Semisimple Cellular Algebras

• Semisimple cellular algebras are finite-dimensional associative algebras with an anti-involution and cell datum that ensures every cell module is irreducible.
• They provide a structured module theory where explicit constructions, criteria for semisimplicity, and clear classification of representations are established.
• This framework underpins diagram algebras and links to Lie theory and categorification, offering practical insights for representation and computational algebra.

Semisimple cellular algebras are finite-dimensional associative algebras equipped with an anti-involution and a combinatorially rich cell datum, which ensure a highly structured module theory in which all simple modules can be constructed explicitly from “cell modules” associated with this datum. In the semisimple case, these cell modules are simple and provide a complete set of pairwise non-isomorphic irreducible representations. The cellular framework encompasses a wide range of classical and modern algebras—most notably diagram algebras such as the Temperley–Lieb, Brauer, partition, and their various extension and deformation families—enabling deep structural analysis, explicit criteria for semisimplicity, and connections to Lie theory and categorification.

1. Cellular Algebras: Structure and Cell Data

A cellular algebra is an associative unital algebra $A$ over a commutative ring $R$ equipped with a quadruple $(\Lambda, M, C, i)$:

• $\Lambda$ is a finite partially ordered set,
• for each $\lambda \in \Lambda$, $M(\lambda)$ is a finite set,
• $$\{C_{st}^\lambda \mid \lambda \in \Lambda,\, s, t \in M(\lambda)\}$$ is an $R$-basis of $A$,
• $i: A \to A$ is an involutive $R$-linear anti-automorphism.

Multiplication satisfies the “cellular triangularity” condition: for all $a \in A$ and all basis elements $C_{st}^\lambda$, the product $aC_{st}^\lambda$ is an $R$-linear combination of elements $C_{s't}^\lambda$ modulo terms indexed by $\mu > \lambda$. The anti-involution satisfies $i(C_{st}^\lambda) = C_{ts}^\lambda$.

Cell modules $W(\lambda)$ are constructed by right action of $A$ on the $R$-span of $\{C_{st}^\lambda \mid t \in M(\lambda)\}$ (fixing a row index $s$), modulo higher terms. The associated Gram matrix is defined via a canonical $R$-valued bilinear form on $W(\lambda)$. If all these forms are nondegenerate, cellular structure yields an explicit complete set of simple modules.

Semisimple cellular algebras are those for which $A$ is semisimple as a ring and every cell module $W(\lambda)$ is simple, i.e., equals its simple quotient.

2. Semisimplicity Criteria

The semisimplicity of a cellular algebra is controlled by the nondegeneracy of cell module forms and, in diagrammatic and affine settings, by the invertibility of certain bilinear or “sandwich” matrices.

• In twisted semigroup algebras (including partition, Brauer, Temperley–Lieb algebras), $R^\alpha[S]$ is semisimple if and only if, for every $\mathcal{D}$-class $D$, the group algebra $R^\alpha[G_D]$ is semisimple and an associated “twisted sandwich matrix” $P^\alpha_D$ is invertible (Wilcox, 2010).
• For affine cellular algebras, $A$ is semisimple if and only if the associated scheme $\operatorname{Spec}[A]$ is reduced and 0-dimensional and the bilinear forms in all layers are isomorphisms. Over a perfect field, this is equivalent to separability (Li et al., 2023).
• For graded symmetric cellular algebras, semisimplicity is equivalent to the graded subspace $L_\mathrm{gr}(A)$ (generated from the homogeneous cellular basis via the symmetrizing trace) filling the centralizer $Z_A(A_0)$ of the degree-zero component (Li et al., 2017).
• In wreath products, $A \wr S_n$ is semisimple if and only if both $A$ and $kS_n$ are semisimple (Green, 2018).
• For KLRW and cyclotomic KLR algebras, semisimplicity is characterized by “entirely semisimple” crystal graphs: the cellular algebra is semisimple if the crystal graph (encoding the combinatorics of the basis) admits no nontrivial face permutations (Mathas et al., 2023).

3. Explicit Constructions and Examples

a) Diagram Algebras as Cellular Algebras

• The Temperley–Lieb, Brauer, cyclotomic Brauer, and partition algebras are all realized as twisted semigroup algebras and inherit cellularity. Partition algebra cell indices are parametrized by pairs $(k, \lambda)$ (number of connections, partition shape), and the Brauer algebra by pairs $(n - 2k, \lambda)$ (Wilcox, 2010, Geetha et al., 2012, Scrimshaw, 2022).
• Wreath products and $A$-Brauer algebras (where $A$ is cyclic cellular with involution and trace) preserve cyclic (hence cellular and, under cell-form nondegeneracy, semisimple) structure (Geetha et al., 2012).
• Cellular bases can be built recursively via Jones basic constructions—using towers of algebras and “branching diagrams,” with path indices on modules and explicit relations for induction and restriction (Enyang et al., 2011).

b) Wreath Product and Cellular Wreath Product

• The cellular wreath product $A \wr S$ generalizes by “decorating” the diagrams of $S$ (a partition algebra subalgebra or diagram algebra) with tensor factors from $A$'s cellular basis (Scrimshaw, 2022). The cell modules factor as products of those from $A$ and $S$.

