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Rational Cherednik Algebras and Their Category O: Examples and Counterexamples

1 · Prerequisites

Rational Cherednik Algebras and Their Category O

2 · Summary

This companion checks the A-page conventions in two small cases. The trivial group gives the first Weyl algebra. In rank one at $c = 1 / 2$ , the polynomial standard module contains $x C [x]$ and the resulting short exact sequence is nonsplit, so category $O$ is not semisimple at that parameter.

Companion to rational-cherednik-algebras-and-category-o.

Examples

For $W = {1}$ and $\dim h = 1$ , the reflection sum vanishes and $[y, x] = 1$ ; the polynomial standard module is $C [x]$ with $y = d / d x$ . The full presentation and verification are in The trivial reflection group gives the first Weyl algebra.

For $W = Z / 2$ acting on $h = C$ by $s (x) = - x$ , the fixed normalization is $[y, x] = 1 - 2 c s$ . At $c = 1 / 2$ , the explicit formula $y x^{n} = {\begin{cases} n x^{n - 1}, & n even, \\ (n - 1) x^{n - 1}, & n odd, \end{cases}$ shows that $x C [x]$ is a submodule. The quotient is the trivial one-dimensional module, and no invariant complement exists because multiplication by $x$ has no nonzero polynomial eigenvector. See Category O is not semisimple at the rank-one parameter c=1/2.

Scope

These finite computations test the sign and normalization of the A-page relations. The rank-one sequence refutes semisimplicity at $c = 1 / 2$ ; it makes no claim about other parameter values. Source locators and the inspected full-text evidence appear in the batch coverage record.

Definition, diagram, and repair contracts

Apply the definition and diagram contracts and prerequisite and repair holds before treating examples as suppliers. No diagram is needed for these finite algebra calculations; any later diagram must state its objects, maps, orientations and algebraic verification. The lane reference is 05-hecke and its source inventory. The full placement record is the placed-pages inventory, entry rational-cherednik-algebras-and-category-o-examples; the original audited research inventory is unchanged.

3 · Logical flowchart

4 · Definitions, theorems and proofs

ExampleConstruction: AI-generatedVerification: AI-generatedOpen item page →

The trivial reflection group gives the first Weyl algebra

Example

Take $W = {1}$ and $\dim h = 1$ . Then $H_{1, 0} (W, h)$ is the first Weyl algebra, with polynomial representation $C [x]$ on which $x$ acts by multiplication and $y$ acts by $d / d x$ .

Facts & Assumptions

Given: The one-dimensional reflection datum with trivial group.

[F1]

The rational Cherednik algebra H_{t,c}(W,h): $H_{t, c}$ is generated by $x, y$ with $[y, x] = t x (y) - \sum_{s} c_{s} α_{s} (y) α_{s}^{\lor} (x) s$ .

[F2]

The PBW theorem for rational Cherednik algebras: the ordered monomials $x^{a} w y^{b}$ form a basis of $H_{t, c}$ .

[F3]

The polynomial representation of a rational Cherednik algebra: $P_{c} = Δ_{c} (1)$ has underlying vector space $C [x]$ .

[F4]

Dunkl operators realize the polynomial standard module: on $P_{c}$ , $y$ acts by $\partial_{y} - \sum_{s} c_{s} α_{s} (y) α_{s}^{- 1} (1 - s)$ .

Proof

technique · direct

Given: [F1]–[F4] and a basis $x \in h^{*}$ with dual $y \in h$ .

1.1F1F2algebra

Since $S = \emptyset$ for $W = {1}$ and $x (y) = 1$ , [F1] reduces to $[y, x] = 1$ with no other generators. This is the defining presentation of the first Weyl algebra; [F2] also gives its basis ${x^{a} y^{b} : a, b \geq 0}$ .

1.2F3F4algebra

In [F4] the reflection sum is empty, so $y = \partial_{y} = d / d x$ and $x$ is multiplication by $x$ on [F3]. Directly, $(d / d x) (x f) - x (d / d x) f = f$ for every polynomial $f$ , verifying the Weyl relation in this representation.

