Axial algebras and groups related to them

This is the original website: axial-algebras.

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2021, Jan-Jun

Star denotes a seminar held in the mixed format

1. 2021.01.26   - Mamontov A.S., axial algebras (Frobenius form, Miyamoto involution) slides video (in Russian)

2. 2021.02.02   - Mamontov A.S., axial algebras (2-generated a.a., radical of the form, Matsuo algebras) slides video (in Russian)

3. 2021.02.09   - Staroletov A.M., 3-generated axial algebras video (in Russian)

4. 2021.02.16   - Staroletov A.M., 3-generated axial algebras and Jordan algebras video (in Russian)

5. 2021.03.02   - Gubarev V.Yu., Jordan algebras: classification and Peirce decomposition slides video (in Russian)

6. 2021.03.09* - Gubarev V.Yu., Jordan algebras: properties of Peirce decomposition video (in Russian)

7. 2021.03.16   - Shpectorov S.V.,  video

   (1) why we can assume bijectivity between axes and Miyamoto involutions (based on Hall, Segev, Sh);

   (2) general construction algorithm;

   (3) how to select  values of the Frobenius form to match the orders of $\tau_a\tau_b$;

   (4) results of our calculation;

   (5) Jordan algebra for a group of reflections (based on De Medts-Rehren)

8. 2021.03.23   - Justin McInroy (University of Bristol), The structure of axial algebras slides video

   Axial algebras are a new class of non-associative algebra, introduced recently by Hall, Rehren and Shpectorov, which have a strong link to groups.  They generalise both a large class of Jordan algebras and also the Griess algebra. In this talk we will discuss various aspects of their structure.  What are their ideals?  How can we easily compute them?  What are the choices for a Frobenius form (a bilinear form which associates with the algebra multiplication)?  Can we decompose the algebra into a direct sum of subalgebras?  When are these subalgebras axial?  What is the best (finest) sum decomposition?

   This is joint work with Sergey Shpectorov (Birmingham) and Sanhan Khasraw (Salahaddin University-Erbil)

9. 2021.03.30   - Discussion of open problems video

10. 2021.04.06 - Shpectorov S.V., double axis construction slides video

11. 2021.04.13 - Shpectorov S.V., double axis construction-II slides video

12. 2021.04.20 - Tkachev V. (Linköping University), Minimal cones and Hsiang algebras-I slides video

    I will discuss a nonassociative algebra approach to certain problems of differential geometry and partial differential equations.  The basic idea comes back to the Freudenthal-Springer-Tits construction of exceptional Jordan algebras: one replaces a study of a cubic form u by the study of a certain commutative algebra $A(u)$ recovering properties of u from the properties of the corresponding algebra, and vice versa. The algebra $A(u)$ is not necessarily associative but it is metrized, i.e. the multiplication operator $L(x)$ by $x$ is self-adjoint. The correspondence $u\rightarrow A(u)$ is natural in the sense that many well-established algebraic concepts can be intrinsically read out from the analytic structure of $u$. Furthermore, the algebra structure $A(u)$ identifies many different geometric and analytic patterns of the corresponding solution u. In my talk, I will explain how this method helps to solve a  long-standing problem on classification of cubic minimal cones, the so-called Hsiang problem. The corresponding algebras have many similarities with axial algebras (any Hsiang algebra is generated by idempotents with the same spectrum and fusion laws).

13. 2021.04.27 - Clara Franchi (Università Cattolica del Sacro Cuore), On the classification of 2-generated primitive axial algebras of Monster type video

    I shall talk about the general strategy for the classification of 2-generated primitive axial algebras of Monster type ($\alpha, \beta$). I’ll define a category for such algebras and construct its universal object. This leads to a natural trichotomy depending whether $\alpha=2\beta, \alpha=4\beta$ and $\alpha\not \in {2\beta,4\beta}$. I shall discuss the main questions and results in these three cases.

