nLab nonunital algebra

Redirected from "nonunital algebras".
Contents

Contents

Idea

A nonunital algebra is like an associative algebra but without specified identity element. The associative algebra version of nonunital ring, see there for more.

Examples

The zero algebra is an example of an algebra that is nonunital as soon as the underlying vector space has dimension greater than 0. Given a vector space VV, simply define its product to be μ:vw0\mu: v\otimes w\mapsto 0 for all vectors v,wVv,w\in V. Note that since a morphism f:(A,μ)(A,μ)f:(A,\mu)\to (A',\mu') is a morphism of vector spaces f:AAf:A\to A' such that f(μ(vw))=μ(f(v)f(w))f(\mu(v\otimes w))=\mu'(f(v)\otimes f(w)), a morphism between zero algebras is the same as linear maps of the underlying vector spaces. Thus, the zero product describes a full embedding of the category of vector spaces into the category of (nonunital) algebras.

References

A definition of algebraic K-theory for nonunital rings is due to

  • Daniel Quillen, K 0K_0 for nonunital rings and Morita invariance, J. Reine Angew. Math., 472:197-217, 1996.

with further developments (in KK-theory) including

  • Snigdhayan Mahanta, Higher nonunital Quillen K’-theory, KK-dualities and applications to topological T-dualities, J. Geom. Phys., 61 (5), 875-889, 2011. (pdf)

Last revised on April 11, 2024 at 19:59:10. See the history of this page for a list of all contributions to it.