Universal-element conjecture for central simple Lie algebras

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Let g\boldsymbol{\mathfrak g} be a central simple Lie algebra over a field, and define its type to be the type of a splitting simple Lie algebra obtained from g\boldsymbol{\mathfrak g} by extension of scalars. Two Lie algebras are assembled from the same elements when their corresponding a-schemas are equivalent and the corresponding terms are proportional. Universal elements are Lie algebra structures over Z\mathbb Z.

Universal-element conjecture. All central simple Lie algebras of the same type can be assembled from the same universal elements, which are Lie algebra structures over Z\mathbb Z.

The conjecture seeks to explain the common assemblage patterns of central simple Lie algebras within a fixed type across different ground fields. It is motivated by the examples and representations discussed in the paper, but no proof or disproof is given.

References

Primary source

Alexandre M. Vinogradov, “Assembling Lie Algebras from Lieons”, arXiv:1205.6096 (2012).

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