nLab paraunital algebra

Idea

Given a commutative ring $R$, a paraunital $R$-algebra is a nonunital $R$-algebra $A$ where there is an element $\iota \in A$ and an involution $x \mapsto \overline{x}$ such that $x \cdot \iota = \iota \cdot x = \overline{x}$ for all $x \in A$. 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 $1$ and the involution is given by the identity function $x \mapsto x$.

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

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 $R$-algebra.

 Generalizations

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

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

$$\array{ I \otimes A &\overset{\iota \otimes \mathrm{id}_A}{\longrightarrow}& A \otimes A \\ \downarrow^{\lambda_A} & & \downarrow^{\pi} \\ A &\overset{j}{\longrightarrow}& A } \,,$$

$$\array{ A \otimes I &\overset{\mathrm{id}_A \otimes \iota}{\longrightarrow}& A \otimes A \\ \downarrow^{\rho_A} & & \downarrow^{\pi} \\ A &\overset{j}{\longrightarrow}& A } \,,$$

where $\lambda_A:I \otimes A \to A$ and $\rho_A:A \otimes I \to A$ 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.

Related concepts

• magma, unital magma

• nonunital algebra, nonunital ring

• composition algebra

• generalized Hurwitz theorem

References

• Alberto Elduque, Composition algebras, Chapter 2 in: Abdenacer Makhlouf (ed.), Algebra and Applications I: Non-associative Algebras and Categories, Sciences-Mathematics, ISTEWiley (2021) 27-57 [arXiv:1810.09979, ISBN:978-1-789-45017-0]