    2021.05.04 - No seminar!

14. 2021.05.11 - Mario Mainardis (Università degli Studi di Udine), The Highwater algebra and its cover in characteristic 5 slides video

    The Highwater algebra is essentially the unique case of an infinite-dimensional 2-generated primitive axial algebra of Monster type over an arbitrary field F of characteristic other than 2 and 3. I shall discuss its construction, its group of automorphisms and describe some of its relevant features. I’ll also show that, in case the ground field has characteristic 5 the Highwater algebra has a proper cover which is still a 2-generated primitive axial algebra of Monster type.

15. 2021.05.18 - Takahiro Yabe (The University of Tokyo), On the classification of 2-generated axial algebras of Majorana type - Seminar was stopped because of the net problems

    I will talk about the details of classification of 2-generated axial algebras which is called Majorana type (Monster type). A 2-generated symmetric axial algebra of Majorana type is isomorphic to one of 18 universal type or their quotients if its axial dimension is less than 6 or it is an algebra over the field of characteristic not 5. I will talk about the definition of some algebras as examples and sketch of my proof.

16. 2021.05.25 - Tkachev V. (Linköping University), Minimal cones and Hsiang algebras-II slides video

    In the 2nd part of my talk I will explain how various methods of nonassociative algebra (including the Springer-Freudenthal construction, the Peirce decomposition etc) help to classify Hsiang algebras of minimal cones. In particular, the classification uses a dichotomy of Hsiang algebras (Clifford vs exceptional) which crucially depends on the Jordan algebra structure on the -1/2 Peirce subspace.

17. 2021.06.01 - Shpectorov S.V., Split spin factor algebras slides video

    Recent classification by Yabe of symmetric 2-generated algebras of Monster type introduced several new classes of examples. Trying to make sense of these new algebras, McInroy and the speaker generalised one of them, III(al,1/2,dl), to any number of generators, obtaining a rich family, singularly similar to the class of spin factor Jordan algebras. In the talk, we will describe the new algebras and their properties, including all idempotents, the fusion law, ideals and factors.

2021, Sep-Dec

18. 2021.09.07 - Takahiro Ybe (The University of Tokyo), On the classification of 2-generated axial algebras of Majorana type slides video chat

    I will talk about the details of classification of 2-generated axial algebras which is called Majorana type (Monster type). A 2 generated symmetric axial algebra of Majorana type is isomorphic to one of 18 universal type or their quotients if its axial dimension is less than 6 or it is an algebra over the field of characteristic not 5. I will talk about the definition of some algebras as examples and sketch of my proof.

19. 2021.09.14 - Roman Kozlov (Sobolev Institute of Mathematics), Introduction to the vertex algebras: basic concepts slides video

    In the first of two talks basic notions and properties will be given. Will be stated and proven basic lemmas and theorems, presented and applied technique to finding Grobner-Shirshov bases and, hence, linear bases. The presentation will be held within two different approaches to treat vertex algebras: as vector spaces with a list of axioms and defined so called “formal distributions” or as algebraic systems endowed with a structure of left-symmetric algebra and conformal algebra at the same time.

20. 2021.09.21 - Roman Kozlov (Sobolev Institute of Mathematics), Introduction to the vertex algebras: lattices and Monster slides video

    In the second talk we introduce a concept of vertex operator algebras (VOA), particular vertex algebras with inner Virasoro structure. Then will be explicitly constructed a lattice VOA for any lattice L. At last, will be constructed and studied a particular lattice VOA, the Monster module, which contains the Griess algebra and has the monster group as the group of automorphisms.

