Cellular algebra

In abstract algebra, a cellular algebra is a finite-dimensional associative algebra A with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of A.

History

The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer.[1] However, the terminology had previously been used by Weisfeiler and Lehman in the Soviet Union in the 1960s, to describe what are also known as coherent algebras. [2][3][4]

Definitions

Let be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also be an -algebra.

The concrete definition

A cell datum for is a tuple consisting of

  • A finite partially ordered set .
  • A -linear anti-automorphism with .
  • For every a non-empty finite set of indices.
  • An injective map
The images under this map are notated with an upper index and two lower indices so that the typical element of the image is written as .
and satisfying the following conditions:
  1. The image of is a -basis of .
  2. for all elements of the basis.
  3. For every , and every the equation
with coefficients depending only on , and but not on . Here denotes the -span of all basis elements with upper index strictly smaller than .

This definition was originally given by Graham and Lehrer who invented cellular algebras.[1]

The more abstract definition

Let be an anti-automorphism of -algebras with (just called "involution" from now on).

A cell ideal of w.r.t. is a two-sided ideal such that the following conditions hold:

  1. .
  2. There is a left ideal that is free as a -module and an isomorphism
of --bimodules such that and are compatible in the sense that

A cell chain for w.r.t. is defined as a direct decomposition

into free -submodules such that

  1. is a two-sided ideal of
  2. is a cell ideal of w.r.t. to the induced involution.

Now is called a cellular algebra if it has a cell chain. One can show that the two definitions are equivalent.[5] Every basis gives rise to cell chains (one for each topological ordering of ) and choosing a basis of every left ideal one can construct a corresponding cell basis for .

Examples

Polynomial examples

is cellular. A cell datum is given by and

  • with the reverse of the natural ordering.

A cell-chain in the sense of the second, abstract definition is given by

Matrix examples

is cellular. A cell datum is given by and

  • For the basis one chooses the standard matrix units, i.e. is the matrix with all entries equal to zero except the (s,t)-th entry which is equal to 1.

A cell-chain (and in fact the only cell chain) is given by

In some sense all cellular algebras "interpolate" between these two extremes by arranging matrix-algebra-like pieces according to the poset .

Further examples

Modulo minor technicalities all Iwahori–Hecke algebras of finite type are cellular w.r.t. to the involution that maps the standard basis as .[6] This includes for example the integral group algebra of the symmetric groups as well as all other finite Weyl groups.

A basic Brauer tree algebra over a field is cellular if and only if the Brauer tree is a straight line (with arbitrary number of exceptional vertices).[5]

Further examples include q-Schur algebras, the Brauer algebra, the Temperley–Lieb algebra, the Birman–Murakami–Wenzl algebra, the blocks of the Bernstein–Gelfand–Gelfand category of a semisimple Lie algebra.[5]

Representations

Cell modules and the invariant bilinear form

Assume is cellular and is a cell datum for . Then one defines the cell module as the free -module with basis and multiplication

where the coefficients are the same as above. Then becomes an -left module.

These modules generalize the Specht modules for the symmetric group and the Hecke-algebras of type A.

There is a canonical bilinear form which satisfies

for all indices .

One can check that is symmetric in the sense that

for all and also -invariant in the sense that

for all ,.

Simple modules

Assume for the rest of this section that the ring is a field. With the information contained in the invariant bilinear forms one can easily list all simple -modules:

Let and define for all . Then all are absolute simple -modules and every simple -module is one of these.

These theorems appear already in the original paper by Graham and Lehrer.[1]

Properties of cellular algebras

Persistence properties

  • Tensor products of finitely many cellular -algebras are cellular.
  • A -algebra is cellular if and only if its opposite algebra is.
  • If is cellular with cell-datum and is an ideal (a downward closed subset) of the poset then (where the sum runs over and ) is a two-sided, -invariant ideal of and the quotient is cellular with cell datum (where i denotes the induced involution and M, C denote the restricted mappings).
  • If is a cellular -algebra and is a unitary homomorphism of commutative rings, then the extension of scalars is a cellular -algebra.
  • Direct products of finitely many cellular -algebras are cellular.

If is an integral domain then there is a converse to this last point:

  • If is a finite-dimensional -algebra with an involution and a decomposition in two-sided, -invariant ideals, then the following are equivalent:
  1. is cellular.
  2. and are cellular.
  • Since in particular all blocks of are -invariant if is cellular, an immediate corollary is that a finite-dimensional -algebra is cellular w.r.t. if and only if all blocks are -invariant and cellular w.r.t. .
  • Tits' deformation theorem for cellular algebras: Let be a cellular -algebra. Also let be a unitary homomorphism into a field and the quotient field of . Then the following holds: If is semisimple, then is also semisimple.

If one further assumes to be a local domain, then additionally the following holds:

  • If is cellular w.r.t. and is an idempotent such that , then the algebra is cellular.

Other properties

Assuming that is a field (though a lot of this can be generalized to arbitrary rings, integral domains, local rings or at least discrete valuation rings) and is cellular w.r.t. to the involution . Then the following hold

  1. is semisimple.
  2. is split semisimple.
  3. is simple.
  4. is nondegenerate.
  1. is quasi-hereditary (i.e. its module category is a highest-weight category).
  2. .
  3. All cell chains of have the same length.
  4. All cell chains of have the same length where is an arbitrary involution w.r.t. which is cellular.
  5. .
  • If is Morita equivalent to and the characteristic of is not two, then is also cellular w.r.t. a suitable involution. In particular is cellular (to some involution) if and only if its basic algebra is.[8]
  • Every idempotent is equivalent to , i.e. . If then in fact every equivalence class contains an -invariant idempotent.[5]

References

  1. Graham, J.J; Lehrer, G.I. (1996), "Cellular algebras", Inventiones Mathematicae, 123: 1–34, Bibcode:1996InMat.123....1G, doi:10.1007/bf01232365, S2CID 189831103
  2. Weisfeiler, B. Yu.; A. A., Lehman (1968). "Reduction of a graph to a canonical form and an algebra which appears in this process". Scientific-Technological Investigations. 2 (in Russian). 9: 12–16.
  3. Higman, Donald G. (August 1987). "Coherent algebras". Linear Algebra and Its Applications. 93: 209-239. doi:10.1016/S0024-3795(87)90326-0. hdl:2027.42/26620.
  4. Cameron, Peter J. (1999). Permutation Groups. London Mathematical Society Student Texts (45). Cambridge University Press. ISBN 978-0-521-65378-7.
  5. König, S.; Xi, C.C. (1996), "On the structure of cellular algebras", Algebras and Modules II. CMS Conference Proceedings: 365–386
  6. Geck, Meinolf (2007), "Hecke algebras of finite type are cellular", Inventiones Mathematicae, 169 (3): 501–517, arXiv:math/0611941, Bibcode:2007InMat.169..501G, doi:10.1007/s00222-007-0053-2, S2CID 8111018
  7. König, S.; Xi, C.C. (1999-06-24), "Cellular algebras and quasi-hereditary algebras: A comparison", Electronic Research Announcements of the American Mathematical Society, 5 (10): 71–75, doi:10.1090/S1079-6762-99-00063-3
  8. König, S.; Xi, C.C. (1999), "Cellular algebras: inflations and Morita equivalences", Journal of the London Mathematical Society, 60 (3): 700–722, CiteSeerX 10.1.1.598.3299, doi:10.1112/s0024610799008212, S2CID 1664006
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