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**Homework Statement:**About semidirect product of Lie algebra

**Relevant Equations:**##\mathfrak{s l}_2=## ##\mathbb{K} F \oplus \mathbb{K} H \oplus \mathbb{K} E##

Hi,

*Please, I have a question about the module of special lie algebra:*Let ##\mathbb{K}## be a field. Let the Lie algebra ##\mathfrak{s l}_2=\mathbb{K} F \oplus \mathbb{K} H \oplus \mathbb{K} E##

is a simple Lie algebra where the Lie bracket is given by the rule: ##[H, E]=2 E,[H, F]=-2 F## and ##[E, F]=H##. Let ##V_2=\mathbb{K} X \oplus \mathbb{K} Y## be the 2-dimensional simple ##\mathfrak{s l}_2##-module with basis ##X## and ##Y##.

Let ##\mathfrak{a}:=\mathfrak{s l}_2 \ltimes V_2## be the semi-direct product of Lie algebras .

The Lie algebra ##\mathfrak{a}## admits the basis ##\{H, E, F, X, Y\}## and the Lie bracket is defined as follows

$$

\begin{array}{lllll}

{[H, E]=2 E,} & {[H, F]=-2 F,} & {[E, F]=H,} & {[E, X]=0,} & {[E, Y]=X,} \\

{[F, X]=Y,} & {[F, Y]=0,} & {[H, X]=X,} & {[H, Y]=-Y,} & {[X, Y]=0 .}

\end{array}

$$

Let ##A=U(\mathfrak{a})## be the enveloping algebra of the Lie algebra ##\mathfrak{a}##.

**Please, I know the basis of ##\mathfrak{s l}_2=##, which is ( as above):**

$$

E=\left[\begin{array}{ll}

0 & 1 \\

0 & 0

\end{array}\right], \quad F=\left[\begin{array}{ll}

0 & 0 \\

1 & 0

\end{array}\right], \quad H=\left[\begin{array}{cc}

1 & 0 \\

0 & -1

\end{array}\right].

$$

A computation in ##M_2(\mathbb{K})## yields the following set of relations:

$$

[H, E]=2 E, \quad[H, F]=-2 F, \quad[E, F]=H.

$$

*I need to know please,***1-**

**what is the matrix of X and Y**,

**2-**

**How we compute the bracket between the elements of ##\mathfrak{s l}_2=## and ##V_2##,**

Where: $$[x, y]=x y-y x$$.Thank you so much in advance,

Where: $$[x, y]=x y-y x$$.Thank you so much in advance,

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