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Jari Desmet

Publications and source records attributed to Jari Desmet.

7 recordsLinked to original sources
arXiv2026

Decompositions of Griess algebras beyond the OZ-setting

To each VOA of strong CFT type, we associate a non-associative algebra which generalizes Griess algebras of One-Zero (OZ) VOAs. We compute the fusion law of Ising vectors in this algebra using techniques from Miyamoto, leveraging the module theory of the corresponding Virasoro VOA. This new class of algebras encapsulates the Chayet-Garibaldi algebras, a family of algebras constructed from absolutely simple linear algebraic groups that includes the $3876$-dimensional algebra for $E_8$.

math.RA↗
arXiv2026

Groups of type $\mathrm{E}_8$ over rings via TKK-algebras and their extremal elements

Over any commutative ring containing $\tfrac16$, we study Lie algebras $L$ of type $\mathrm{E}_8$ that arise from the Tits--Kantor--Koecher (TKK) construction on a Brown algebra, and their twisted forms. We construct a smooth scheme $\mathbf{Y}$ of pairs of extremal elements in $L$. When $L$ arises from the TKK-construction, we express the automorphism group, of type $\mathrm{E}_8$, as an $\mathrm{E}_7$-torsor over $\mathbf{Y}$. We show that twisting by this torsor produces the graded isomorphism classes of those algebras isomorphic to $L$, and parametrize these classes by using $\mathbf{Y}$. We show that this torsor is non-trivial, yielding isomorphic Lie algebras of type $\mathrm{E}_8$ that are not graded isomorphic, as opposed to the behaviour over fields.

math.RA↗
arXiv2024

Solid lines in axial algebras of Jordan type $\tfrac{1}{2}$ and Jordan algebras

We show that a primitive axial algebra of Jordan type $η= \tfrac{1}{2}$ is a Jordan algebra if and only if every $2$-generated subalgebra is \emph{solid}, a notion introduced recently by Ilya Gorshkov, Sergey Shpectorov and Alexei Staroletov. As a byproduct, we show that a subalgebra generated by axes $a,b$ is solid if and only if the associator $[L_a,L_b]$ is a derivation. Moreover, we show that $2$-generated subalgebras that are not solid contain precisely $3$ axes.

math.RA↗
arXiv2024

Non-associative Frobenius algebras of type $E_7$

Recently, Maurice Chayet and Skip Garibaldi introduced a class of commutative non-associative algebras. In previous work, we gave an explicit description of these algebras for groups of type $G_2,F_4$ and certain forms of $E_6$ in terms of octonion and Albert algebras. In this paper, we extend this further by dealing with $E_7$ in terms of generalised Freudenthal triple systems.

math.RA↗
arXiv2023

Non-associative Frobenius algebras of type $^1E_6$ with trivial Tits algebras

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). In a previous paper, we gave an explicit description of these algebras for groups of type $G_2$ and $F_4$ in terms of the octonion algebras and the Albert algebras, respectively. In this paper, we attempt a similar approach for type $E_6$.

math.RA↗
arXiv2022

Non-associative Frobenius algebras of type $G_2$ and $F_4$

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). We give an explicit description of these algebras for groups of type $G_2$ and $F_4$ in terms of the octonion algebras and the Albert algebras, respectively. As a byproduct, we determine all possible invariant commutative algebra products on the representation with highest weight $2ω_1$ for $G_2$ and on the representation with highest weight $2ω_4$ for $F_4$. It had already been observed by Chayet and Garibaldi that the automorphism group for the algebras for type $F_4$ is equal to the group of type $F_4$ itself. Using our new description, we are able to show that the same result holds for type $G_2$.

math.RT↗