HDB1 said:
Thank you so much, I really appreciate your time and help,
please, I have a question, here, how you get the isomorophism here:
\begin{align*}
\mathfrak{gl}(2) &\twoheadrightarrow \mathfrak{gl}(2)/\mathfrak{sl}(2)\cong \mathbb{K}\\[16pt]
\mathfrak{gl}(2) &\twoheadrightarrow \mathfrak{gl}(2)/ \mathbb{K}\cdot \mathbb{1}\cong \mathfrak{sl}(2)
\end{align*}
could you please, give me example how we compute: ##\mathfrak{g l}(2) / \mathfrak{s l}(2)##
Quotient spaces are sets of equivalence classes. Given ##A/B## we consider its elements as ##a+B.## This means that ##a+B =a'+B \Longleftrightarrow a-a' \in B.##
The easiest way to get used to it is to consider, say ##\mathbb{Z}/3\cdot \mathbb{Z}.## It consists of elements ##z_0+3\cdot \mathbb{Z}.## So the elements themselves are sets, represented by ##z_0.## ##3\mathbb{Z}=\{\ldots,-6,-3,0,3,6,9,12,\ldots\}.## In ##\mathbb{Z}/3\cdot \mathbb{Z}## we identify
$$
a+3\mathbb{Z}=a'+3\mathbb{Z} \Longleftrightarrow a-a' \in 3\mathbb{Z}
$$
This results in three elements:
\begin{align*}
\bar 0=0+3\mathbb{Z}&=\{\ldots,-6,-3,0,3,6,9,12,\ldots\}\\
\bar 1=1+3\mathbb{Z}&=\{\ldots,-5,-2,1,4,7,10,13,\ldots\}\\
\bar 2=2+3\mathbb{Z}&=\{\ldots,-4,-1,2,5,8,11,14,\ldots\}
\end{align*}
Every integer is in one of these sets and represented by one of the elements ##\bar 0\, , \,\bar 1,\bar 2.## So $$
\mathbb{Z}/3\mathbb{Z}=\mathbb{Z}_3=\mathbb{F}_3=\{\bar 0\, , \,\bar 1,\bar 2\}=\{[0],[1],[2]\}=\{0,1,2\}
$$
The letter ##\mathbb{F}## signals that ##\mathbb{Z}_3## is a field (of characteristic ##3##) so it is sometimes written ##\mathbb{F}_3## and the set ##\{0,1,2\}## comes from lazy people who do not want to carry the bar, or sometimes brackets. The brackets here have nothing to do with Lie algebras. They are only a possible notation for equivalence classes. But as soon as it is clear that we speak about a quotient, i.e. equivalence classes as elements, as soon people simplify notation by just writing, here ##0,1,2## although they mean ##0+3\mathbb{Z}\, , \,1+3\mathbb{Z}\, , \,2+3\mathbb{Z}.##
This is the construction plan behind quotients. In this example, we have in ##0,1,2## the possible remainders by an integer division by ##3##. And these three numbers (with ##2+1=0\, , \,2\cdot\otimes =1##) form a field. The set under the quotient is the new zero. Here all numbers divisible by ##3## become zero.
The quotients for vector spaces or rings or algebras or Lie algebras work accordingly.
