In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, a cellular algebra is a
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
''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.
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 algebra
A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication, Schur product, transposition, and contains both the identity matrix I and the all-ones matrix J.
Definitions
A subspace \mathcal ...
s.
Definitions
Let
be a fixed
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
with unit. In most applications this is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, but this is not needed for the definitions. Let also
be an
-
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
.
The concrete definition
A ''cell datum'' for
is a tuple
consisting of
:* A finite
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
.
:* A
-
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
anti-automorphism
In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From b ...
with
.
:* For every
a
non-empty
In mathematics, the empty set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by inclu ...
finite set
of indices.
:* An
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
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:
# The image of
is a
-
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
of
.
#
for all elements of the basis.
# 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.
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
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
such that the following conditions hold:
#
.
# There is a left ideal
that is
free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
as a
-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
and an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
:::
::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
#
#
is a two-sided ideal of
#
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.
Every basis gives rise to cell chains (one for each
topological ordering
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge ''uv'' from vertex ''u'' to vertex ''v'', ''u'' comes before ''v'' in the ordering. For ins ...
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 unit
In linear algebra, a matrix unit is a matrix (mathematics), matrix with only one nonzero entry with value 1. The matrix unit with a 1 in the ''i''th row and ''j''th column is denoted as E_. For example, the 3 by 3 matrix unit with ''i'' = 1 and ' ...
s, 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 algebra
In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group.
Hecke algebras are quotients of the group rings of Artin braid groups. This ...
s of finite type are cellular w.r.t. to the involution that maps the standard basis as
. This includes for example the
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
group algebra of the
symmetric groups as well as all other
finite Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
s.
A basic Brauer tree algebra over a field is cellular
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
the Brauer tree is a straight line (with arbitrary number of exceptional vertices).
Further examples include q-
Schur algebra In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of ...
s, the
Brauer algebra In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the ...
, the
Temperley–Lieb algebra In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quant ...
, the
Birman–Murakami–Wenzl algebra, the blocks of the Bernstein–Gelfand–Gelfand category
of a
semisimple Lie algebra.
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 module In mathematics, a Specht module is one of the representations of symmetric groups studied by .
They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of ''n'' form a complete set of irreducible representations of t ...
s 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
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
is a field. With the information contained in the invariant bilinear forms one can easily list all
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
-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.
Properties of cellular algebras
Persistence properties
*
Tensor products
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of finitely many cellular
-algebras are cellular.
* A
-algebra
is cellular if and only if its
opposite algebra In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring w ...
is.
* If
is cellular with cell-datum
and
is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
(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
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
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:
#
is cellular.
#
and
are cellular.
* Since in particular all
blocks of
are
-invariant if
is cellular, an immediate
corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
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
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
, 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 ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
s or at least
discrete valuation rings) and
is cellular w.r.t. to the involution
. Then the following hold
*
is split, i.e. all simple modules are
absolutely irreducible In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the integers ...
.
* The following are equivalent:
#
is
semisimple.
#
is split semisimple.
#
is simple.
#
is
nondegenerate
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
.
* The
Cartan matrix of
is
symmetric and
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
.
* The following are equivalent:
#
is
quasi-hereditary (i.e. its
module category
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring ...
is a
highest-weight category In the mathematical field of representation theory, a highest-weight category is a ''k''-linear category C (here ''k'' is a field) that
*is ''locally artinian''
*has enough injectives
*satisfies
::B\cap\left(\bigcup_\alpha A_\alpha\right)=\bigcup_\ ...
).
#
.
# All cell chains of
have the same length.
# All cell chains of
have the same length where
is an arbitrary involution w.r.t. which
is cellular.
#
.
* If
is
Morita equivalent
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ...
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.
* Every idempotent
is equivalent to
, i.e.
. If
then in fact every
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
contains an
-invariant idempotent.
References
Algebras
Representation theory