Frobenious property, N-Koszul algebras / Quantized trace rings (& others)

0:00 This piece is written as an NOI for every line. So this summarizes the classical case, the case where the algebras are also in the Euclid articulation. We can also express the quantity in terms of some of the cases. So we will look at the first, then the next, for the concept of the following sequence of spaces in apps. The spaces are given by this intersection. So here we can see the products are always in the case. And the last two here are used by injection of the WM into H in some way, and then complexed into that area, and here, the lower end, where this is the complex. And when we have such an L-context, we can always contract it into a two-context, so we can do it here in the following way. So we take the first line, M, and we take the good position of the following vector M minus 1, So this is a way of constructing a 2-complex, so an ordinary complex. So we denote it by KL 4.8 and we call it the QC2-complex of it. So from this talk, every time there is a new complex in the contract, this new complex will repeat itself. Ok, then we have three compositions. A is L2U, KF1A is the level resolution of K by L2U.

2:30 And we have also some analytic descriptions of the complexes in terms of the dual algebra, right? So, we see this today, the quadratic dual algebra, the quadratic algebra. Here this is the same, so we take the tensile algebra of the dual key. We take the quotient by R to the number we know is X to the power of M. So we are creating an multiplication. And we can write our M complex like this, where we can show the multiplication by this term Pm. So having this, we can also contract it and rewrite KM prime. And here, everywhere, we can also do the same thing, write the same thing with KM. For example, there is a twistor algebra. Consider the artificial regular algebras. They were introduced in the late 80s by Anthony Chester in order to have some analogs of the polynomial algebra in this case. They are refined by the following properties. Their global direction is finite. They have polynomial work and a minimal inclusion of k as a magnitude. As a property of symmetry, which is given by Newton. So this property is called the A.S. Bernstein property of Bernstein. In case where the global dimension is 3, at the very beginning of the classification, they showed that there are two possible minimum resolutions for k, either of this type or this one. So in this type, we can see that there are two types. The torus is concentrated in degrees 0, 1, 2 and 3, so that means that the energy bar is consumed in a gas process, and it has three generators and three relations of degree 2. And in the second type, that means that the torus is concentrated in degrees 0, 1, 3 and 4, meaning that A is pre-consumed, because there are two generators and two relations of degree 2.

5:00 So this is the first class of examples introduced in one. So we consider two integrals, small n and capital N. And N is the algebra with n integrals of x and y. And relation is the antisymmetrical degree N. So here we have n integrals of a symmetric group. So if capital N is equal to 2, this is just the polynomial algebra of the entire group. Then, using some combinatoric properties of cosine algebras, we can show that A is n cosine and a finite global dimension. Indeed, we can give some basis of the dual algebra. So we have some basis if n is between capital N and the symbol N. And we can show that A n is equal to zero if N is greater than N. But now, because of the local series, In fact, we know that the global dimension rate is equal to the height, such that a timeshift in the Greek dynamite is one zero, and a y plus one, this is equal to a minimum, depends on the height, so we have the following conditions. So when we have such algebra as the planet's global dimension, we can compensate for some of our usual algebra by writing the unit algebra. So we consider the algebra of EOI, which is the algebra of the subset, the extension of K, and this is the algebra of A. But simplicity in the formulas, I assume that K is the culture of A, and if A is atrocious, we get this X by using the minimal resolution, positive resolution. But now, every term of this context is on the folio form, which is concentrated in the green, minus EOI. But now, if we look at the differential in this context, it is of degree zero. Therefore, the differential is zero, and we will miss Wicker space.

7:30 So we know E of A as K-Wicker space, and we need also to know its product. So its product is given by the trigonometric computation. If we take two elements, F and G, and we denote that F is the product in the unit algebra, this product is zero. I and J are odd, and this is equal to fg if you i or j is even and here is the same product in the neural algebra. Algebra is not extra-algebra. Neural algebra, this is why it looks good. Let me give you a sketch of it. So if we take two co-cycles and a g, we denote for simplicity the likelihood of oscillation. What we need to do to compute the product in the neural algebra is to compute g to the cycle g to the dot of this complex. So we do this in a different way. Here we consider the resolution key. Here we consider the same complex but with chains in the middle. And also we twist the middle. We shift the degree of the left side. And here we put the map G. Therefore we have a resolution, a complex of projectives. So we can construct the map G to the left, G to the right. This is the notion of complexes. So we can compute the unit of product using the integration of f and g i, t, la. And now if we just watch the degrees, g i, t, la starts from a space which is generated in g , we take the amount f that goes to k, and therefore the composition will be zero if this integrals are different. And this integrals are different for i and j alpha, just by using this definition.

