arXiv subjects

Simon W. Rigby

Publications and source records attributed to Simon W. Rigby.

6 recordsLinked to original sources

arXiv2021

Galois cohomology of algebraic groups acting on the tensor product of two composition algebras

This is a study of algebras with involution that become isomorphic over a separable closure of the base field to a tensor product of two composition algebras. We classify these algebras, provide criteria for isomorphism and isotopy, and determine their automorphism groups, structure groups, and norm-similitude groups. The most interesting cases of these algebras are the $64$-dimensional bi-octonion algebras, on which groups of absolute type $(G_2 \times G_2)\rtimes \mathbb{Z}/2\mathbb{Z}$ and $\mathbf{Spin}_{14}$ act faithfully by automorphisms and norm-preserving isotopies, respectively. We classify the cohomological invariants of these algebras, characterise when they are division algebras, and study their 14-dimensional Albert forms and 64-dimensional octic norms. The main application that we reach is a classification of the mod 2 cohomological invariants of 14-dimensional quadratic forms in $I^3$, and of the group $\mathbf{Spin}_{14}$. This extends Garibaldi's classification of cohomological invariants for lower-dimensional Spin groups.

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arXiv2021

A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

We give a one-to-one correspondence between ideals in the Steinberg algebra of a Hausdorff ample groupoid $G$, and certain families of ideals in the group algebras of isotropy groups in $G$. This generalises a known ideal correspondence theorem for Steinberg algebras of strongly effective groupoids. We use this to give a complete graph-theoretic description of the ideal lattice of Leavitt path algebras over arbitrary commutative rings, generalising the classification of ideals in Leavitt path algebras over fields.

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arXiv2021

The Allison-Faulkner construction of $E_8$

We show that the Tits index $E_8^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. We construct two cohomological invariants, in degrees 6 and 8, of the Tits construction and the more symmetric Allison-Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.

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arXiv2019

The groupoid approach to Leavitt path algebras

When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph's boundary path groupoid. This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras.

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arXiv2018

Tensor products of Steinberg algebras

We prove that $A_R(G) \otimes_R A_R(H) \cong A_R(G \times H)$, if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R} \otimes L_{3,R}$ and $L_{2,R} \otimes L_{2,R}$. Indeed, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $*$-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}} \otimes L_{3,\mathbb{Z}} \not \cong L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}}$ (as $*$-rings).

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arXiv2018

Strongly graded groupoids and strongly graded Steinberg algebras

We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $κ: \mathcal G \to Γ$, to a discrete group $Γ$, such that $κ^{-1}(γ)κ^{-1}(δ) = κ^{-1}(γδ)$, for all $γ, δ\in Γ$. We introduce the category of graded $\mathcal G$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal G$ is strongly graded if and only if every graded $\mathcal G$-sheaf is induced by a $\mathcal G_ε$-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. Applying this result, we obtain a complete graphical characterisation of strongly graded Leavitt path and Kumjian-Pask algebras.

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