Intereting Posts

How can I solve this question on Quadratic Equations and Arithmetic Progression?
Could someone explain aleph numbers?
Picking Multiples of 4
Variable naming convention in mathematical modeling
Simple module and homomorphisms
Is it possible to define a ring as a category?
Find the elements of $S_6$ that commute with $(1234)$.
Solution of $ax=a^x$
Integral of $1/z$ over the unit circle
Does a non-abelian semigroup without identity exist?
Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$
Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?
A question about the fixed point and group action.
Metrizable compactifications
Cardinalities of $\mathbb{R^{2}}$ and $\mathbb{C}$ and isomorphisms

**Fact** : $\phi : L_1\rightarrow L_2$ is $surjective$

Lie algebra

homomorphism. If $h\in L_1$ and ${\rm ad}_h$ is diagonalizable

then ${\rm ad}_{\phi(h)}$ is diagonalizable

**Defn** $x\in {\rm gl}\ ({\bf C}^n)$ has **Jordan** decomposition if $$

x= d+n $$ where $[d,n]=0$, $d$ is diagonalizable and $n$ is

nilpotent.

**EXE** (cf. 90page in Erdmann and Wildon’s book) If under the same assumption in the above fact $L_i$ are complex semisimples and if

$$ x=d+n$$ is Jordan, then we have Jordan $$ \varphi x=

\varphi d + \varphi n $$

- Reference for Lie-algebra valued differential forms
- Constructing faithful representations of finite dim. Lie algebra considering basis elements
- On surjectivity of exponential map for Lie groups
- The diffential of commutator map in a Lie group
- Lie algebra isomorphism between ${\rm sl}(2,{\bf C})$ and ${\bf C}^3$
- Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

**Proof** : Commutativeness : $$ [\phi d, \phi n] = \phi

[dn]=0$$

**Question** : I cannot prove “Diagonalizability of $\varphi d$ and nilpotency of $\varphi n$”

**My try** : $ {\rm ad}_x$ has Jordan That

is $$ {\rm ad}_x = {\rm ad}_{d’} + {\rm ad}_{n’},\ d’,\ n’\in L_1 $$

Note that we have Lie algebra homomorphism $$ \Phi : {\rm ad} \ L_1

\rightarrow {\rm ad}\ L_2,\ \Phi\ ({\rm ad}_y) = {\rm ad}_{\phi y}$$

and $$ {\rm ad}_{\phi x} = {\rm ad}_{\phi d’} + {\rm ad}_{\phi n’}

$$ where $[{\rm ad}_{\phi d’}, {\rm ad}_{\phi n’}]=0$, and ${\rm ad}_{\phi d’}$ is diagonalizable.

And note that $ {\rm ad}_{\varphi n’} $ is nilpotent : Since $$ 0=

\varphi (0) = \varphi [ ({\rm ad}_{n’})^k(y) ] = \varphi [ n’,\

\cdots, [n’,y]\cdots]=[\varphi n’,\cdots, [ \varphi n’, \varphi

y]\cdots ] = ({\rm ad}_{\varphi n’})^k (\varphi y) $$ ${\rm

ad}_{\varphi n’}$ is nilpotent.

My try is right ? I need helpful comment Thank you !

- Are the $C$-points of a simply connected algbraic group simply connected?
- Are Lie algebras $u_n$ and $su_n$ simple?
- On the relationship between the commutators of a Lie group and its Lie algebra
- Is every element of a complex semisimple Lie algebra a commutator?
- Basis for adjoint representation of $sl(2,F)$
- Exponential of powers of the derivative operator
- Three-dimensional simple Lie algebras over the rationals
- Calculating the Lie algebra of $SO(2,1)$
- How to differentiate a homomorphism between two Lie groups
- Two Definitions of the Special Orthogonal Lie Algebra

- Ultrafilters and measurability
- Discriminant of $x^n-1$
- Sufficient conditions to conclude that $\lim_{a \to 0^{+}} \int_{0}^{\infty} f(x) e^{-ax} \, dx = \int_{0}^{\infty} f(x) \, dx$
- Any left ideal of $M_n(\mathbb{F})$ is principal
- Is there a base in which $1 + 2 + 3 + 4 + \dots = – \frac{1}{12}$ makes sense?
- Compactness of Algebraic Curves over $\mathbb C^2$
- Applying Arzela-Ascoli to show pointwise convergence on $\mathbb{R}$.
- Picard's existence theorem, successive approximations and the global solution
- Problem about limit of Lebesgue integral over a measurable set
- Let $A = \mathbb{Z}$, $B = $, $C = (2, 7)$. List all elements of $A \cap (B^c \cap C)$.
- How many integers less than $1000$ can be expressed in the form $\frac{(x + y + z)^2}{xyz}$?
- Isometry in compact metric spaces
- A series problem by Knuth
- Applications of Probability Theory in pure mathematics
- Exponential Diophantine: $2^{3x}+17=y^2$