nLab paraunital algebra

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Idea

Given a commutative ring RR, a paraunital RR-algebra is a nonunital R R -algebra AA where there is an element ιA\iota \in A and an involution xx¯x \mapsto \overline{x} such that xι=ιx=x¯x \cdot \iota = \iota \cdot x = \overline{x} for all xAx \in A. A paraunital \mathbb{Z}-algebra is also called a paraunital ring.

A unital algebra is a paraunital algebra in multiple different ways:

  • where the element is given by the unit 11 and the involution is given by the identity function xxx \mapsto x.

  • where the element is given by the negation of the unit 1-1 and the involution is given by negation xxx \mapsto -x

These were first defined in the specific context of composition algebras in the generalized Hurwitz theorem in Elduque 2021 but could be generalized from composition algebras to any RR-algebra.

 Generalizations

This concept could be generalized from the category of RR-modules to any monoidal category:

A paraunital algebra object in a monoidal category (C,I,)(C, I, \otimes) is an object ACA \in C with morphisms ι:IA\iota:I \to A, π:AAA\pi:A \otimes A \to A, and j:AAj:A \to A such that jj=id Aj \circ j = \mathrm{id}_A and the following diagrams commute:

IA ιid A AA λ A π A j A, \array{ I \otimes A &\overset{\iota \otimes \mathrm{id}_A}{\longrightarrow}& A \otimes A \\ \downarrow^{\lambda_A} & & \downarrow^{\pi} \\ A &\overset{j}{\longrightarrow}& A } \,,
AI id Aι AA ρ A π A j A, \array{ A \otimes I &\overset{\mathrm{id}_A \otimes \iota}{\longrightarrow}& A \otimes A \\ \downarrow^{\rho_A} & & \downarrow^{\pi} \\ A &\overset{j}{\longrightarrow}& A } \,,

where λ A:IAA\lambda_A:I \otimes A \to A and ρ A:AIA\rho_A:A \otimes I \to A are the left and right unitors of the monoidal category.

In the category of sets, paraunital algebra objects are called paraunital magmas, and in the category of abelian groups, paraunital algebra objects are called paraunital rings.

References

Last revised on October 4, 2023 at 15:18:14. See the history of this page for a list of all contributions to it.