Started typing the equivalence of the concrete and abstract PBW theorem.

basics/universal_enveloping_algebra.tex

@@ -212,7 +212,7 @@ \subsubsection{Graded $k$-algebras}
 
 
 \begin{defi}
- Let $A$ and $B$ be graded $k$-algebras. A homomorphism of $k$-algebras $\varphi \colon A \to B$ is called a \emph{homomorphism of graded $k$-algebras} if $\varphi(A_n) \subseteq B_n$ for every $n \in \N$. An homomorphism of graded $k$-algebras is called an isomorphism if it is bijective.
+ Let $A$ and $B$ be graded $k$-algebras. A homomorphism of $k$-algebras $\varphi \colon A \to B$ is called a \emph{homomorphism of graded $k$-algebras} if $\varphi(A_n) \subseteq B_n$ for every $n \in \N$, and the induced homomorphisms of vector spaces are denoted by $\varphi_n \colon A_n \to B_n$ for every $n \in \N$. An homomorphism of graded $k$-algebras is called an isomorphism if it is bijective.
 \end{defi}
 
 
@@ -396,53 +396,118 @@ \subsubsection{Filtered $k$-algebras}
 \subsubsection{The PBW theorem}
 
 
-\begin{thrm}[Poincar\'{e}-Birkhoff-Witt (abstract version)]
- Let $\g$ be a Lie algebra over $k$ and
+\begin{thrm}[Poincar\'{e}-Birkhoff-Witt (abstract version)] \label{thrm: pbw abstract}
+ Let $\g$ be a Lie algebra over $k$ and denote by $\pi$ the canonical projection
  \[
   \pi \colon T(\g) \to \Ue(\g), \quad
   x_1 \otimes \dotsb \otimes x_n \mapsto x_1 \dotsm x_n
   \quad \text{for all $x_1, \dotsc, x_n \in \g$}.
  \]
- the canonical projection. Then the homomorphisms of graded $k$-algebras
+ Then the two homomorphisms of graded $k$-algebras $\gr(\pi) \colon T(\g) \to \gr(\Ue(\g))$ and
  \[
-  \varphi \colon T(\g) \to S(\g), \quad 
+  \pi' \colon T(\g) \to S(\g), \quad 
   x_1 \otimes \dotsb \otimes x_n \mapsto x_1 \dotsm x_n
-  \quad \text{for all $x_1, \dotsc, x_n \in \g$}.
+  \quad \text{for all $x_1, \dotsc, x_n \in \g$}
  \]
- and $\gr(\pi) \colon T(g) \to \gr(\Ue(\g))$ have the same kernel and thus induce an isomorphism of graded algebras
- \[
-  \psi \colon \gr(\Ue(\g)) \to S(\g), \quad
-  [x_1 \dotsm x_n] \mapsto x_1 \dotsm x_n
+ have the same kernel and thus induce an isomorphism of graded algebras
+ \begin{equation}\label{eqn: induced isomorphism of graded algebras}
+  \varphi \colon S(\g) \to \gr(\Ue(\g)), \quad
+  x_1 \dotsm x_n \mapsto [x_1 \dotsm x_n]
   \quad \text{for all $x_1, \dotsc, x_n \in \g$}.
- \]
+ \end{equation}
 \end{thrm}
 
 
-\begin{thrm}[Poincar\'{e}-Birkhoff-Witt (concrete version)]
+\begin{rem}
+ Notice that the two multiplications in \eqref{eqn: induced isomorphism of graded algebras} live in different $k$-algebras.
+\end{rem}
+
+
+\begin{thrm}[Poincar\'{e}-Birkhoff-Witt (concrete version)] \label{thrm: pbw concrete}
  Let $\g$ be a $k$-Lie algebra and $(x_i)_{i \in I}$ a $k$-basis of $\g$ where $(I,\leq)$ is a totally ordered index set. Then the familiy
- \[
+ \begin{equation}\label{eqn: concrete pbw basis}
   \left(
-  x_{i_1}^{p_1} \dotsm x_{i_n}^{p_n}
+   x_{i_1} \dotsm x_{i_n}
   \mid
-  n \in \N, i_1, \dotsc, i_n \in I, i_1 < \dotsb < i_n, p_1, \dotsc, p_n \geq 1
+   n \in \N,\;
+   i_1, \dotsc, i_n \in I,\;
+   i_1 \leq \dotsb \leq i_n
   \right)
- \]
+ \end{equation}
  is a $k$-basis of $\Ue(\g)$.
 \end{thrm}
 