Example Algebra	Cell Indices	Semisimplicity Criterion
Brauer, TL, Partition	$(k, \lambda)$	Cell forms nondegenerate, sandwich invert.
$A \wr S_n$	multipartitions	$A$ and $kS_n$ semisimple
Affine cellular (e.g., KLR)	crystal paths	Crystal entirely semisimple

c) Cyclotomic and Graded Algebras

• Graded cellular structures exist for KLR, KLRW, and their cyclotomic quotients (Mathas et al., 2021, Mathas et al., 2023). Graded decomposition numbers can be explicitly computed in level one using the cellular basis constructed from crystal combinatorics. Weighted KLRW and cyclotomic KLRW algebras are quasi-hereditary or affine quasi-hereditary except in types where nontrivial face permutations occur in the crystal.

d) Centralizer and Endomorphism Algebras

• For tilting modules in highest weight or standard categories (e.g., category $\mathcal{O}$ for rational Cherednik algebras), endomorphism algebras are cellular. The cellular basis is constructed from morphisms factorizing through summands, with cell modules given by hom-spaces from standard and costandard objects. In the case of Hecke algebras for real reflection groups, this construction recovers known cellularity (Bellamy et al., 2021).

4. Representation Theory and Cell Modules

Cell modules $W(\lambda)$ provide the essential building blocks for representation theory:

• In the semisimple case, cell modules are irreducible and exhaust all simple modules.
• The dimension of the algebra is given by $\sum_\lambda (\dim W(\lambda))^2$. For the cellular wreath product $A \wr S$, the dimension formula involves both factors (Scrimshaw, 2022).
• Simple modules are indexed by those labels $\lambda$ (or pairs, multipartitions, or paths, depending on the algebra) where the associated bilinear form is nondegenerate.

The explicit construction of cell modules—via diagram combinatorics, Murphy/branching basis formulas, or path representations—allows for linear-algebraic checks of simplicity and Cartan decompositions, and in cases such as the $q$-Brauer algebra gives rise to closed-form criteria for simple modules and semisimplicity parameters (Nguyen, 2012, Nguyen, 2013).

5. Lie-Theoretic Connections: The Plesken Lie Algebra

The Plesken Lie algebra associated to a semisimple cellular algebra $A$ with anti-involution $i$ is the $(-1)$-eigenspace,

$L(A) = \{ a \in A : i(a) = -a \}.$

In the semisimple setting, $$A \cong \bigoplus_{\lambda \in \Lambda'} \mathrm{M}_{d_\lambda}(\mathbb{C})$$, where $d_\lambda = \dim W(\lambda)$. The Plesken Lie algebra decomposes canonically:

$$L(A) \cong \bigoplus_{\lambda \in \Lambda'} \mathfrak{o}(d_\lambda, \mathbb{C})$$

where $\mathfrak{o}(d_\lambda, \mathbb{C})$ is the orthogonal Lie algebra of $d_\lambda \times d_\lambda$ skew-symmetric matrices. For diagram algebras, this yields explicit decompositions—planar rook and Temperley–Lieb algebras, when semisimple, correspond to direct sums of orthogonal Lie algebras with dimensions determined by the underlying combinatorics of the cell modules (Holm et al., 28 Aug 2025).

6. Extensions and Generalizations

• Affine cellular algebras, graded cellular algebras, and relative cellular algebras extend the theory to settings with schemes of higher complexity, gradings, or multiple partial orders localized to idempotents (Ehrig et al., 2017, Li et al., 2017, Li et al., 2023).
• In positive characteristic and quantum settings, many diagram and modular analogues are only “relative” cellular; their Cartan matrices are positive semi-definite, not definite, but much of the cellular machinery—and, in the semisimple locus, full structure—remains (Ehrig et al., 2017).
• Connections to Schur–Weyl dualities and categorification are prevalent, with cellular structure often controlling the essential representation-theoretic and homological invariants.

7. Applications and Impact

Semisimple cellular algebras provide:

• Explicit module-theoretic control (structure, classification, dimensions) for large classes of algebras in representation theory, statistical mechanics, knot theory, and categorification.
• Powerful, combinatorial bases enabling algorithmic calculations of irreducibles, Cartan invariants, and decomposition numbers.
• Decompositions of associated Lie algebras (via Plesken's construction) into direct sums of orthogonal Lie algebras tightly controlled by cell module dimensions.

This framework not only unifies key results for classical objects such as symmetric groups and diagram algebras, but also offers systematic methods for constructing and verifying semisimple structures in new algebraic and categorified settings.