Algebra objects and internal modules

The conventions are those of Etingof et al. (2015), Chapter 7. The explicit construction and the algorithms implemented here are developed in Mäurer (2026), Chapter 3. An algebra object has maps $m:A\otimes A\to A$ and $u:\mathbb 1\to A$, satisfying

\[\label{eq:algebra-associativity} m\circ(m\otimes\mathrm{id}_A) =m\circ(\mathrm{id}_A\otimes m)\circ a_{A,A,A}.\]

With the normalized unit constraints used by the package, the unit equations are

\[\label{eq:algebra-unit} m\circ(u\otimes\mathrm{id}_A)=\mathrm{id}_A =m\circ(\mathrm{id}_A\otimes u).\]

These are Etingof et al. (2015), Definition 7.8.1 and Eqs. (7.10)–(7.11), after using the package's strict unit constraints. AlgebraObject(C,A,m,u) stores the data; is_algebra checks the equations. The constructor does not verify that C, $A$, and the domains and codomains of $m$ and $u$ are compatible. Supply both maps in the parent category of $A$, with exactly the indicated sources and targets.

A right action $r:M\otimes A\to M$ and a left action $\ell:A\otimes M\to M$ satisfy

\[\label{eq:right-module-associativity} r\circ(r\otimes\mathrm{id}_A) =r\circ(\mathrm{id}_M\otimes m)\circ a_{M,A,A},\]

and

\[\label{eq:left-module-associativity} \ell\circ(m\otimes\mathrm{id}_M) =\ell\circ(\mathrm{id}_A\otimes\ell)\circ a_{A,A,M},\]

respectively. Their unit equations are $r\circ(\mathrm{id}_M\otimes u)=\mathrm{id}_M$ and $\ell\circ(u\otimes\mathrm{id}_M)=\mathrm{id}_M$ under the same normalized identifications Etingof et al. (2015), Definition 7.8.5 and Eqs. (7.12)–(7.13). The current is_right_module and is_left_module methods check the displayed associativity equations. The is_bimodule method also checks the compatibility equation below. None of these predicates checks the separate unit equations for manually supplied actions. Together with is_algebra, they compare morphisms by structural equality and do not provide a ball-overlap branch for independently supplied numerical structure maps.

For an $(A,B)$-bimodule, the compatibility between the left action $\ell$ and right action $r$ is

\[\label{eq:bimodule-compatibility} r\circ(\ell\otimes\mathrm{id}_B) =\ell\circ(\mathrm{id}_A\otimes r)\circ a_{A,M,B}.\]

This is Etingof et al. (2015), Definition 7.8.25 and Eq. (7.18). The following example uses the unit algebra, so its modules recover the original category:

using TensorCategories, Oscar
C = ising_category()
U = one(C)
A = AlgebraObject(C,U,id(U),id(U))
@assert is_algebra(A)
M = category_of_right_modules(A)
F = free_right_module(C[3],A)
@assert is_isomorphic(object(F),C[3])[1]
object(F)

category_of_left_modules(A), category_of_right_modules(A), and category_of_bimodules(A) construct the corresponding parents. Free modules use tensoring with the algebra. The relative tensor product of a right and left module is a coequalizer; bimodules acquire their tensor product this way Etingof et al. (2015), Definition 7.8.21 and Eq. (7.16). A category of right modules is not automatically monoidal.

An algebra is separable when its multiplication splits as an $A$-bimodule map Etingof et al. (2015), Definition 7.8.29. This is the hypothesis used by the simple-module algorithm below.

Computing simple modules

If $\mathcal C$ is fusion and $A$ is separable, the category of right $A$-modules is semisimple Etingof et al. (2015), Proposition 7.8.30. Every simple module occurs in a free module $X_i\otimes A$ for some simple $X_i\in\mathcal C$. Accordingly, simples(M) constructs the free modules on the simple objects, computes their minimal subquotients, and removes duplicate isomorphism classes. Under the separability hypothesis these subquotients are direct summands. This is Algorithm 7 of Mäurer (2026), §3.4. The analogous construction applies to left modules and, with both algebras separable, to bimodules.

Separability is the hypothesis that makes this an exhaustive semisimple decomposition algorithm. Constructing category_of_right_modules(A) itself does not assert separability.

Commutativity and structure searches

Commutativity requires a supplied braiding and means $m\circ c_{A,A}=m$. In the terminology used by etale_algebra_structures and Etingof et al. (2015), §8.27.2, p. 263, an étale algebra is commutative and separable; connectedness is a separate condition. This differs from Mäurer (2026), Definition 3.3.23, where “étale” includes connectedness.

algebra_structures, separable_algebra_structures, commutative_algebra_structures, and etale_algebra_structures search for structures on a fixed underlying object. The polynomial systems can have positive-dimensional solution spaces. These searches do not generally classify all algebras up to isomorphism, and an empty sampled result is not a proof of nonexistence.

Continue with Group actions and equivariantization.