# derivations's questions - English 1answer

13 derivations questions.

### 9 Why should we study derivations of algebras?

3 answers, 831 views ra.rings-and-algebras derivations
Some authors have written that derivation of an algebra is an important tools for studying its structure. Could you give me a specific example of how a derivation gives insight into an algebra's ...

### 1 Partial derivatives of spherical harmonics

Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?

### 3 Locally nilpotent derivation on $A[X,Y]$ whose kernel is $A$; where $A$ is an affine $k$ domain, $char k=0$

Let $k$ be a field of characteristic zero and $A$ be a $k$-algebra. A derivation on $A$ is a $k$-linear map $D: A \to A$ such that $D(ab)=aD(b)+bD(a), \forall a,b \in A$. A derivation is called ...

### 2 Lie algebra of derivations for a transcendental field extension and intersection fields

0 answers, 37 views galois-theory fields derivations
Suppose that $L$ is a finite Galois extension of the field $K$. If $L_1$ and $L_2$ are subfields of $L$ containing $K$ then $L_1\cap L_2=L^H$ where $H$ is the group generated by ${\rm Aut}_{L_1}(L)$ ...

### 4 Restricted universal extensions and lifting of derivations

Let $L$ be a perfect Lie algebra. Then it is well-known that $L$ has a universal central extension $\hat{L}$ and every derivation of $L$ can be lifted to a derivation of $\hat{L}$. (See e.g. Section 2 ...

### 3 Derivations of central extensions of simple Lie algebras

Let $L$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is not difficult to see (and also follows from Theorem 4.4 of [G. Hochschild: Semi-simple algebras and generalized ...

### 2 Formal Cauchy-Riemann equations for formal power series without complex analysis

Consider the ring $\mathbb{C}[[X,Y]]$ and its subring $\mathbb{C}[[X+iY]]$, where $i=\sqrt{-1}$. One can show that $f(X,Y):=u(X,Y)+iv(X,Y)\in \mathbb{C}[[X,Y]]$ lies in $\mathbb{C}[[X+iY]]$ iff $u$ ...

### The limitation of derivation of modified Bessel function of second kind

The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all \$I = \...