Let $L = n(n, F)$, the Lie algebra of strictly upper triangular $n \times n$ matrices over a field $F$. Show that $L_k$ has a basis consisting of all the matrix units $e_{ij}$ with $j −i &gt; k$. Hence show that $L$ is nilpotent. What is the smallest m such that Lm = 0?  I understand that nilpotent means that when applying the bracket operator repeatedly the contents of the matrix $\rightarrow 0$ after some number of applications, but I'm a little unsure of how to approach this proof. Any help would be appreciated since I'm still very new to Lie algebras.
I have been attempting this problem for a while, it is an assignment problem so I don't want somebody to just post the answer, I'm just looking for hints.  Let $\mathfrak{u}(n,\mathbb{C})$ be the Lie algebra of strictly upper triangular $n\times n$ matrices over $\mathbb{C}$.  I am trying to show that $\mathfrak{u}(n,\mathbb{C})$ is a nilpotent Lie algebra for $n\geq 2$.  This is what I have looked at so far.  I have done some explicit calculations (I shall omit the bulk of them and just summarise below) for $n=2,3,4$ and have come up with some ideas.  Let $L=\mathfrak{u}(n, \mathbb{C})$.  If $n=2$,  $L^1 = [L, L] = &lt;0&gt;_{\mathbb{C}}$  So $L$ is nilpotent.  If $n=3$,  $L^1 = [L, L] = &lt;e_{13}&gt;_{\mathbb{C}}$  $L^2 = [L, L^1] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  If $n=3$  $L^1 = [L,L] = &lt;e_{13},e_{14},e_{24}&gt;_\mathbb{C}$  $L^2 = [L,L^1] = &lt;e_{14}&gt;_{\mathbb{C}}$  $L^3 = [L, L^2] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  So my hypothesis so far is that for an arbitrary $m\in\mathbb{N}$, $L^{m-1}=\{0\}$  My instinct is to do this by induction. My base case, $n=2$ has already been shown. Now I assume that $\mathfrak{u}(n,\mathbb{C})$ is nilpotent for $n=k$ ($k\in\mathbb{N}$) and consider the case for n=k+1.  However, I have no idea how to link these two cases together. I have read in Introduction to Lie Algebras and Representation Theory - J.E. Humphreys that $L^k$ should be spanned by basis vectors $e_{ij}$ where $j-i=k+1$ - however in my calculations for $L^1$, I obtain an $e_{14}$ which does not satisfy 4-1=2.  Am I missing something completely obvious? Or have I made a mistake somewhere? (I have checked my calculations numerous times, but cannot find any errors!)  Thanks,  Andy.
Could the missing antimatter lie outside the observable universe? Why don't we have an isomorphism between $R[x]$ and $ R[[x]]$? $R$ is a ring. Why don't we have an isomorphism  $$ R[x] \cong R[[x]]\ ? $$ Positive integer multiples of an irrational mod 1 are dense I'm not sure how to solve this one. Thank you!  $2.$ For any $\alpha\in \mathbb R$ we define $$\lfloor \alpha \rfloor = \max_{n\in\mathbb Z}\{\,n\mid n\leq \alpha\,\}$$ and $$\alpha\bmod 1 = \alpha - \lfloor \alpha \rfloor$$ Let $\alpha$ be irrational. (a) Given $n\in\mathbb N$ show that $\{\,k\alpha\bmod1\mid k\in\mathbb N\,\}\cap\left[0,\frac{1}{n}\right]\neq\emptyset$ (b) Prove that $\{\,n\alpha\bmod 1\mid n\in\mathbb N\,\}$ is dense in $[0,1]$. The set of functions which map convergent series to convergent series Suppose $f$ is some real function with the above property, i.e.  if $\sum\limits_{n = 0}^\infty {x_n}$ converges, then $\sum\limits_{n = 0}^\infty {f(x_n)}$ also converges.   My question is: can anything interesting be said regarding the behavior of such a function close to $0$, other than the fact that $f(0)=0$? What's the 1 inch square knob sticking out of the wall? Please consider gathering Meta feedback before the "ship" stage How to prove that $p-1$ is squarefree infinitely often? How to prove that $p-1$ is squarefree infinitely often, where $p$ is prime?  I had thought of using Dirichlet's theorem on arithmetic progressions.  The number of squareful values of $p-1$ less than or equal to $n$ is bounded by $\sum_{q}{\pi_{q^2,1}(n)}$, each term $\pi_{q^2,1}(n) \approx \frac{\pi(n)}{q \cdot (q-1)}$, and $\sum_{q}{\frac{1}{q \cdot (q-1)}} \lt 1$.  But using for example , the possibility is open that the error in each approximation for $\pi_{q^2,1}(n)$ may continue to increase with $q$, and since the sum runs up to $\sqrt{n}$, this argument is invalid.  Is there a more refined version of Dirichlet's theorem that can be used to circumvent this issue?  Or an entirely different way to prove this statement? I'm a student and I'm working on a low power communication project. I am trying to design a PCB using the . There is a MC-306 (32.768kHz, 12.5pf, and 20/50ppm).   