Английская Википедия:Cellular algebra
Шаблон:Short description Шаблон:About 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 <math>R</math> be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also <math>A</math> be an <math>R</math>-algebra.
The concrete definition
A cell datum for <math>A</math> is a tuple <math>(\Lambda,i,M,C)</math> consisting of
- A finite partially ordered set <math>\Lambda</math>.
- A <math>R</math>-linear anti-automorphism <math>i:A\to A</math> with <math>i^2 = \operatorname{id}_A</math>.
- For every <math>\lambda\in\Lambda</math> a non-empty finite set <math>M(\lambda)</math> of indices.
- An injective map
- <math>C: \dot{\bigcup}_{\lambda\in\Lambda} M(\lambda)\times M(\lambda) \to A</math>
- The images under this map are notated with an upper index <math>\lambda\in\Lambda</math> and two lower indices <math>\mathfrak{s},\mathfrak{t}\in M(\lambda)</math> so that the typical element of the image is written as <math>C_\mathfrak{st}^\lambda</math>.
- and satisfying the following conditions:
- The image of <math>C</math> is a <math>R</math>-basis of <math>A</math>.
- <math>i(C_\mathfrak{st}^\lambda)=C_\mathfrak{ts}^\lambda</math> for all elements of the basis.
- For every <math>\lambda\in\Lambda</math>, <math>\mathfrak{s},\mathfrak{t}\in M(\lambda)</math> and every <math>a\in A</math> the equation
- <math>aC_\mathfrak{st}^\lambda \equiv \sum_{\mathfrak{u}\in M(\lambda)} r_a(\mathfrak{u},\mathfrak{s}) C_\mathfrak{ut}^\lambda \mod A(<\lambda)</math>
- with coefficients <math>r_a(\mathfrak{u},\mathfrak{s})\in R</math> depending only on <math>a</math>, <math>\mathfrak{u}</math> and <math>\mathfrak{s}</math> but not on <math>\mathfrak{t}</math>. Here <math>A(<\lambda)</math> denotes the <math>R</math>-span of all basis elements with upper index strictly smaller than <math>\lambda</math>.
This definition was originally given by Graham and Lehrer who invented cellular algebras.[1]
The more abstract definition
Let <math>i:A\to A</math> be an anti-automorphism of <math>R</math>-algebras with <math>i^2 = \operatorname{id}</math> (just called "involution" from now on).
A cell ideal of <math>A</math> w.r.t. <math>i</math> is a two-sided ideal <math>J\subseteq A</math> such that the following conditions hold:
- <math>i(J)=J</math>.
- There is a left ideal <math>\Delta\subseteq J</math> that is free as a <math>R</math>-module and an isomorphism
- <math>\alpha: \Delta\otimes_R i(\Delta) \to J</math>
- of <math>A</math>-<math>A</math>-bimodules such that <math>\alpha</math> and <math>i</math> are compatible in the sense that
- <math>\forall x,y\in\Delta: i(\alpha(x\otimes i(y))) = \alpha(y\otimes i(x))</math>
A cell chain for <math>A</math> w.r.t. <math>i</math> is defined as a direct decomposition
- <math>A=\bigoplus_{k=1}^m U_k</math>
into free <math>R</math>-submodules such that
- <math>i(U_k)=U_k</math>
- <math>J_k:=\bigoplus_{j=1}^k U_j</math> is a two-sided ideal of <math>A</math>
- <math>J_k/J_{k-1}</math> is a cell ideal of <math>A/J_{k-1}</math> w.r.t. to the induced involution.
Now <math>(A,i)</math> 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 <math>\Lambda</math>) and choosing a basis of every left ideal <math>\Delta/J_{k-1}\subseteq J_k/J_{k-1}</math> one can construct a corresponding cell basis for <math>A</math>.