10:00 The two figures are given. We can give a formula for G theta. By doing the following thing, we can write Q prime as the contraction of an N-context. So this is more or less the given by the complex. And then if you consider the action of G, where G this time is considered, I think, in delta space, that gives a motion of N-contexts from Q to K-level A. We can contract it. So we get an explicit formula for the example of such a unilateral algebra. So we go back to the case of artificial regular algebras, a global energy stream with cubic relations. It can be written in a different way. Then, it's easy to see that the unilateral algebra as a great creative workspace is as follows. So here, remember that... We go to the intersection of these and solve R and R and so on. Then it is known that W, this W is the basis of W4. Moreover, we can write it as G1x and G2y, where G1 and G2 are generated in the space of relations. So we can write G1 and G2 in terms of an invertible matrix at x1 and x2. And then we can express the Hilbert approach. For example, zero or five-year-old, as we have already said, it could take any amount of big words and any amount of small words. So we can do a formal computation here. This gives an example of the unit algebra. We will show that, under certain conditions, we have a formula for this unit algebra. So first I record the definition of the cohomology algebra. And next we have the following theorem. Let A be an end-procedure to a finite global dimension, and we will need the second part of the Goldstein property, then we will give our position.

12:30 So with these assumptions, we can show that the dimension of this space is one, and if L is bigger than 2, D is all. This is one, that's why we could fix the generator of it, and we did that in this map, 5R, as we did in validation of new stuff. And then we can show that A is branched A, if and only if the maps 5R are biometric, but only in the degrees LR, so only for certain maps. So this gives a condition if we know for the algebra to be branched A. So we consider another random flex LR of A, which is the following. So the differential is given by the left multiplication by 2R, also in terms of the basis of D. Then, it's easy to see that this same context is isomorphic to the n-context from k-L over k-A. And if we contract it, we find a context which is isomorphic to this one. So this one is the one we use to confuse the x from k to k, so that's why this is interesting. Though we can show that if we take the number 5-R and we consult with the identity array, we have an n-context function from k-R over k-A. And next we say that if A is Bernstein, but A is Bernstein, if and only if this complex is a minimal resolution of K, but if this is the case, since phi is a multiple term between 2 and K, this is a... Now again, translated, that's not a common property, but it's probably just if this is a multiple term between 2 and K, this is equivalent to the existence of... Digitized by Landier-Ford, which is also balanced, and we say that the DC matrix is quite a DC matrix.

15:00 We have the following corollary. If A is N plus U of N at global dimension B, then A is the Einstein. If N or B, then your A is the problem. Your data should be less, because the problem is property. Let us go back to the idea of how oscillations are given by DC matrix results. Then we can show that there is much light different if either capital N is equal to 2, this is known in that case as a polynomial algebra, or we add the formula N to N. So the idea of proof to obtain this relation is to use the fact that it is odd and that the Gorinstein property implies symmetry of the dimensions. And in the other sense, we show that the maps are directed and we use the theory. Now for the cubic algebraic algebras, we know that they have multiple properties, they are non-shining, so they are diverse algebraic properties. And the great compute individual properties are here, because we know the unit number. So in that case, A on B and W. And we can see here that there are asymmetries between the matrix Q and the identity. And in that case, they correspond to the classifications of Hartmann and Shekhar. In the case where A is of the so-called type A. In this talk, we can pose the following question. Is H probing S? So we have seen that in some assumptions, the unit angle for A is probing S. This is not a sub-agent graph of A and B, but this is quite close to the sub-agent graph.

17:30 So why is it probing S? And also, we have the technical condition that states that the polynomial algebra is not valuable. Then, the dual algebra is provenious if and only if the biggest i such as a shape is not 0, is i equal to 0, and d, and all the maps 5 are, in this time, are by g. So, for instance, if we go back to the abuture term, polygram, cubic algebraic, type a, So in that case, we know that the correlations are as follows, where A, B and C are in P2, K, minus Mx, etc. So here we know, because these algebras are Einstein, that 5R is an element between 1, 0, 1, 3 and 4. So we have to look at 5R and 5R2. So we can compute this matrix and we can... This gives a condition for this matrix to be natural and this gives a condition for the algebraic shape to be impervious. As a remark, if we assume that this condition is satisfied, we can write specifically the 3-complex equation 5 that I've introduced in the book of theorem. So for the left version of the n-complexes, that's the growing simple expression, right here this is the equation. So this is the end here of the webinar on the multiplication value and the kind of computations for every type of problem that we have to recapitulate. For every type we have to learn something new.

