# Some examples of Lie algebras

Last time, we defined Lie algberas and basic notions related to them. This time, I’ll show you some actual Lie algebras.

First, remember that given an associative algebra we can get a Lie algebra by defining the bracket, using multiplication in as This gives us a ton of examples already, the most important being that of the *general linear Lie algebra:*

Given a vector space the space of all linear maps from to noted is an algebra over Then define the *general linear algebra * as the Lie algebra you get when you redefine multiplication as the bracket

Similarly, is the corresponding Lie algebra of matrices with entries in All subalgebras of or are called *linear Lie algebras*. For example, we have the space of skew-symmetric matrices with coefficients in :

This is indeed a subalgebra because if are skew-symmetric,

so is also skew-symmetric. It is as easy to show that these are also linear Lie algebras :

where

There are also the linear Lie algebras of upper-triangular matrices, stricly upper-triangular matrices and diagonal matrices, respectively noted and It is easy to see that these are linear subspaces of to see that the diagonal matrices are closed under the bracket operation, note that for any two commuting elements we have so actually

Such a Lie algebra is called *Abelian. *For the two others, introduce the standard basis elements the matrices consisting of a in position and of elsewhere. This is a basis of and we calculate

meaning that is the zero matrix if and is if So for this basis, the bracket multiplication reads

Then for with and (those are the basis elements appearing in the decomposition of a matrix in ), only one of the two terms in the bracket expansion can appear. Indeed, if , it is because but then we can’t have because On the other hand, if it is because and then forces We showed that for two strictly upper triangular matrices the expression of involves only elements of the basis of and so this subspace is closed under the bracket operation. A similar argument shows that is a linear Lie algebra.

In the next post I will hint at the link between Lie algebras and the differential geometry of Lie groups.

**References:**

- Humphreys –
*Introduction to Lie Algebras and Representation Theory* - Knapp –
*Lie Groups Beyond an Introduction*

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