations, you have this additional piece of data, space of relations. You have two extended products of this category, which is a black product and white product. And you have one extension of the notion of the dual object, the factorial, and so you have a funny monoidal structure, this actually a pair of monoidal structures which are interchanged by the duality function.

7:30 And then, in both monoidal categories of this way, in one you get an internal homomorphism, in another you have an internal homomorphism. They're given by very beautiful formulas, it should be just considered as generalization of tensor product of one linear space by the dual of another. This is essentially the meaning of the matrices. This is essentially the meaning of quantum matrices whenever the algebras are . It took me some time to realize that the most robust part of this construction is this internal homomorphism, and not this beautiful triple of objects, two tons of products, between them. But whenever one can get them, one gets something very beautiful. And that was the insider that one can define a direct analog of quadratic algebras. They called them binary quadratic operas. So they are classical operas. One can extend this . And in this case, you get this explicit formula for the dual operas, for the ,, which is quite beautiful. kind of very and there exists one example where this object deserves to be started very carefully for two very concrete of the routes which are not covered by the And if you make a product like that, you will get some better Very, very beautiful opera art, which started in connection with language.

10:00 So we have a construction, but not yet real properties. And this object is just our several references for the end of this story. And I'll stop that and ask, thank you for your tolerance. Thank you very much. Are there questions? In the case of quadratic algements, yeah, that was a definition. So essentially, you always take the tensor product of generators that, as an In one case, you take tens of products of spaces of relations, and in other case, you essentially take the union. I have one question concerning your terminology, you're talking about school from a border. Yeah. The use of how to do a life. Consider the. Generally, no. Sometimes as in quadratic case and with different analog of black product. Generally, no, I cannot do that. Because it's natural, because co-morphisms of algebra should be measured as morphisms of underlying non-community spaces. So in this context, it's co-natural to that. So, thank you very much. Marina theory, for the usual operands, we did in our paper with Caprano, it's not complete,

12:30 but a good chunk of Marina theory, yes. In a generalized setting, they didn't try it, but I think, yes, probably some additional Thank you very much. abstract general, algebra general, normally it's just a representation, a calcul, like graph, like an arbre. And the idea, it's very interesting, you know, it's very It's an idea that you can put in place, let's say, an arbre or a card, and look at what's going on. It's a bit like that. Yes, that's to say that you can just do the theory of presentation, but in this case, it's more generalised when you do the presentation. But it becomes more interesting. The other idea of Tenzo, for example. This white product, I don't understand it, but it's something that replaces Tenzo. Thank you.