20:00 We have a break of one minute. The session chairman stimulated the discussion. Oops, I stimulated, but you see I was looking at something and it didn't react, so that was stimulating for my side, but it felt all right. I will say this at the end of this class, I was stimulating, and please put it in your bedroom. There are also a number of different fields of study, such as mathematics, geometry, algebra, mathematics, and physics. There are also a number of different fields of study, such as mathematics, geometry, algebra, mathematics, and physics. There are also a number of different fields of study, such as mathematics, geometry, algebra, mathematics, and physics. There are also a number of different fields of study, such as mathematics, geometry, mathematics, and physics. There are also a number of different fields of study, such as mathematics, geometry, mathematics, and physics.

22:30 There are also a number of different fields of study, such as mathematics, geometry, mathematics, and physics. There are also a number of different fields of study, such as mathematics, geometry, mathematics, and physics. There are also a number of different fields of study, such as mathematics, geometry, mathematics, and physics. So, our aim was to define a model. So, for example, three of the generic matrices, space, and energy, and the contest of quantum mechanics. We'll go over this here and then again. Let's rewrite the generic matrix. So, we start with space. This is a land finder of the simultaneous conjugation action SLM. Its coordinate entry is just an n times n squared variable, a little bit more linear algebra, but not by SIDR, the coordinate function that maps an n-tuple to the IJ entry of its R component. We collect coordinate functions belonging to the fixed matrix component and arrange them in an n-byte matrix. In the obvious way, then, we get a subordinary matrix, known by the capital X of R. So this matrix is an element of the n-by-n matrix, where it is large, and it is polynomial. And then, we define the ring of G as the subalgebra, the algebra of n-by-n matrices, where it is large, polynomial ring, regenerated by the generic matrix X of 1, X of n. I assume that this, and the identity, so I assume that this thing is unique, and the real matrix invariance is just the sub-algebra, but we know that invariance, especially in the algorithm which acts on this coordinate ring, by reaction induced by the equal simultaneous conjugation, which on the n-th of the n-th of the matrices, it is well known that this algorithm is generated by the form, the trace, the product of some generic matrices.

25:00 And then, one defines the so-called red strings, which is what we refer to as algebra, and the subalgebra, which is defined by the matrices over here, generated by the generic matrices over a group of matrices. So, originally, we were generic in the variety of associative algebras, satisfying the same polynomial identities, satisfying all the polynomial identities satisfied by online matrices. It was an observational approach and in that moment I realized string of generic matrices in such a specific way. It may be a general observation, but it opened the way to apply the methods of classical invariant theory, because if we enlarge the string of generic matrices to obtain the so-called trace tree, this is an object which can be investigated by the standard methods of invariant theory and also via the trace method. From Tn to Rn, between Tn and Rn and Rn, or the substance functionalizing of generic matrices, for example, the place to which the theory under is, the link of generic matrices is not, or the place to which the binary represents is not. So it is nice to have algebra, and on the other hand, it is not too far from the link of generic matrices. For example, there is a central element in the trace tree.

27:30 which multiply the two of these three variable generic matrices. So these algebras have a rich literature of names. And now, we want to find what is functional across, since we have quantum matrices and quantum SNL and so on. So let me go over the definition of it. This is an associative C-algebra generated by n square generated by Tij, subject to some quadratic relations which can be written in this complex form using this R matrix notation. We know that chemicals are obtained by the arrangement of the generators in the matrix, and T sub 1 denotes the chronicle of the product of the matrix T and the identity matrix, and T sub 2 denotes the chronicle of the product of the identity matrix, and so T sub 1 is a large, heterogeneous matrix, which kind of spins, which somehow grows up, so it centers on the identities of T and T parts of the same matrix in various positions, for example. The key is what we call, in two books we have half of the four homogenous, quadratic, and heliophysic lectures in which we have to choose two different relations, we have a connotation relation for each pair of the generators, and they are not necessarily associated with S and M, but especially the one which is two, I wrote down the stages of this debate, so Q is the non-zero contest on there.

