Minicourse
Axial algebras
Sergey Shpectorov received his PhD from Lomonosov Moscow State University in 1990. He is currently a Professor in the School of Mathematics at the University of Birmingham, United Kingdom. His research focusses on group theory, including actions of groups on various combinatorial, geometric and algebraic objects, most recently on non-associative algebras. He is also interested in purely combinatorial and geometric questions.
The course
Axial algebras are a recently developed class of non-associative algebras, which are inherently related to groups. The motivating examples include the Jordan algebras, related to classical and some exceptional algebraic groups, Matsuo algebras, related to 3-transposition groups, and the Griess algebra, which was used to realise the Monster sporadic simple groups. This minicourse focusses on the general theory of axial algebras, as well as on examples and classification of the two most interesting classes: algebras of Jordan and Monster type.
Outline:
Lecture 1. Introduction into axial algebras
Lecture 2. Structure theory (radical, sum decompositions)
Lecture 3. Algebras of Jordan type, Matsuo algebras
Lecture 4. Classification for eta not equal to ½
Lecture 5. The case of eta=½
Lecture 6. Algebras of Monster type, initial examples
Lecture 7. Double axes, the flip construction
Lecture 8. Classification of the 2-generated case
Outline:
Lecture 1. Introduction into axial algebras
Lecture 2. Structure theory (radical, sum decompositions)
Lecture 3. Algebras of Jordan type, Matsuo algebras
Lecture 4. Classification for eta not equal to ½
Lecture 5. The case of eta=½
Lecture 6. Algebras of Monster type, initial examples
Lecture 7. Double axes, the flip construction
Lecture 8. Classification of the 2-generated case
Bibliography
J.I. Hall, F. Rehren, and S. Shpectorov, Primitive axial algebras of Jordan type, J. Algebra, 437 (2015) 79−115.
S. Khasraw, J. McInroy and S. Shpectorov, On the structure of axial algebras, Trans. Amer. Math. Soc., 373 (2020) 2135−2156.
A. Galt, V. Joshi, A. Mamontov, S. Shpectorov and A. Staroletov, Double axes and subalgebras of Monster type in Matsuo algebras, Comm. Algebra 49 (2021) 4208−4248.
J. McInroy, S. Shpectorov, Split spin factor algebras, J. Algebra, 595 (2022) 380−397.
C. Franchi, M. Mainardis and S. Shpectorov, An infinite-dimensional 2-generated primitive axial algebra of Monster type, Annali di Matematica, (2021), https://doi.org/10.1007/s10231−021−1 157−8.
S. Khasraw, J. McInroy and S. Shpectorov, On the structure of axial algebras, Trans. Amer. Math. Soc., 373 (2020) 2135−2156.
A. Galt, V. Joshi, A. Mamontov, S. Shpectorov and A. Staroletov, Double axes and subalgebras of Monster type in Matsuo algebras, Comm. Algebra 49 (2021) 4208−4248.
J. McInroy, S. Shpectorov, Split spin factor algebras, J. Algebra, 595 (2022) 380−397.
C. Franchi, M. Mainardis and S. Shpectorov, An infinite-dimensional 2-generated primitive axial algebra of Monster type, Annali di Matematica, (2021), https://doi.org/10.1007/s10231−021−1 157−8.