21. 2021.09.28 - Justin McInroy (University of Bristol), Axets and shapes in axial algebras slides video

    The shape of an axial algebra was first introduced by Ivanov and is the configuration of 2-generated subalgebras on the axes. This is analogous to a group amalgam. Similarly to the group case, we want to be able to talk about shapes without an algebra. To do this, we introduce an axet, which abstracts the properties of a closed set of axes, and shapes on an axet. In this talk, we will introduce axets and shapes and describe some of their properties. We will classify the 2-generated axets which will give us a new family which has a hitherto unseen configuration of axes. As an example, we will calculate the axet, and hence the size of a closed set of axes, for the 2-generated algebras of Jordan type 1/2. This is joint work with Sergey Shpectorov (Birmingham).

22. 2021.10.05 - Tendai M. Mudziiri Shumba (University of Johannesburg), Axial algebras for the sporadic simple group HS slides video

    We present constructions of axial algebras for the Higman-Sims sporadic simple group via Norton algebras. Fusion laws are presented as well as the extensions of these algebras by unit. This is work that was part of a PhD thesis supervised by Bernardo Rodrigues and Sergey Shpectorov.

23. 2021.10.12 - Shpectorov S.V., How to find the full automorphism groups of Matsuo algebras slides video

    I will discuss a possible approach to finding the full automorphism group of a general Matsuo algebra with the parameter eta not equal to 1/2. Equivalently, it means finding the complete set of axes of Jordan type in such algebras. I’ll start with reviewing the paper by Hall and the speaker on the spectra of the diagrams of 3-transposition groups focussing on the simplest case of symmetric groups. I will demonstrate how the tables in this paper can be used to solve the above problem.

24. 2021.10.19 - Gubarev V. (joint with Gorshkov I.), Quasi-definite axial algebras of Jordan type 1/2 slides video

    We consider axial algebras of Jordan type 1/2 which satisfy additional conditions on the Frobenius form and on the property of idempotents. Under such conditions we may state when an axial algebra of Jordan type 1/2 is unital or finite-dimensional.

25. 2021.10.26 - Segev Y. (Ben-Gurion University), Axes in non-commutative algebras slides video

    Let A be a non-associative (i.e. not necessarily associative) non-commutative algebra, with or without an identity element over a field $F$ (usually of characteristic not 2). We start experimenting with the notion of an axis in such an algebra, in the simplest possible way. I will present results about 2-generated such algebras. There are open problems, and, indeed, this is sort of an experiment. One possible goal, is that in generalizing away from commutativity, one can come back to commutative primitive axial algebras of Jordan type (for example) and get results there. This is joint work with Louis Rowen.

26. 2021.11.02 - Yunxi Shi (University of Birmingham), Axial algebras of Monster type (2$\eta$, $\eta$) for symplectic and orthogonal groups over $F_2$ slides video

    In this talk, we investigate the classes of involutions of the isometry group of a non-degenerate symplectic space over the field with two elements and we apply the double axis construction to build new axial algebras of Monster type $(2\eta, \eta)$. We also consider the case of the orthogonal space and describe the involutions and flip subalgebras in this case.

27. 2021.11.09 - Pilar Paez Guillán (University of Santiago de Compostela), On central extensions of axial algebras slides video

    In this talk, we will describe a method for constructing axial algebras from a given one, basing on the method of Skjelbred-Sund for classifying nilpotent Lie algebras by means of central extensions. We will discuss the usefulness of this technique and apply it to some examples, such as all axial algebras of dimension 2 over an algebraically closed field.

28. 2021.11.16 - Staroletov A.M., On 3-generated groups of Jordan type slides video

    Inspired by the work of Gorshkov and Staroletov on 3-generated primitive axial algebras of Jordan type, we study the corresponding 3-generated Miyamoto groups. Of particular interest are the groups for algebras over quadratic fields: in these cases, the order of two Miyamoto involutions lies in a small list of values (if finite). As a consequence, we get a class of groups that extends the class of 3-transposition groups.

2021.11.23 - No seminar!