30:00 The functional algebra of the session is in our group, by every one of our relations, so the coordinate problem of the m by m matrices contains a central group like l and m, the quantum determinant, and we impose the additional relation that the quantum determinant gives us. And in this way we get a whole file algebra, so the coordinate of the m by m matrices is a binary algebra, the commodification is formally the same with modula as the commoditism. The matrix multiplication is going to be phased. So this implies that for the unfold S, we have this problem because if we apply the unfold for each of the generators in the range of matrix 1, then there is a plus in the matrix to be the inverse of the matrix capital D. If the matrix 1 comes from R and 1, so this is just the algebra. There is a sum of algebraic 1 to the 4th ring of n-line matrices lying above this algebra. Then we take the sum of algebraic 2 to the 3rd line of algebra. Then we specialise Q to 1. This corresponds to the ring of invariants 1 to the 3rd line of algebra. It turns out that there is no sub-architecture of quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum-quantum

32:30 If we would make the benzoyl carbon 2 copies of quantum matrices, then we would need to make the benzoyl carbon plasma structure on the below. But we can't change that. Rn2 is not a plasma, but it is actually a plasma membrane. So we need to do something else. But it turns out that there are, that can be used, service bases for constructing the object theory. One could ask, well, anyway, instead of the coordinate angle, quantum coordinate angle of M-I-M matrices should take the reflection of a given algebra and put it in the same algebra as the algebra of plane matrices by a machine. M-Q-M is again defined in terms of generators and relations, it's a generic sign J. Subject to some homogenous quadratic relations which are given in this compact form using the S-R matrix notation. So here, we think of R as the random R of the N-sperm and S-Sperm matrix. We think of R as the random autism victims or big birds. These seem to be in an electrical vector space. And we compose R in the fluid. And, of course, in a change, we have to consider each one of these, and that's how we get R-hat, and then those are here in the reflection equation algebra. And R, but then the logarithms will be 1, 2, and... So in this way, we can make it work. This algebraic element brings us to the point of view, and looking at it so by standard, it means that AQL is in this way a domain. And it has the PDW basis where you take an elementary notation of the variables to the S-variable variables and the ordinary commutative equations go into the monomial form of the basis.

35:00 So it has the same Hilbert space as the ordinary S-variable variables. And why it is better? Because it turns out that a-humanity conjugation action. All of this goes to FXT, where we think of the elements of AQM and WSLM, which suppress the tensor-scientific representation, seeing that the elements from FQSLM and AQM come in at all, and then we estimate that the FXT can be considered as an element of AQM and WSLM. It's a similar collection on the coordinating of quantum matrices, but it won't be a correct algebra. So, we can take this on the edge of the equation, for example, the quantum PLM, it is a correction, but the quantum matrix is only the corresponding correction that is not a mathematical model. So, if we are dealing with the co-integration action, then it seems that the reflection equation on the enterprise is better, the formation, than the usual quantum quadrilateral dynamics.

37:30 We want to define the quantum coordinates of pairs and like matrices, so we just take two copies of the reflection equation algebra, so the matrix is generated for one of them by x, for the other by y, and then we impose some external relations, some connotational relations between the x's and the y's, which again can be written in a compact form into this. So this algebra was defined exactly this way by a condition that's not up here, but it also appears in Martin's work as the braiding. So this is nothing else than the braiding, the so-called braiding, in terms of production, the reflection of these lines. This braiding is actually because of the volume problem. If you take B and D, but it won't be in general, but it shows that if we are dealing with a dual model, That's all about the quantum structure, but introducing a new multiplication. So what can introduce a new multiplication in the same vector space that will make the common larger graph of the tensor product whole? So this new multiplication can be explained in terms of this so-called universal algorithmic form. It appears in the definition of Newton's last example, or Hoppe's last example, as the algorithmic multiplication between B and B and B. And then, for an arbitrary m, you just iterate this process, you take the m for the break-through tensor for the whole m of the deflection equation of the algebra, it goes on the trigonometric brackets.