2.1F1F2F3F4step 1.1step 1.2∎

Thus the trivial-group rational Cherednik algebra is the first Weyl algebra and its polynomial standard module is the usual polynomial representation.

CounterexampleConstruction: AI-generatedVerification: AI-generatedOpen item page →

Category O is not semisimple at the rank-one parameter c=1/2

Statement

Let $W = {1, s} ≅ Z / 2$ act on $h = C$ by $s = - 1$ . At $t = 1$ and $c = 1 / 2$ , the polynomial standard module has a nonsplit exact sequence $0 ⟶ x C [x] ⟶ C [x] ⟶ C ⟶ 0.$ Thus category $O$ is not semisimple at this parameter.

Facts & Assumptions

Given: The rank-one reflection representation, parameter $c = 1 / 2$ , and the polynomial representation.

[F1]

Reflection data and parameters for a rational Cherednik algebra: choose $α = x$ and $α^{\lor} (x) = 2$ , so $s (x) = - x$ and $s^{2} = 1$ .

[F2]

The rational Cherednik algebra H_{t,c}(W,h): $[y, x] = 1 - 2 c s$ in this rank-one normalization.

[F3]

The PBW theorem for rational Cherednik algebras and Standard modules for rational Cherednik category O: the polynomial standard module is $C [x]$ as a vector space.

[F4]

Category O for a rational Cherednik algebra and Standard modules lie in rational Cherednik category O: $C [x] \in O_{c}$ and its submodules and quotients are objects of $O_{c}$ .

[F5]

Dunkl operators realize the polynomial standard module: $y = \frac{d}{d x} - \frac{c}{x} (1 - s)$ on $C [x]$ , and $s f (x) = f (- x)$ .

Proof

technique · direct

Given: [F1]–[F5].

1.1F1F2F5algebra

By [F2] and [F5], $y x^{n} = (n - c (1 - (- 1)^{n})) x^{n - 1}$ for every $n \geq 1$ , while $y 1 = 0$ . At $c = 1 / 2$ , this is $n x^{n - 1}$ for even $n$ and $(n - 1) x^{n - 1}$ for odd $n$ . In particular, $y x = 0$ .

2.1F3F4step 1.1algebra

The subspace $I = x C [x]$ is nonzero and proper. Multiplication by $x$ preserves $I$ , $s (x^{n}) = (- 1)^{n} x^{n}$ preserves $I$ , and step 1.1 shows $y (x^{n}) \in I$ for every $n \geq 1$ : for $n = 1$ it is zero, while for $n \geq 2$ its polynomial degree is $n - 1 \geq 1$ . By [F3], this is the polynomial standard module; [F4] places it and its submodules in $O_{c}$ .

3.1F4step 2.1algebra∎

The quotient $C [x] / I$ is one dimensional; $x$ and $y$ act by zero and $s$ acts trivially, so it is simple. If the displayed exact sequence split, its complement would be a one-dimensional submodule $J$ mapping isomorphically to the quotient. A generator $f \in J$ would have nonzero constant term; rescale so $f (0) = 1$ . Since $x J \subseteq J$ , there would be a scalar $λ$ with $x f = λ f$ , or $(x - λ) f = 0$ . The polynomial ring $C [x]$ is an integral domain and both factors are nonzero, a contradiction. The sequence is nonsplit, so $O_{c}$ is not semisimple.

5 · Examples, counterexamples and false statements

None yet.

Rational Cherednik Algebras and Their Category O: Examples and Counterexamples

1 · Prerequisites

2 · Summary

Examples

Scope

Definition, diagram, and repair contracts

3 · Logical flowchart

4 · Definitions, theorems and proofs

The trivial reflection group gives the first Weyl algebra

Example

Facts & Assumptions

Proof

Category O is not semisimple at the rank-one parameter c=1/2

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources