Intereting Posts

Embedding manifolds of constant curvature in manifolds of other curvatures
If N is elementary nilpotent matrix, show that N Transpose is similar to N
Induction based on sum of $kth$ powers.
Let $a>0$ and $x_1 > 0$ and $x_{n+1} = \sqrt{a + x_n}$ for $n \in \mathbb{N}$. Show that $\{x_n\}_{n\ge 1}$ converges
Examples of mathematical induction
Finite Groups with exactly $n$ conjugacy classes $(n=2,3,…)$
If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
$ a $ and $b$ are real numbers with $0 < b < a$. Prove that if $n$ is a positive integer, then $a^n – b^n \leq na^{n-1}(a – b)$.
What is the distribution of a random variable that is the product of the two normal random variables ?
Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$
What kind of matrix norm satisifies $\text {norm} (A*B)\leq \text {norm} (A)*\text {norm} (B)$ in which A is square?
How to find solutions of linear Diophantine ax + by = c?
Can the intersection of open or closed balls be empty, if their radii are bounded from below?
Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
Proof of dividing fractional expressions

In the classification of Coxeter groups, or equivalently root systems:

$$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$

with $p \geq 7$, the last four fail to generate any simple finite dimensional Lie algebras over fields of characteristic zero because of the crystallographic restriction theorem.

- Is every finite group of isometries a subgroup of a finite reflection group?
- Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$…
- What is the Coxeter diagram for?

However, I know that there exist well defined “projections” (foldings in the context of Coxeter-Dynkin diagrams) $A_4 \rightarrow H_2, D_6 \rightarrow H_3, E_8 \rightarrow H_4$ and $A_n \rightarrow I_2(n+1)$, and that they have been used in certain situations to extend some of the objects related to crystallographic root systems to noncrystallographic ones, or to better understand them (e.g., quasicrystals, icosians, aperiodic tilings, affine Toda field theories, etc.). This leads me to wonder whether one could define a generalization of Lie group/Lie algebra such as to include the remaining four groups in the classification. So my question is:

Is there an “almost-Lie algebra” mathematical structure corresponding to the $H_n$ and $I_2(p)$ noncrystallographic root systems, in the same way that simple Lie algebras correspond to crystallographic ones? Has this been explored somewhere?

- When is the Killing form null?
- How to obtain a Lie group from a Lie algebra
- Construction of an Irreducible Module as a Direct Summand
- — Cartan matrix for an exotic type of Lie algebra --
- Is the Lie bracket of a Lie Algebra of a Matrix Lie Group always the commutator?
- Reference for Lie-algebra valued differential forms
- Is every skew-adjoint matrix a commutator of two self-adjoint matrices
- Decomposing tensor product of lie algebra representations
- Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?
- Defining an isomorphism that respects the Lie bracket: is my work correct?

There is a program around this called Spetses. The aim is to find some Lie theoretic object whose `Weyl group’ is a complex reflection group (which includes the non-crystallographic reflection groups above). I am not entirely clear on what has been accomplished. The notes linked allude to some results for general Coxeter groups, but I haven’t looked into those results.

- In how many ways can $1000000$ be expressed as a product of five distinct positive integers?
- Proving that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$
- can any continuous function be represented as a sum of convex and concave function?
- Is Inner product continuous when one arg is fixed?
- Weird measurable set
- Prove if $(a,p)=1$, then $\{a,2a,3a,…,pa\}$ is a complete residue system modulo $p$.
- Understanding the Definition of a Differential Form of Degree $k$
- Does there exist a continuous function from to R that has uncountably many local maxima?
- Adjoint of an Operator in $l^2$
- Does Riemann integrable imply Lebesgue integrable?
- Problem in Skew Symmetric Matrix
- Please explain inequality $|x^{p}-y^{p}| \leq |x-y|^p$
- Is every sigma-algebra generated by some random variable?
- Proof showing there exists a sequence of $m$ consecutive natural numbers which contains exactly $n$ primes.
- Holder's inequality $ \sum_{i=1}^n |u_i v_i| \leq (\sum_{i=1}^n |u_i|^p )^{\frac{1}{p}}(\sum_{i=1}^n |v_i|^q )^\frac{1}{q} $