Cellular Algebras Research Articles

Idempotents and homology of diagram algebras

AbstractThis paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the ‘invertible-parameter’ cases of the Temperley–Lieb and Brauer results of Boyd–Hepworth and Boyd–Hepworth–Patzt. We are also able to give a new proof of Sroka’s theorem that the homology of an odd-strand Temperley–Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham–Lehrer.

Skew cellularity of the Hecke algebras of type 𝐺(ℓ,𝑝,𝑛)

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G ( ℓ , p , n ) G(\ell ,p,n) are graded skew cellular algebras. In the special case when p = 2 p = 2 this implies that the Hecke algebras of type G ( ℓ , 2 , n ) G(\ell ,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G ( ℓ , p , n ) G(\ell ,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

Semisimplicity of affine cellular algebras

In this paper, we prove that an affine cellular algebra [Formula: see text] is semisimple if and only if the scheme associated to [Formula: see text] is reduced and 0-dimensional, and the bilinear forms with respect to all layers of [Formula: see text] are invertible. Moreover, if the ground ring is a perfect field, then [Formula: see text] is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple.

Schurifying quasi‐hereditary algebras

We study new classes of quasi-hereditary and cellular algebras which generalize Turner's double algebras. Turner's algebras provide a local description of blocks of symmetric groups up to derived equivalence. Our general construction allows one to “schurify” any quasi-hereditary algebra A $A$ to obtain a generalized Schur algebra S A ( n , d ) $S^A(n,d)$ which we prove is again quasi-hereditary if d ⩽ n $d\leqslant n$ . We describe decomposition numbers of S A ( n , d ) $S^A(n,d)$ in terms of those of A $A$ and the classical Schur algebra S ( n , d ) $S(n,d)$ . In fact, it is essential to work with quasi-hereditary superalgebras A $A$ , in which case the construction of the schurification involves a non-trivial full rank sub-lattice T a A ( n , d ) ⊆ S A ( n , d ) $T^A_\mathfrak {a}(n,d)\subseteq S^A(n,d)$ .

Handlebody diagram algebras

In this paper we study handlebody versions of some classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer, BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory.

Cellularity of endomorphism algebras of tilting objects

We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer [4]. This result raises the question of whether all cellular algebras can be realized in this way. The construction also works without the presence of a duality and yields standard bases, in the sense of Du and Rui, which have similar combinatorial features to cellular bases. As an application, we obtain standard bases—and thus a general theory of “cell modules”—for Hecke algebras associated to finite complex reflection groups (as introduced by Broué, Malle, and Rouquier) via category O of the rational Cherednik algebra. For real reflection groups these bases are cellular.

Irreducibility of Certain $FC_n$-Modules

Irreducibility for cellular algebras can be investigated by solving problems in linear algebra. In this work, we discuss a large family of cell modules over Fuss-Catalan algebras and describe dimensions and basis diagrams of these cell modules. Also, conditions for these modules to be irreducible are determined by their Gram matrices and determinants.

On derived equivalences and homological dimensions

Abstract Unlike Hochschild (co)homology and K-theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for many finite-dimensional algebras arising in algebraic Lie theory. Both dimensions then can be characterised intrinsically inside certain derived categories. On the way, a restriction theorem is proved, and used, which says that derived equivalences between algebras with positive ν-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, which under this assumption do exist.

Some properties for almost cellular algebras

<p style='text-indent:20px;'>In this paper, we will investigate some properties for almost cellular algebras. We compare the almost cellular algebras with quasi-hereditary algebras, which are known to carry any homological and categorical structures. We prove that any almost cellular algebra is the iterated inflation and obtain some sufficient and necessary conditions for an almost cellular algebra <inline-formula><tex-math id="M1">$ \mathrm{A} $</tex-math></inline-formula> to be quasi-hereditary.</p>

Radicals of weight one blocks of Ariki–Koike algebras

Let K be a field and Let be an Ariki–Koike algebra, where the cyclotomic parameter with For a weight one block B of we prove in this article that where I is the nilpotent ideal constructed for a symmetric cellular algebra [Radicals of symmetric cellular algebras, Colloq. Math. 133 (2013) 67–83]. We also give some applications of this result.

