$\mbox{SL}(n;\mathbb{R})$ and $\mbox{SL}(n;\mathbb{C})$ are $n$ dimensional Special (meaning [determinants](Determinants.md) are $1$) [[Matrix Lie group]]s over $\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ invertible matrices with real or complex entries respectively. They are subgroups of [[GL(n;F)]]. We can see that these are Lie groups since the determinant is a [continuous function,](Determinants.md#^ea894a) and since this also implies that these are [continuous groups.](Lie%20groups.md#^60524f) ([see proof.](SL(n;R)%20and%20SL(n;C)#Proof%20that%20mbox%20SL%20n%20mathbb%20R%20and%20mbox%20SL%20n%20mathbb%20C%20are%20Lie%20Groups)) # Lie algebras of $\mbox{SL}(n;\mathbb{R})$ and $\mbox{SL}(n;\mathbb{C})$ The [[Lie algebras]], $\mathfrak{sl}(n;\mathbb{R})$ and $\mathfrak{sl}(n;\mathbb{C})$ together contain all real and complex matrices with [trace](Trace.md) $0$, which can be shown from the [matrix exponentials](Matrix%20exponentials.md) property that $\det{(e^{tX})}=e^{t\mbox{tr}(X)}$ where $X$ is in $\mathfrak{sl}(n;\mathbb{R})$ or $\mathfrak{sl}(n;\mathbb{C})$. # Dimensionality of $\mbox{SL}(n;\mathbb{R})$ and $\mbox{SL}(n;\mathbb{C})$ $\mbox{SL}(n;\mathbb{R})$ is a Lie Group of dimension $n^2-1$. The dimensionality of a matrix group is not necessarily the same thing as the dimensionality of its matrix elements. * __Example__: We can start to see how dimensionality works if, for example, we take $\mbox{SL}(2;\mathbb{R})$. It is clearly inside $\mathbb{R}^4$ since it's a $2\times2$ matrix, meaning it contains 4 elements. * In general we may define a [[Smooth function]] on $\mathbb{R}^k$ where $f(\mathbf{x}) = c$ on some set $\mbox{E}$, where $\mbox{E}$ is a smooth surface in $\mathbb{R}^k$ of dimension $k-1$. * $\det( \begin{pmatrix} a & b \\ c & c \\ \end{pmatrix}) = ab - cd = 1$ * which also a smooth function $f(a,b,c,d) = 1$. One of its partial derivatives ($\frac{\partial f}{\partial a} = d$, $\frac{\partial f}{\partial b} = -c$, $\frac{\partial f}{\partial c} = -b$, $\frac{\partial f}{\partial d} = a$) must always be non-zero for the function to be valid, fulfilling the criteria for the determinant of a $2\times2$ to be a smooth surface in 3 dimensions. And Thus the set $\mbox{SL}(2;\mathbb{R})$ forms a 3 dimensional Lie Group. --- # Proofs and examples ## Proof that $\mbox{SL}(n;\mathbb{R})$ and $\mbox{SL}(n;\mathbb{C})$ are Lie groups %%See 1.1. of Hall's Lie Group text for this then either find or try to devise a better proof or find it somewhere else since it's very informal here.%% %%This might actually be better written for fields in general not R or C. Check this works for other fields.%% #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups