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The standard two-dimensional ${s l}_{2}$ module extends to the enveloping algebra

Example

Let $V = C^{2}$ and let ${s l}_{2} (C)$ be the Lie algebra of traceless $2 \times 2$ complex matrices with commutator bracket. The standard action is matrix multiplication. In the usual basis, put $E = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}), F = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}), H = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) .$

Facts & Assumptions

Given: $V = C^{2}$ , the subspace ${s l}_{2} (C)$ of ${End}_{C} (V)$ consisting of trace-zero matrices, and the commutator bracket $[A, B] = A B - B A$ .

[F1]

Lie algebras over a field: a Lie algebra is a vector space with a bilinear alternating bracket satisfying the Jacobi identity.

[F2]

Representations of Lie algebras: ${End}_{k} (V)$ with commutator is a Lie algebra, and a representation is a Lie-algebra homomorphism into it.

[F3]

Lie representations are U(g)-modules: restriction along $g \to U (g)$ and extension by the enveloping-algebra universal property give mutually inverse correspondences between Lie representations and scalar-compatible unital left $U (g)$ -modules.

Verification

1.1givenF1F2algebra

The matrices $E, F, H$ are linearly independent and every trace-zero matrix is $a H + b E + c F$ for unique $a, b, c \in C$ . Direct multiplication gives $[H, E] = 2 E$ , $[H, F] = - 2 F$ , and $[E, F] = H$ . By bilinearity the span is closed under commutators, and it inherits alternation and Jacobi from ${End}_{C} (V)$ ; hence it is the stated Lie algebra.

2.1givenF2step 1.1

The inclusion $ρ : {s l}_{2} (C) ↪ {End}_{C} (V)$ is linear and preserves the bracket because both brackets are the same matrix commutator. Thus $V$ is a Lie-algebra representation.

3.1givenF3step 2.1∎

By [F3], this representation extends uniquely to a scalar-compatible unital left $U ({s l}_{2} (C))$ -module on $V$ . This uses neither PBW nor injectivity of the canonical map into the enveloping algebra.

Depends on

Used by

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Sources

Pavel Etingof et al., Introduction to Representation Theory, §§1.13–1.14, printed pp. 20–21 (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras, §§4.1 and 5.1, printed pp. 49–50 and 71–72 (standard reference, not scraped)

The standard two-dimensional sl2sl2​ module extends to the enveloping algebra