nLab
associative unital algebra

Definitions

For $R$ a commutative ring, an associative unital $R$-algebra is equivalently

• a monoid internal to $R$Mod equipped with the tensor product of modules $\otimes$;

• a pointed one-object category enriched over $(R\mathrm{Mod},\otimes )$;

• a pointed $R$-algebroid with one object;

• an $R$-module $V$ equipped with linear maps $p:V\otimes V\to V$ and $i:k\to V$ satisfying the associative and unit laws;

• a ring $A$ under $R$.

If there is no danger for confusion, one often says simply ‘associative algebra’, or even only ‘algebra’.

More generally, a (merely) associative algebra need not have $i:R\to V$; that is, it is a semigroup instead of a monoid.

Less generally, a commutative algebra (where associative and unital are usually assumed) is an abelian monoid in $\mathrm{Vect}$.

Variants

• A cosimplicial algebra is a cosimplicial object in the category of algebras.

• A dg-algebra is a monoid not in Vect but in the category of (co)chain complexes.

• A smooth algebra is an associative $ℝ$-algebra that has not only the usual binary product induced from the product $ℝ\times ℝ\to ℝ$, but has a $n$-ary product operation for every smooth function ${ℝ}^n\to ℝ$.

  This may be understood as a special case of an algebra over a Lawvere theory, here the Lawvere theory CartSp.

Examples

• function algebra

Properties

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	${\mathrm{Mod}}_A$
$R$-algebra	${\mathrm{Mod}}_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	${\mathrm{Mod}}_A$
$R$-2-algebra	${\mathrm{Mod}}_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category	3-category of module 2-categories
$A$	${\mathrm{Mod}}_A$
$R$-3-algebra	${\mathrm{Mod}}_R$-4-module

Related concepts

• augmented algebra

• differential algebra

• differential graded algebra, A-infinity algebra