40:00 Or, you can say, in terms of generations and relations, so we take our m over, this is actually a equation of algebra, we take these generators, x1 and so on, xn, and for each, I call it, each pair, x1 and j, we call this cross-transition between m and xi and xj as the relation between x and y. So, the native generators are all of them, otherwise this is unique. There are more of these than two. One is the invariance. We have this quantum coordinate theorem. And the other is an n-line matrix with the multiplication and the correction of the quantum SLM. So let's define the quantized quantum matrix invariance as the invariance of the algebra of this correction. So these are just the elements of AQMM. There are many different types of equations, from f to h to n to f times or more. And let's define quantizing. Grayscreen has the epsilon and beta for each angle of the semi-n matrix, and the resulting matrix is the same. The fpIq is an element of the semi-n matrix with energies of h times or more. This is a suggested way to do that. In particular, using the S of P as the matrix in where the D is immediate, it shows that DQ is a subalgebra in the algebra of lambda and the matrices of the AQM.

42:30 Just to give you the name, that's for the one-time derivative of the generic matrices G2 , the subalgebra in Mn. If you don't make a distinction for generic, you are good. By the way, these definitions I don't know. But I'm afraid you would call it generic. There is some problem with these identities. Yes, just to speak about this, because I don't have much evidence for this. This theory, can they call it generic mathematics? We have some evidence for the other two. The key is not the root of anything, but first of all, the algebra aqnm is z-thumb-graded, so we count the degrees, we just add to each of the matrix components, and the co-action is z-thumb-graded, so it's what I do with matrix invariants, it's also z-thumb-graded sum algebra, so it makes sense to speak about it's n-rand or n-thumb series, and it is the same as the corresponding input series in the classical case. And the same goes for PQM, for Hilbert space for PQM. And this follows more or less the definitions from the construction because we are responsible, we have a collection of documents on the appearance of the previous representation. We compute essentially the same representation series as SLM. We can calculate this Hilbert series by... All of these are completely formal characters and formed in the same combination with the Hilbert series in the quantum cases in the classical.

45:00 And it was the same for two, we just had to observe it. Five elements, P and M, we just... So we have the action of S and M by conjugation and M by matrices. The three-year representation plus an x-per-minus one value reduces the representation, and we want to locate these two representations between the coordinators, in the common purpose of pulling out this case. And because I've shown them the non-economic correspondence for each location of such a representation, it gives us up to scale a really good idea of the trace. We can give access to generators of big grains, so the quantum-real matrix in grains is generated by so-called monotone traces of products of generic matrices where the quantum trace of an end-by-end matrix is not the sum of the direct monotone traces of its direct matrix of powers to a particular monotone trace of the identity matrix, the end-by-end identity matrix. Most times square just comes from the best known of our policy, the classic graph system. And quantum placement is generated as an argument and a monument by products of generic physics. Again, we have more on the balance of these products. And this comes out by standard arguments of transcendentality once we omit the cognitive. In general, we can't show it in general, except for a two-by-two case where an inspection of the concrete defining relations may be possible to show that this is an ethereum, and then, automatically, this implies a huge number of probability that one of them will make this.

47:30 There are various invariants that we might look at in the theory, and it is quite likely that it is an algebra. By a theory, we have an argument to prove that Reynolds space was created by our function. And a similar thing for quantum physics. But what comes up in the left-hand side is that we think that it is quite likely that it is a left-hand view. So somehow we look at it like symmetry. We did some... Exit calculations in the 2x2 case are possible because we have a basis in the reflection equation of algebra and the weight and the rewriting and the algorithm to rewrite an algebra in terms of the basis elements and these cross-relations for aqm2 are given in the form that there immediately is some rewriting which is wrong for that, so in principle it is possible to perform calculations. At least not as to a conclusion which surprised us, but we are looking at the pairs of 2x2 matrices. And for now I will give 2x2 matrices. We have capital S and capital Y, these generators. They are the generators of a, q, and 2x. From there, capital S to S. The scalar marking of the quantum trace is small x.

50:00 That's why I'm going into it. Mathematics tests on algebras was quite surprising for us. First of all, G2 is a non-trivial, non-electrolytic object. This is a non-trivial, non-electrolytic object. It's only the case of two by two generatives. The results are known about the events just like G2. This is not a financially connected algebra. So it's very strange that they involve this. The issue of commutativity involves these strange relations, the reality. These are the relations for the two-by-two regression equation algebra. So I think there are different relations for the two-by-two quantum matrices compared to the relation before. And the cross-relations for pairs of two-by-two quantum matrices are written there. So if you remember the middle there, it's an arbitrary relation. So we impose these strange relations for the estimate device, but once we arrange them in two-by-two matrices... They all carry the same algebra as you are assuming, but they are commuting between themselves and the picture there. Thank you very much for your time. In March, I am supposed to stimulate this year. I thought, while we were just talking about that, you can use this to raise an estimate of the overall product. So, does it make sense to define the right and the wrong as meeting in that category?