 
+\begin{rem}
+ The basis in \eqref{eqn: concrete pbw basis} can also be written as
+ \[
+  \left(
+   x_{i_1}^{p_1} \dotsm x_{i_n}^{p_n}
+  \mid
+   n \in \N,\;
+   i_1, \dotsc, i_n \in I,\;
+   i_1 < \dotsb < i_n,\;
+   p_1, \dotsc, p_n \geq 1
+  \right).
+ \]
+\end{rem}
+
+
 \begin{expl}
- If $\g$ is a finite $k$-Lie algebra with basis $x_1, \dotsc, x_n$ then $\Ue(\g)$ has a basis given by $(x_1^{p_1} \dotsm x_n^{p_n} \mid p_1, \dotsc, p_n \in \N)$. In particular a basis of $\Ue(\sll_2(k))$ is given by $(e^\ell h^m f^n \mid \ell, m ,n \in \N)$.
+ If $\g$ is a finite dimensional $k$-Lie algebra with basis $x_1, \dotsc, x_n$ then $\Ue(\g)$ has a basis given by $(x_1^{p_1} \dotsm x_n^{p_n} \mid p_1, \dotsc, p_n \in \N)$. In particular a basis of $\Ue(\sll_2(k))$ is given by $(e^\ell h^m f^n \mid \ell, m ,n \in \N)$.
 \end{expl}
 
 
+\begin{lem}
+ Let $\g$ be a $k$-Lie algebra. Then the collection defined by \eqref{eqn: concrete pbw basis} generates $\Ue(\g)$ as a vector space.
+\end{lem}
+\begin{proof}
+ The collection
+ \begin{equation}\label{eqn: generators of Ug_n as filtration}
+  \mc{B}_n \coloneqq (x_{i_1} \dotsm x_{i_m} \mid m \leq n, \; i_1, \dotsc, i_m \in I, \; i_1 \leq \dotsb \leq i_m)
+ \end{equation}
+ generates $\Ue(\g)_{(n)}$ as a vetor space by induction over $n \in \N$: For $n = 0$ this is clear as $\Ue(\g) = k$ is one-dimensional and thus spanned by the monomial given by the empty product $\prod_{i \in I} x_i^0$.
+
+ Suppose that the statement holds for some $n \in \N$. Then $\Ue(\g)_{(n+1)}$ is generated by $x_{i_1} \dotsm x_{i_m}$ with $m \leq n+1$ and $i_1, \dotsc, i_m \in I$ as a vector space, hence it sufficies to express these monomials in terms of $\mc{B}_{n+1}$.
+\end{proof}
+
+
 \begin{prop}
  The concrete version and the abstract versions of the PBW-theorem are equivalent.
 \end{prop}
 \begin{proof}
- (concrete $\Rightarrow$ abstract)
+ For $x, y \in \g$ it follows from the definition of $\gr(\pi)$ that
+ \[
+  \gr(\pi)(x \otimes y - y \otimes x) = [xy-yx] \in \gr(\Ue(\g))_2,
+ \]
+ with representative $xy-yx \in \Ue(\g)_{(2)}$. By the definition of $\Ue(\g)$ it follows that already \mbox{$xy-xy = [x,y] \in \Ue(\g)_{(1)}$}. Hence it follows for the residue class of $xy-yx$ in $\gr(\Ue(\g))_2 = \gr(\Ue(\g))_{(2)}/\gr(\Ue(\g))_{(1)}$ that $[xy-yx] = [[x,y]] = [0] = 0$. Hence $\gr(\pi)(x \otimes y - y \otimes x) = 0$.
+
+ As the kernel of $\pi'$ is generated by the element $x \otimes y - y \otimes x$ with $x,y \in \g$ it follows that $\pi'$ factorizes through a homomorphism of graded $k$-algebras $\varphi$ as in Theorem~\ref{thrm: pbw abstract}.
+
+ (concrete $\Rightarrow$ abstract) The algebra $\Ue(\g)$ has a basis
+ \[
+  \left(
+   x_{i_1} \dotsm x_{i_n}
+  \mid
+   n \in \N, \;
+   i_1, \dotsc, i_n \in I, \;
+   i_1 \leq \dotsb \leq i_n
+  \right).
+ \]
+ It follows that $\gr(\Ue(\g))_n$ has a basis given by the residue classes
+ \[
+  \left(
+   [x_{i_1} \dotsm x_{i_n}]
+  \mid
+   i_1, \dotsc, i_n \in I, \;
+   i_1 \leq \dotsb \leq i_n
+  \right).
+ \]
+ The linear subspace $S(\g)_n$ has a basis $(x_1 \dotsm x_n \mid i_1, \dotsc, i_n \in I \; i_1 \leq \dotsb \leq i_n)$ which is mapped by $\varphi_n$ to the above basis of $\gr(\Ue(\g))_n$. Hence $\varphi_n$ is an isomorphism for every $n \in \N$, which is why $\varphi$ is an isomorphism.
 
+ (abstract $\Rightarrow$ concrete) A
 
+ (abstract $\Rightarrow$ concrete) Because $\gr(\pi)$ is surjective it follows that the same goes for $\varphi$. As the basis $(x_{i_1} \dotsm x_{i_n} \mid n \in \N, \; i_1, \dotsc, i_n \in I, \; i_1 \leq \dotsb \leq i_n)$ of $S(\g)$ is mapped to \eqref{eqn: concrete pbw basis} by $\varphi$ it follows that \eqref{eqn: concrete pbw basis} generates $\gr(\Ue(\g))$ as a vector space.
 \end{proof}