I don't know what the 20/50ppm rating is. For me, the size is very important, so I decided to replace it with the , but its ppm is -20/+20. Will it cause a problem if I use this one instead?  What is the ppm rating in the crystal oscillator? difference between over and for I have a simple question.     over centuries / for centuries   what is the difference?  Thank you. I find I always have trouble scaling the amount of time the eggs should be boiled, cooled, etc. for different-sized batches of hard-boiled eggs.  I've heard a variety of "folk lore"-type rules for how it should be done, but what's really the right way? Proving any number of the form $2^n$ can't be written as the sum of k consecutive numbers. I've tried different ways to install iTunes, but to no avail. I tried with Wine, but it didn't work. Is it possible to have iTunes on Ubuntu? Hot and Week tabs not shown to non-logged in users (except on SO) How to put a full-width table at the top or bottom of the same two-column page as the reference text? Are other players able to see you on their map in Pokemon Go? I am concerned that if I use the game, stalkers and other such people would be able to find me. I have searched for an answer for this question on other forums, but have yet to find the answer. Why was Edo Tensei Madara young, when he had died at an old age? From Naruto manga and anime, I know that Uchiha Madara died as an old person after he helped Uchiha Obito.   But it was bit weird for me, when Kabuto summoned Uchiha Madara-sama with Edo Tensei, Madara was a younger person.  Again, when Chiyo was summoned, she was older as when she died.  How can it be possible? How to get the spaces properly on the x axis, how to place the y values on the corresponding bars? I am using the following commands to draw a barchart.  \begin{tikzpicture}         \pgfplotstableread{ % Read the data into a table macro             Label       Female   Male               2006-2008    3     1              2007-2009    2     0             2008-2010    9     2             2019-2011    3     3             2010-2012    11    1              2011-2013    6     0             2012-2014    12    7             2013-2015    10    7             2014-2016    10    3         }\datatable          \begin{axis}[         compat=newest, %Better label placement %        axis on top,         ybar stacked,   % Stacked horizontal bars         xmin=0,              % Start x axis at 0         ymax=25,         ymin=0, ylabel={Number of students}, xlabel={Batches},bar width=10pt,         y axis line style={opacity=0},         yticklabels={\empty},         ytick style={draw=none},enlarge y limits={upper, value=0.2},         xtick=data,     % Use as many tick labels as x coordinates         enlarge x limits = .2,         xticklabels from table={\datatable}{Label}  % Get the labels from the Label column of the \datatable         ]         \addplot [fill=yellow] table [y=Male, x expr=\coordindex] {\datatable};    % Plot the "First" column against the data index         \addplot [fill=green!70!blue]table [y=Female, x expr=\coordindex] {\datatable}; %        \addplot [fill=red!80!yellow] table [y=Third, x expr=\coordindex] {\datatable}; \legend{Male,Female}         \end{axis}         \end{tikzpicture}   I got the picture as,     How to get the spaces properly on the x axis? How to place the y values on the corresponding bars? How to rotate the label 90 degree? How to take out the ticks on the top? Is there any other way of representing this data(two dim only)? (optional)   I hope all the questions can be answered in a single answer? Force newlines with cat wildcard printing I want to use cat with wildcards in bash in order to print multiple small files (every file is one sentence) to standard output. However, the separate file contents are not separated by a newline, which I'd like to for ease of reading.  How can I add a file delimiter of some sort to this command? What kind of node is this? We consider $g : (t,u,v)\in \mathbb{R}^3 \mapsto (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)\in\mathbb{R}^6$. I have to prove that $g(\mathbb{S}^2)$ is a submanifold of $\mathbb{R}^6$.  $dg_{(t,u,v)}=\begin{pmatrix}2t&amp;0&amp;0\\  0&amp;2u&amp;0\\  0&amp;0&amp;2v\\  0&amp;\sqrt{2}v&amp;\sqrt{2}u\\ \sqrt{2}v&amp;0&amp;\sqrt{2}t\\ \sqrt{2}u&amp;\sqrt{2}t&amp;0\\ \end{pmatrix} $  So it proves that $g$ is an immersion (the rank is $3$ clearly) for $(\mathbb{R}^3-\{(0,0,0)\})$.  Now how to prove that $\mathbb{S}^2$ is homeomorphic to $g(\mathbb{S}^2)$ ? I must prove that $g$ is injective  ?  Thanks in advance !