RELATIVE CELLULAR ALGEBRAS

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are relative cellular, but not cellular. Most prominently, the restricted enveloping algebra and the small quantum group for $\mathfrak{sl}_{2}$, and an annular version of arc algebras.

Quiver mutations and Boolean reflection monoids

In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type A). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of Dynkin diagrams with frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova respectively. Surprisingly, inner by diagram automorphisms of irreducible Weyl groups and Boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams. Besides, we show that semigroup algebras of Boolean reflection monoids are cellular algebras.

Inductions and restrictions on towers of cellularly stratified diagram algebras

Hartmann et al. defined the concept of cellularly stratified algebras that combine the features of both cellular algebras and stratified algebras. Many important diagram algebras in mathematics and physics, such as some Brauer, partition and BMW algebras, are cellularly stratified algebras, and each of these forms a tower of algebras. This article gives the concept of towers of cellularly stratified algebras in an axiomatic manner, and studies it in terms of induction and restriction functors. In particular, for certain towers of cellularly stratified algebras, we provide a criterion for semi-simplicity by using the cohomology groups of cell modules.

Based quasi-hereditary algebras

A notion of a split quasi-hereditary algebra has been defined by Cline, Parshall and Scott. Du and Rui describe a based approach to split quasi-hereditary algebras. We develop this approach further to show that over a complete local Noetherian ring, one can achieve even stronger basis properties. This is important for ‘schurifying’ quasi-hereditary algebras as developed in our subsequent work. The schurification procedure associates to an algebra A a new algebra, which is the classical Schur algebra if A is a field. Schurification produces interesting new quasi-hereditary and cellular algebras. It is important to work over an integral domain of characteristic zero, taking into account a super-structure on the input algebra A. So we pay attention to super-structures on quasi-hereditary algebras and investigate a subtle conforming property of heredity data which is crucial to guarantee that the schurification of A is quasi-hereditary if so is A. We establish a Morita equivalence result which allows us to pass to basic quasi-hereditary algebras preserving conformity.

Correction to: Self-injective Cellular Algebras Whose Representation Type are Tame of Polynomial Growth

The original version of this article unfortunately contains mistakes introduced during the production phase.

Self-injective Cellular Algebras Whose Representation Type are Tame of Polynomial Growth

We classify Morita equivalence classes of indecomposable self-injective cellular algebras which have polynomial growth representation type, assuming that the characteristic of the base field is different from two. This assumption on the characteristic is for the cellularity to be a Morita invariant property.

Borelic pairs for stratified algebras

We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley–Lieb and Jones algebras. For this, we develop and apply a general theory of finite dimensional algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of König. Another application of the theory leads to a proof that Auslander–Dlab–Ringel algebras admit exact Borel subalgebras.

Iterated inflations of cellular algebras

We present a result characterising iterated inflations of cellular algebras, derived from the work of König and Xi. This result is intended to replace an incorrect proposition in the literature, and gives explicit and readily checked conditions which establish that an algebra is an iterated inflation of cellular algebras, and hence is cellular, with cellular data directly related to the cellular data of the constituent cellular algebras.

Ring theoretical properties of affine cellular algebras

As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced in [12] in order to treat affine Hecke algebras of type A and affine versions of diagram algebras like affine Temperley–Lieb algebras in a unifying fashion. Affine cellular algebras include Kleshchev's graded quasihereditary algebras, Khovanov–Lauda–Rouquier algebras and various other classes of algebras. In this paper we will study ring theoretical properties of affine cellular algebras. We show that any affine cellular algebra A satisfies a polynomial identity. Furthermore, we show that A can be embedded into its asymptotic algebra if the occurring commutative affine k-algebras Bj are reduced and the determinants of the swich matrices are non-zero divisors. As a consequence, we show that the Gelfand–Kirillov dimension of A is less than or equal to the largest Krull dimension of the algebras Bj and that equality holds, in case all affine cell ideals are idempotent or if the Krull dimension of the algebras Bj is less than or equal to 1. Special emphasis is given to the question when an affine cell ideal is idempotent, generated by an idempotent or finitely generated.