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All irreducible finite-dimensional sl2 modules

Example

Let ${s l}_{2} = {s l}_{2} (C)$ have its standard basis $e, f, h$ with $[e, f] = h$ , $[h, e] = 2 e$ , $[h, f] = - 2 f$ (The special linear Lie algebra sl_2). For every integer $n \geq 0$ let $V (n)$ be the vector space with basis $v_{0}, v_{1}, \dots, v_{n}$ and let $h \cdot v_{k} = (n - 2 k) v_{k}, f \cdot v_{k} = v_{k + 1}, e \cdot v_{k} = k (n - k + 1) v_{k - 1},$ where $v_{- 1} = v_{n + 1} = 0$ . Then:

(i) these formulas define a representation of ${s l}_{2}$ on $V (n)$ ;

(ii) $V (n)$ is irreducible of dimension $n + 1$ ;

(iii) every finite-dimensional irreducible ${s l}_{2}$ -module is isomorphic to exactly one $V (n)$ .

Facts & Assumptions

Given: The Lie algebra ${s l}_{2}$ with basis $e, f, h$ (The special linear Lie algebra sl_2), the displayed operators $E, F, H$ on the basis $v_{0}, \dots, v_{n}$ of $V (n)$ , and the defining relations $[e, f] = h$ , $[h, e] = 2 e$ , $[h, f] = - 2 f$ . Weights are taken with respect to the Cartan subalgebra $C h$ , so a vector of $H$ -eigenvalue $μ \in C$ has weight the functional $h \mapsto μ$ (Weight and weight space).

[L1]

A finite-dimensional ${s l}_{2}$ -module is a direct sum of irreducible submodules; an irreducible submodule has a top weight $m \geq 0$ and $h$ -eigenvalues $m, m - 2, \dots, - m$ , each on a one-dimensional subspace (Finite-dimensional representations of sl_2).

[L2]

A nonzero submodule of an irreducible module is the whole module, and irreducibility means the absence of nonzero proper submodules (Irreducible, completely reducible, and faithful representations, Representations of Lie algebras).

Verification

technique · direct

1.1

The operators define a representation: on each basis vector, $H F v_{k} - F H v_{k} = (n - 2 k - 2) v_{k + 1} - (n - 2 k) v_{k + 1} = - 2 v_{k + 1} = - 2 F v_{k}$ , and similarly $H E v_{k} - E H v_{k} = 2 E v_{k}$ ; moreover $E F v_{k} - F E v_{k} = (k + 1) (n - k) v_{k} - k (n - k + 1) v_{k} = (n - 2 k) v_{k} = H v_{k}$ , with both sides zero for $k = n$ and $k = 0$ respectively. This verifies the three bracket relations on every basis vector, hence (i).

L2given

2.1

The eigenvalues $n - 2 k$ , $k = 0, \dots, n$ , of $H$ are pairwise distinct, so every $H$ -eigenspace of $V (n)$ is one-dimensional, spanned by the corresponding $v_{k}$ .

givenstep 1.1

3.1

$V (n)$ is irreducible: if $0 \neq W \subseteq V (n)$ is a submodule, then $W$ is $H$ -stable and contains a nonzero $H$ -eigenvector, hence some $v_{k}$ ; applying $E$ exactly $k$ times gives $E^{k} v_{k} = k! (n - k + 1) (n - k + 2) \dots n v_{0} \neq 0$ because each coefficient $j (n - j + 1)$ with $1 \leq j \leq n$ is nonzero, so $v_{0} \in W$ ; applying $F$ repeatedly then gives $v_{1}, \dots, v_{n} \in W$ ; hence $W = V (n)$ by [L2].

givenL2step 2.1

4.1

Every finite-dimensional irreducible ${s l}_{2}$ -module $W$ is isomorphic to some $V (n)$ : by [L1] its top weight is an integer $n \geq 0$ , and it has a highest weight vector $w_{n}$ with $E w_{n} = 0$ and $H w_{n} = n w_{n}$ ; the commutation identity $[E, F^{j}] = j F^{j - 1} (H - (j - 1))$ , proved by induction, gives $E F^{j} w_{n} = j (n - j + 1) F^{j - 1} w_{n}$ and $H F^{j} w_{n} = (n - 2 j) F^{j} w_{n}$ ; the span of $F^{j} w_{n}$ , $j = 0, \dots, n$ , is nonzero and stable under $E, F, H$ , hence equals $W$ by irreducibility, and the assignment $F^{j} w_{n} \mapsto v_{j}$ is an isomorphism $W \to V (n)$ .

L1L2step 1.1step 3.1

5.1

Steps 1.1, 3.1 and 4.1 establish (i), (ii) and (iii), and the modules $V (n)$ for distinct $n$ are non-isomorphic because $H$ has different eigenvalue sets.

step 1.1step 3.1step 4.1∎

Depends on

Used by

Clebsch–Gordan decomposition for sl2 Example
Verma modules for sl2 Example
Weyl character and dimension formulas for sl2 Example

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)