29. 2021.11.30 - Fox D. (Universidad Politécnica de Madrid), Partial associativity conditions and trace-forms slides video chat

    There will be explained, for general not necessarily associative algebras, notions of partial and quantitative associativity, called projective and sectional nonassociativity - for which the Norton inequality is motivating - and their interaction with the invariance of certain trace forms (e.g. the Killing type trace form) will be discussed through some simple examples. As a toy model, there will be described the characterization in these terms of the algebra structure on the standard representation of the symmetric group as the unique up to isomorphism projectively associative, Killing metrized, exact commutative algebra. These notions are complementary to the axial algebra framework, and it is hoped they suggest some interesting questions.

30. 2021.12.07 - Ivanov A.A., Towards Majorana representation of $U_3(5)$ slides video

    This work is with contributions of Andries Brouwer, Clara Franchi, Willian Giuliano, Mario Mainardis. We are aimed to construct a Majorana representation of the group $U_3(5)$. The shape is uniquely determined and the Majorana axis are arranged in a distance regular graph. We estimate the dimension of the representation based on embedding into the Monster and calculate the inner product to get the rank. There is still a step to be completed.

2022, Sep-Dec

31. 2022.09.26 - Tendai Shumba (Mathematical Center in Akademgorodok), On the automorphism groups of axial algebras slides video

    Let $A := (X, F)$ be an axial algebra with Miyamoto group G. We are interested in finding the full automorphism group, $\mathrm{Aut}(A)\geq G$, of $A$. If there is a grading, this is equivalent to finding all the axes in $A$. We discuss the techniques used and report on results obtained for some algebras of modest dimensions. In particular, we take $G = S_4$, and $F$ to be the Monster fusion law.

32. 2022.10.10 - Shpectorov S.V., A “solid” approach to algebras of Jordan type half slides video

    Classification of algebras of Jordan type half remains an important open problem. We will discuss an approach to the classification via the concept of a solid subalgebra. Namely, let $A$ be an algebra of Jordan type half. A (2-generated) subalgebra $B$ of $A$will be called solid when every primitive idempotent from $B$ is an axis of Jordan type half in $A$.

    In the talk we show that the 2-generated subalgebras $B$ of $A$ are almost always solid. In fact, if the ground field is of characteristic 0, $B=<<a,b>>$ can be non-solid only in two specific situations, where the order of $\tau_a\tau_b$ is 3 or 4. Hence, in characteristic 0, an algebra $A$ without infinite solid lines should have a 4-transposition group for its Miyamoto group.

    We also discuss some examples, found by Gorshkov and Staroletov, of Matsuo algebras which are not Jordan algebras and yet contain infinite solid lines.

33. 2022.11.21 - Faarie Alharbi (Birmingham University), Automorphism groups of Matsuo algebras and aligned 3-transposition groups slides video

    In this talk, we look for the exceptions to the general situation that the automorphism group of a Matsuo algebra (with $\eta$ not 1/2) is the same as the automorphism group of the underlying group of 3-transpositions. We will describe some examples where a Matsuo algebra has additional axes and a larger group of automorphisms. Further, we will introduce a method to investigate all cases of pairs of 3-transposition groups that are in cross characteristic and deduce that no new examples arise from this situation.

34. 2022.12.05 - Jari Desmet (Ghent University), Representations of algebraic groups with an interesting axial structure slides

    In an effort to understand more about the representations of algebraic groups (and in particular $E_8$), a paper by Tom De Medts and Michiel van Couwenberghe and a paper by Maurice Chayet and Skip Garibaldi independently constructed algebras for a simple adjoint algebraic group $G$, on which $G$ acts as automorphisms. In this talk, we give an overview of the construction by Maurice Chayet and Skip Garibaldi, and describe how it leads to idempotents with surprisingly simple fusion laws for types $B_n$, $C_n$, $F_4$ and $G_2$.

2024, Feb-Jul

35. 2024.02.27 - Albert Gevorgyan (Imperial College London), Monster embeddings of 3-transposition groups via standard Majorana representations video

    The Monster group $M$ is the largest sporadic simple group, which has more than 8 · 1053 elements. In addition, it is the group of automorphisms of the 196, 884-dimensional Fischer-Griess algebra VM, which is equipped with a positive definite inner product (·, ·), and a commutative, non-associative algebra ·, which satisfy to the relation $(x·y, z) = (x, y·z)$. The algebra VM is generated by a set of axial vectors A. In 2009, A. A. Ivanov axiomatized some properties of the axes $a\in A$ and introduced the notions of Majorana algebra and Majorana representation. Later, Majorana theory proved itself to be a powerful machinery to study the subgroup structure of M, and the subalgebra structure of VM.

    The 3-transposition groups with a trivial center and a simple derived subgroups are categorized by $B$. Fischer. In addition, the Monster group $M$ contains subgroups isomorphic to quite big 3-transposition groups, or their subgroups of index 2. Therefore, there is a motivation to study Majorana representations of 3-transposition groups.

    Firstly, we find the sizes of the maximal symmetric subgroups of the groups from the Fischer list, generated by the transpositions. Then, we use this information to find all pairs of 3-transposition groups from the Fischer list, which can be embedded into each other. Furthermore, we find groups from the Fischer list, which admit a standard Majorana representation. The main result is that a group from the Fischer list, except possibly F i24, admits a standard Majorana representation if and only if it can be embedded in the Monster group.

36. 2024.03.12 - Michael Turner (University of Birmingham), Binary Axial Algebras video

    For an axial algebra with a $C_2$ graded fusion law, we can define a natural automorphism for each axis called the Miyamoto involution. Binary axial algebras have pairs of axes with the property that the pairs are invariant under the Miyamoto involutions. We will start by a quick introduction to assure that everyone is on the same page. Defining binary axial algebras formally, we can produce a binary diagram to represent how each pair acts on each other. Further, we can say a few results for general binary axial algebras before focusing the rest of the talk on the fusion law being Monster type. Firstly, we will present generalisations of some 2-generated axial algebras to n-generated algebras which are binary. Secondly, we will look at 3-generated binary axial algebras, giving examples and partial classifications as well as open problems.

37. 2024.03.26 - Jari Desmet (Ghent University), A characterization of Jordan algebras using solid lines video slides

    The classification of primitive axial algebras of Jordan type half is still an open problem. Recently, Gorshkov, Shpectorov and Staroletov introduced solid subalgebras to tackle this problem. In this talk, I give a method to prove that a primitive axial algebra of Jordan type half is a Jordan algebra if and only if all its 2-generated subalgebras are solid, over fields of characteristic not equal to 2. Using similar techniques, we extend the result by Gorshkov, Shpectorov and Staroletov that 2-generated subalgebras are solid whenever they contain more than 3 axes to positive characteristic.

38. 2024.04.09 - Vsevolod Gubarev (joint with F. Mashurov and A. Panasenko), Generalized sharped cubic form and split spin factor algebra video slides

    There is a well-known construction of a Jordan algebra via a sharped cubic form. We introduce a generalized sharped cubic form and prove that the split spin factor algebra is induced by this construction and satisfies the identity $((a,b,c),d,b) + ((c,b,d),a,b)$ $+$ $((d,b,a),c,b)$ $=$ $0$.

39. 2024.04.23 - Sergey Shpectorov, The universal baric algebra of Jordan type half video slides

    By the result of De Medts, Rowen and Segev, 4-generated algebras of Jordan type half have dimension at most 81. In a joint project with Yunxi Shi, we tried to see what the universal 4-generated algebra of Jordan type half looks like in the simplest case where all values of the Frobenius form on pairs of axes are equal to one. It turns out that in this case the dimension 81 is only reached in characteristic 3, while in all other characteristic there is a much tighter bound. In the final part of the talk, we will discuss the possibility of extending our methods to arbitrary values of the Frobenius form.

40. 2024.07.22 - Sergey Shpectorov, On the structure of axial algebras of Jordan type half video slides

2024, Sep-Dec

41. 2024.09.24 - Louis Rowen, Weakly primitive axial algebras (jointly with Y. Segev) slides video

    In earlier work we studied the structure of primitive axial algebras of Jordan type (PAJ’s), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and various noncommutative examples. Also we broaden our investigation to the case of 2-generated algebras for which only one axis satisfies the fusion rules.

    As an example we describe precisely the 2-dimensional axial algebras and the 3-dimensional and 4-dimensional weakly primitive axial algebras of Jordan type (weak PAJ’s), and we see, in contrast to the case for PAJ’s, that there are higher dimensional weak PAJ’s generated by two axes.

    We also obtain a Frobenius form.

42. 2024.10.08 - Sergey Shpectorov, Radicals in Matsuo algebras and their flip subalgebras (joint project with B. Rodrigues) slides video

    Matsuo algebras, introduced by Matsuo in 2007 are an important class of algebras of Jordan type. Every flip (an automorphism of order 2) $\sigma$ of a Matsuo algebra M defines a flip subalgebra of M generated by all single and double axes fixed by $\sigma$. These flip subalgebras can be viewed as twisted versions of Matsuo algebras and they belong to the class of algebras of Monster type $(2\eta,\eta)$. There is currently a project underway focussing on the classification of flips and properties of flip subalgebras.

    One of the key properties of an axial algebra is whether it is (semi)simple and if not then what is its radical. It turns out that both Matsuo algebras and their flip subalgebras are generically (i.e., for all but finitely many values of the parameter $\eta$) (semi)simple, that is, they have trivial radical. The exceptional values of $\eta$, for which the radical is non-zero, are called critical. Hall and Shpectorov suggested a method of finding the critical values for an arbitrary Matsuo algebra and finding the dimension of the radical.

    In the talk we will present a generalisation of this method to the class of flip subalgebra. It turns out that flip subalgebras have the same critical values as their ambient Matsuo algebras and The dimension of the radical can be found by solving a simple system of linear equations.

43. 2024.10.22 - Ching Hung Lam, Some 3-transposition groups arising from VOA theory video

    We will discuss several examples of 3-transposition groups that can be realized as automorphism subgroups of vertex operator algebras and explain the ideas behind such constructions. In particular, we will discuss some generalizations of the theory of Miyamoto involutions associated with simple Virasoro VOA of central charge 1/2 to other Virasoro VOA in the unitary series.

44. 2024.11.05 - Michael Turner, Double axes and constructing axial algebras of Monster type slides video

    For two orthogonal axes, we can produce a double axis by taking their sum. This area has been applied to produce subalgebras of Matsuo algebras by Galt, Joshi, Mamontov, Shpectorov, and Staroletov in 2021 as well as other papers. Similar work can be applied to axial algebras of Monster type and using double axes to help construct new algebras. We will begin with the basics around double axes and at the specific conditions we desire. Our work is only concerned with shapes of an axial algebra which have certain subalgebras of 4A, 4J, and their quotients of codimension one. Using double axes, we can produce a subalgebra which is of Jordan type and then expand to the whole algebra. Looking at each possible case, we will present classifications and open questions which depend on understanding axial algebras of Jordan type. This is joint work with Justin McInroy.

45. 2024.11.19 - Hans Cuypers, Axial algebras related to polar spaces and Jordan algebras of quasi-Clifford algebras video

    Abstract

46. 2024.12.17 - Bernardo Rodrigues, Axial algebras of Monster type from unitary groups slides video

    Abstract

If you have any questions please do not hesitate to write by e-mail: [email protected]

Bibliography:

1. Axial algebras (non-official lecture notes), D. Craven, S. Shpectorov, 2017.
2. Axial algebras, J. McInroy, 2018 (update 2020).