52:30 Basically, look at the generic object there, and then call it the generic plane. It's supposed to be the same thing, and might also explain, sometimes, what is generic. No other questions then we thank the speakers for the stipend of the interview because we have two breaks in this long series of short talks and eventually we will have another break later so we should be glad that we should continue. But let us without break now continue with the next lecture by Alexander Sibyl. Yeah, okay, we take one minute break for a few minutes.

55:00 And biodegradable k-algebra, which is going to go from I to I as a biodegradable k-algebra. Then such an algebra is called symmetric, if there is a y in the form of this algebra, which is symmetric, and if there is an h in the algebra, linear forms. So this is a place, this is a link, where you can use the symmetric algebra. Mathematical algebras, of course, are good algebras, but they are not good. There are many more. Sixth, Richard Brouwer. He developed and then he defined the DTA.

57:30 This is a patient. This is a set of support agents such that some people get out with a good algebras, a mathematical algebras, a general algebras for papers. Fixes the aim. This algebras is fine in the mental. It will stop at some point. And A is supposed to be just a radical. plus this commutator. Symmetric algebras, as you can see, we have a binary-expanded uniform. The intersection of this autoconon space of the end is the soccer and the center.

1:00:00 And this ideal is called Rayner's ideal. Increasing sequence, taking autoconon, we get a decreasing sequence. Center contains the T1 autoconon, T2 autoconon and so on. And the intersection, as I said, is just this. The most interesting is that all these A's are alphabets, and this Rayleigh's ideal contains another ideal, which is the Hickman ideal, which is the image of a trace, and this contains, let's say, 0 and a simple plot of this alphabet. If you take the square root, then T1A is complemented. So that is HA, but if you take the square root idea, then you are in between C0A and sigma idea. If P is odd, then you get the equality of these simple blocks. If P is equal to 2, then the difference is 0 to the z0, or you can use T1A, or you can use T2A.

1:02:30 And the symmetry of the three forms is what we are asked for, then this COH is really different to this one, we can express this into real-value characters, this is possible to verify that actually here you really have a difference. Square, just the number of blocks in the carbon matrix. I mentioned here, just count the number of blocks with an odd diameter in the carbon matrix. Now, I would like to see how this works actually, the basic algebra, and if you have two more diagrams, then the basic algebra are really isomorphic and of course different, but nevertheless, the result here looks like p, points in there, so between 1997 and 1919, and after more diagrams, and then this is a cosmological declassified and after derived equations.

1:05:00 The top two in this simulation that you have on the right hand side are equivalent as triangulating factories. But the problem there is that still in this relation there are some problems. And these problems, the possibilities which occur, but basically this problem is not fixed. And now, in recent research from last week, we could show...

1:07:30 We could show that actually these powers could be detected for so-and-so, so-and-so, and so we can detect it for the legal defect group, for semi-legal defect group, but for general defect group we could not see anything about this. Think of boobing, actually. It's clear that these ideals are invariable with respect to nerve equivalence. So, this comes from Kuhlsamer's observation with this Kuhlsamer noise. In the center, and for any integer, there is a unit, ZnA, in the center, with the empowerment, Kb, active, Zn is a TnA, so Zn for some choice, so for biological forming.

1:10:00 Okay, I guess we have a... Sorry? Yeah. In which all calculus can be derived in variance. In which all calculus can be derived in variance and coincide, except for... It's really quite hard to distinguish these two, these two blocks, these two blocks. On the side of this semi-diagonal type, there's two symbols that we need. On the diagonal type, there's another, another possibility, even by, you know, a power, and so on and on and on again. So then we have a first language. I think we're more or less on push of a star. At what time? At what time? At the right time. Which means 45 hours on the X. 44, actually. The initial results on this.

