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.
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.
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]
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.
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 a finite-dimensional -algebra with an involution and a decomposition in two-sided, -invariant ideals, then the following are equivalent:
is cellular.
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
All cell chains of have the same length where is an arbitrary involution w.r.t. which is cellular.
.
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]
^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.
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.
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.
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]
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.
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 a finite-dimensional -algebra with an involution and a decomposition in two-sided, -invariant ideals, then the following are equivalent:
is cellular.
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
All cell chains of have the same length where is an arbitrary involution w.r.t. which is cellular.
.
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]
^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.