Examples
Polynomial examples
<math>R[x]/(x^n)</math> is cellular. A cell datum is given by <math>i = \operatorname{id}</math> and
- <math>\Lambda := \lbrace 0,\ldots,n-1\rbrace</math> with the reverse of the natural ordering.
- <math>M(\lambda) := \lbrace 1\rbrace</math>
- <math>C_{11}^\lambda := x^\lambda</math>
A cell-chain in the sense of the second, abstract definition is given by
- <math>0 \subseteq (x^{n-1}) \subseteq (x^{n-2}) \subseteq \ldots \subseteq (x^1) \subseteq (x^0)=R[x]/(x^n)</math>
Matrix examples
<math>R^{\,d \times d}</math> is cellular. A cell datum is given by <math>i(A)=A^T</math> and
- <math>\Lambda := \lbrace 1 \rbrace</math>
- <math>M(1) := \lbrace 1,\dots,d\rbrace</math>
- For the basis one chooses <math>C_{st}^1 := E_{st}</math> the standard matrix units, i.e. <math>C_{st}^1</math> 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
- <math>0 \subseteq R^{\!d \times d}</math>
In some sense all cellular algebras "interpolate" between these two extremes by arranging matrix-algebra-like pieces according to the poset <math>\Lambda</math>.
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 <math>T_w\mapsto T_{w^{-1}}</math>.[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 <math>\mathcal{O}</math> of a semisimple Lie algebra.[5]
Representations
Cell modules and the invariant bilinear form
Assume <math>A</math> is cellular and <math>(\Lambda,i,M,C)</math> is a cell datum for <math>A</math>. Then one defines the cell module <math>W(\lambda)</math> as the free <math>R</math>-module with basis <math>\lbrace C_\mathfrak{s} \mid \mathfrak{s} \in M(\lambda)\rbrace</math> and multiplication
- <math>aC_\mathfrak{s} := \sum_{\mathfrak{u}} r_a(\mathfrak{u},\mathfrak{s}) C_\mathfrak{u}</math>
where the coefficients <math>r_a(\mathfrak{u},\mathfrak{s})</math> are the same as above. Then <math>W(\lambda)</math> becomes an <math>A</math>-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 <math>\phi_\lambda: W(\lambda)\times W(\lambda)\to R</math> which satisfies
- <math>C_\mathfrak{st}^\lambda C_\mathfrak{uv}^\lambda \equiv \phi_\lambda(C_\mathfrak{t},C_\mathfrak{u}) C_\mathfrak{sv}^\lambda \mod A(<\lambda)</math>
for all indices <math>s,t,u,v\in M(\lambda)</math>.
One can check that <math>\phi_\lambda</math> is symmetric in the sense that
- <math>\phi_\lambda(x,y) = \phi_\lambda(y,x)</math>
for all <math>x,y\in W(\lambda)</math> and also <math>A</math>-invariant in the sense that
- <math>\phi_\lambda(i(a)x,y)=\phi_\lambda(x,ay)</math>
for all <math>a\in A</math>,<math>x,y\in W(\lambda)</math>.
Simple modules
Assume for the rest of this section that the ring <math>R</math> is a field. With the information contained in the invariant bilinear forms one can easily list all simple <math>A</math>-modules:
Let <math>\Lambda_0:=\lbrace \lambda\in\Lambda \mid \phi_\lambda\neq 0\rbrace</math> and define <math>L(\lambda):=W(\lambda)/\operatorname{rad}(\phi_\lambda)</math> for all <math>\lambda\in\Lambda_0</math>. Then all <math>L(\lambda)</math> are absolute simple <math>A</math>-modules and every simple <math>A</math>-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 <math>R</math>-algebras are cellular.
- A <math>R</math>-algebra <math>A</math> is cellular if and only if its opposite algebra <math>A^{\text{op}}</math> is.