Consider an arbitrary matrix ##\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \mathfrak{gl}(2).## Then we write it as
$$
A=\begin{pmatrix}a&b\\c&d\end{pmatrix}=\underbrace{\begin{pmatrix}\dfrac{a+d}{2}&0\\0&\dfrac{a+d}{2}\end{pmatrix}}_{=\lambda \mathbb{1}}
+ \underbrace{\begin{pmatrix}\dfrac{a-d}{2}&b\\c&\dfrac{d-a}{2}\end{pmatrix}}_{=\;X\;\in \mathfrak{sl}(2)}
$$
The first part is a multiple of the identity matrix ##\mathbb{1}##, and the second part is in ##\mathfrak{sl}(2).##
A) The multiples of the identity matrix, ##\mathbb{K}\cdot \mathbb{1}## commute with all other matrices, so they are in the center or ##\mathfrak{gl}(2).## They are in fact the entire center (Schur's lemma, I think). The center of a Lie algebra is an ideal. So we can not only build ##\mathfrak{gl}(2)/\mathbb{K}\cdot \mathbb{1} =\mathfrak{gl}(2)/\mathfrak{Z}(\mathfrak{gl}(2)),## the ideal property also guarantees that this quotient is again a Lie algebra. (Try to prove: If ##I\triangleleft L## is an ideal, then ##L/I=\{a+I\,|\,a\in L\}## is again a Lie algebra.) Here we have
$$
\bar A = A+\{\lambda \cdot \mathbb{1}\,|\,\lambda \in \mathbb{K}\}= X
$$
The first part of the decomposition of ##A=\lambda\cdot \mathbb{1}+X## is swallowed in ##\{\lambda \cdot \mathbb{1}\,|\,\lambda \in \mathbb{K}\}## which is nothing else as the center of ##\mathfrak{gl}(2),## or a copy of the scalar field ##\mathbb{K}## because we have only one scalar as a parameter, it is a one-dimensional vector space:
$$
\mathfrak{gl}(2)/\mathfrak{Z}(\mathfrak{gl}(2))=\mathfrak{gl}(2)/\mathbb{K}\cdot \mathbb{1}\cong \{X\} =\mathfrak{sl}(2)
$$
The isomorphism is ##\varphi (A)=\bar A=X+\mathfrak{Z}(\mathfrak{gl}(2))## and since nobody wants to write this, we simply say ##\varphi (A)=X## where ##X## is the part of ##A## that is in ##\mathfrak{sl}(2).##
B) The second possibility is accordingly. We have
$$
[\mathfrak{gl}(2),\mathfrak{gl}(2)]=\mathfrak{sl}(2)
$$
so ##\mathfrak{sl}(2)\triangleleft \mathfrak{gl}(2)## is also an ideal in ##\mathfrak{gl}(2).## Therefore
$$
\mathfrak{gl}(2)/\mathfrak{sl}(2) \cong \mathbb{K}
$$
is again a Lie algebra. This time, we make ##\mathfrak{sl}(2)## our new zero in the quotient space. It swallows the parts of ##A## that is in ##\mathfrak{sl}(2),## i.e. it swallows ##X##. The isomorphism is thus ##\varphi (A)=\bar A=\lambda \cdot\mathbb{1}+\mathfrak{sl}(2),## and again if we strip all redundant notation, we get ##\varphi (A)=\lambda \in \mathbb{K}.## Strictly speaking, it would have been ##\mathbb{K}\cdot \mathbb{1}## but this is the same as just writing ##\mathbb{K}.##
Note: The decomposition of ##A## also shows, that ##\mathfrak{gl}(2)=\mathbb{K}\cdot \mathbb{1}\oplus \mathfrak{sl}(2)## as mere vector spaces, just by the addition of ##A=\lambda \cdot \mathbb{1}+X.## This allows us to calculate
\begin{align*}
[\mathfrak{gl}(2),\mathfrak{gl}(2)]&=[\mathbb{K}\cdot \mathbb{1}\oplus \mathfrak{sl}(2),\mathbb{K}\cdot \mathbb{1}\oplus \mathfrak{sl}(2)]\\
&=\underbrace{[\mathbb{K}\cdot \mathbb{1},\mathbb{K}\cdot \mathbb{1}]}_{=\{0\}}+
\underbrace{[\mathbb{K}\cdot \mathbb{1},\mathfrak{sl}(2)]}_{=\{0\}}+\underbrace{[\mathfrak{sl}(2),\mathbb{K}\cdot \mathbb{1}]}_{=\{0\}}+[\mathfrak{sl}(2),\mathfrak{sl}(2)]\\
&=[\mathfrak{sl}(2),\mathfrak{sl}(2)]=\mathfrak{sl}(2)
\end{align*}