1:12:30 We have two elements in the extension of the X. Small x, 18. This is going to be the minimum value of dimensions of extensions between A and B, A minus A minus B, and this is the general value, so this is the value taken on both of the subsets of A and B, and then the other. So the actual theorem works a bit more generally than this, but it's slightly easier to write down. So the general model of X manages all of A's and all of the other potentials on B as well. We have a nice formula for the co-dimensions of A star B. In terms of the co-dimensions of A and B, we have just the general X useful results when trying to study these. So for key thinking, this was studied by Marcus.

1:15:00 Actually, the year before the generalist definition came out, and this was the initial idea, and then we realised that the correct formulations in this more general context, in this case, they might be really clear, and of course, each provides a different model, this is because there are only quite a lot of many for any representation across, there are only quite a lot of many representations. And people, artists themselves, are going to use them for some of these sort of things. You also can sort of show that they're not quite in that make-believe. The more normal thing is to find the relations, the sort of specialisations, the quantum serarations. So here I'm actually using a sort of twisted version of the quantum serarations.

1:17:30 This is the quantum group. What's the quantum group? So, here, you can see it's in the literature, which gives you the information. So, we have an alloy of an I to J, general equations, quantum serrations, I squared, J to J, Q plus 1, IJR, that's equal to 0, and I to X squared, J to J, T to X squared, and these are the quantum serrations. There's no arrow at the end, they're just commuting. One more. Questions closed, by the way. There were no explicit examples, but it was fully calculated. So this is why I chose to do the quencher of it. It's sort of the next most difficult. And also, there are examples of what happens to the chronically unexpected, but similar things happen for most of the main groups. It's a quiver of two birth speeds and two hours. So here we go. There are lots of other modules that you have to give me, but those will go to some of the major ones, to be fun, to be projectives. So here, the first one is about check.

1:20:00 The minimum of algebra equals I, and three of the values, and this is for the next and then the next thing. And in this case, early in Q, which I'm going to re-write, because I think it's a good question because it's part of the big theorem. So this is every element of the composition of one, right? We can be experts uniquely in the form of a model of an MA. So these are all pre-projected, and some of them are pre-injected.

1:22:30 So we have a list of pre-projected and some of them are pre-injected. Just so you know, the module has to be pre-projected. So the proof, so some of the remarks are not going to be too much of a question. M is generated by two elements, R and I. So it's quite easy to see this. For any dimension vector, R, D, any module here, any module, if you can always say, submodule of dimension vector d, j here, The first part is we want to show some commutation between these various things. So the commutation could be in two pre-projectives and we could use this field when we're matching x with matching x, the final code I mentioned is 0. We can work with the commutation between a certain element here and a certain element here, so we can put everything into the required form. Any element here, besides the closure of the orbital part, it's quite clear that we have V1, Vr, this indistinguishable pre-projection of dimension, nr, then this first part is the same as V, and then it's not too difficult to see that this is in fact the orbital closure of the direct sun's model.

1:25:00 Keep in mind, this first bit is the orbital closure of some pre-rejective, this is the orbital closure of some pre-injective, and then we just need to show these varieties again when we need them. So, it works. So it's not too surprising that we don't expect that the... Well, I think one of the more obvious, and it wasn't clear at all what, whether there's even a map from the composition algebra, Defining relations becomes minimal defining relations. So this is two name and projective models. And we can write this as equal state commutes for all the quantities. Two is similar to the three objectives. And the third one, which this was already in just in the world of scope theory,

1:27:30 is expansion and recovery. And this is for all the surprising results. There's infinitely many unneeded, so it's very different from quantum really, because in finance there's just the quantum set of actions, and that's fine, but here we actually need an infinite number, so then what happens with the composition of the algebra, so it's like the second layer, the third layer, and this is the analysis version, not the mathematical version of maths or algebra, but the next layer, either comes from the function theory or something. I'll just write one of them down, it's a funky piece of paper, so we'll just put it on. And you'll see, when we take a look at it, you'll see a cube, sort of a composition algebra. It's a sort of lattice that's formed over the polynomial range, and so it's generated 5, 2, 3, 7, 8. And then we have, if we specialise 0, then we actually get, and every month we want to do a composition. The basic idea is to show that C is very similar to this, so again the infinite connection I need is, and the kernel we can describe is generated by a connection with the composition 1, if you interpret the composition 1 as being this way.

1:30:00 The algebra, I don't want to know your name, but sitting inside the corner, shows a connection with Schubert's, a collection of Schubert's and canonical decompositions. These are Schubert's, a representation of Marxism and anamorphism, just to say.