- If <math>A</math> is cellular with cell-datum <math>(\Lambda,i,M,C)</math> and <math>\Phi\subseteq\Lambda</math> is an ideal (a downward closed subset) of the poset <math>\Lambda</math> then <math>A(\Phi):=\sum RC_\mathfrak{st}^\lambda</math> (where the sum runs over <math>\lambda\in\Lambda</math> and <math>s,t\in M(\lambda)</math>) is a two-sided, <math>i</math>-invariant ideal of <math>A</math> and the quotient <math>A/A(\Phi)</math> is cellular with cell datum <math>(\Lambda\setminus\Phi,i,M,C)</math> (where i denotes the induced involution and M, C denote the restricted mappings).
- If <math>A</math> is a cellular <math>R</math>-algebra and <math>R\to S</math> is a unitary homomorphism of commutative rings, then the extension of scalars <math>S\otimes_R A</math> is a cellular <math>S</math>-algebra.
- Direct products of finitely many cellular <math>R</math>-algebras are cellular.
If <math>R</math> is an integral domain then there is a converse to this last point:
- If <math>(A,i)</math> is a finite-dimensional <math>R</math>-algebra with an involution and <math>A=A_1\oplus A_2</math> a decomposition in two-sided, <math>i</math>-invariant ideals, then the following are equivalent:
- <math>(A,i)</math> is cellular.
- <math>(A_1,i)</math> and <math>(A_2,i)</math> are cellular.
- Since in particular all blocks of <math>A</math> are <math>i</math>-invariant if <math>(A,i)</math> is cellular, an immediate corollary is that a finite-dimensional <math>R</math>-algebra is cellular w.r.t. <math>i</math> if and only if all blocks are <math>i</math>-invariant and cellular w.r.t. <math>i</math>.
- Tits' deformation theorem for cellular algebras: Let <math>A</math> be a cellular <math>R</math>-algebra. Also let <math>R\to k</math> be a unitary homomorphism into a field <math>k</math> and <math>K:=\operatorname{Quot}(R)</math> the quotient field of <math>R</math>. Then the following holds: If <math>kA</math> is semisimple, then <math>KA</math> is also semisimple.
If one further assumes <math>R</math> to be a local domain, then additionally the following holds:
- If <math>A</math> is cellular w.r.t. <math>i</math> and <math>e\in A</math> is an idempotent such that <math>i(e)=e</math>, then the algebra <math>eAe</math> is cellular.
Other properties
Assuming that <math>R</math> 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 <math>A</math> is cellular w.r.t. to the involution <math>i</math>. Then the following hold
- <math>A</math> is split, i.e. all simple modules are absolutely irreducible.
- The following are equivalent:[1]
- <math>A</math> is semisimple.
- <math>A</math> is split semisimple.
- <math>\forall\lambda\in\Lambda: W(\lambda)</math> is simple.
- <math>\forall\lambda\in\Lambda: \phi_\lambda</math> is nondegenerate.
- The Cartan matrix <math>C_A</math> of <math>A</math> is symmetric and positive definite.
- The following are equivalent:[7]
- <math>A</math> is quasi-hereditary (i.e. its module category is a highest-weight category).
- <math>\Lambda=\Lambda_0</math>.
- All cell chains of <math>(A,i)</math> have the same length.
- All cell chains of <math>(A,j)</math> have the same length where <math>j:A\to A</math> is an arbitrary involution w.r.t. which <math>A</math> is cellular.
- <math>\det(C_A)=1</math>.
- If <math>A</math> is Morita equivalent to <math>B</math> and the characteristic of <math>R</math> is not two, then <math>B</math> is also cellular w.r.t. a suitable involution. In particular <math>A</math> is cellular (to some involution) if and only if its basic algebra is.[8]
- Every idempotent <math>e\in A</math> is equivalent to <math>i(e)</math>, i.e. <math>Ae\cong Ai(e)</math>. If <math>\operatorname{char}(R) \neq 2</math> then in fact every equivalence class contains an <math>i</math>-invariant idempotent.[5]
References
