Questions tagged [derivations]

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4
votes
1answer
108 views

Categorical Kähler differentials and the Leibniz rule

From nlab, the module of Kähler differentials over some category $\mathcal{C}$ is the free functor: $$\Omega: \mathcal{C} \to \mathsf{Mod_{\mathcal{C}}}$$ left-adjoint to the (forgetful) embedding: $$...
-4
votes
0answers
51 views

partial differential inequality [duplicate]

I want to prove that if $f\leq l$ and $\lim_{a\to \infty}\frac{f(a)}{l(a)}=1$ (we can also suppose that $\lim_{a\to \infty}f(a)=\lim_{a\to \infty}l(a)=1$) where $f$ is a positive continuous bounded ...
9
votes
2answers
253 views

The relation between t-structures and derived category

Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap ...
5
votes
0answers
74 views

Kac-Moody Lie algebra as derivations of associative algebras

The set of derivations of an algebra $\Bbb A$ forms a Lie algebra. This is one aspect of why Lie algebras are interesting. When $\Bbb A$ is polynomial algebra in $n$ variable then $\text{Der } \Bbb A$ ...
13
votes
2answers
619 views

Does any derivation of commutative algebra preserve its nil-radical?

Given a commutative associative unital algebra over a field of characteristic zero. Is it true that any derivation of it preseves its nil-radical? More explicitly, let $D$ be a derivation of an ...
4
votes
1answer
120 views

Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations

A correspondence $_{N} H_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras ...
1
vote
1answer
213 views

Katz's paper on $p$-curvature – help with proof understanding

I am studying N. Katz's paper "Nilpotent connections and the monodromy theorem: applications of a result of Turrittin" where I found a fairly good account on $p$-curvatures. I don't understand the ...
1
vote
1answer
145 views

The derivative of a filter with respect to a output singal [closed]

I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e. $$ d(t)*w(t)=p(t) $$ where $*$ denotes convolution.The impulse response $w(t)$ may be ...
3
votes
1answer
151 views

How to prove monotonicity of such function?

Let $0<a \le 1, \alpha<0$ and $\beta>0$. How to prove that the function: $$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\...
-2
votes
1answer
270 views

Why this function is monotonic?

Let $a> 0, \alpha<0$ and $\beta>0$. How to prove that the function: $$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \...
-2
votes
3answers
205 views

What's the relation between Lipschitz constant and the determinant of Jacobian matrix?

For a vector fuction $y=f(x)$ where $x, y$ are vectors. What's the relation between Lipschitz constant and the determinant of Jacobian matrix?
1
vote
0answers
58 views

References: Lie derivations of Full matrix algebra [closed]

I want, if possible a list of references that trait Lie derivations of Full matrix algebras. Other than Lie derivations of generalized matrix algebras . Thanks
0
votes
0answers
47 views

Generalized derivations in the sense of Brešar and Gölbaşı

When I search about the definition of generalised derivations, I found there are two definitions. The first definition by Brešar On the distance of the composition of two derivations to the ...
7
votes
1answer
131 views

Structure of the module of derivations on the space of Holomorphic functions

Maybe this is well-known, maybe not. Let $\Omega\subset \mathbb{C}$ be connected open and non-empty. It can be shown that if $d\in\mathfrak{Der}(\mathcal{H}(\Omega))$ (i.e. $d$ is a derivation of ...
4
votes
3answers
703 views

Does there exist another form of the derivative for polynomials?

Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that $$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$ for all $P, Q \in \mathbb{R}[X]...
2
votes
1answer
191 views

When $\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)$? [closed]

Let $\phi:G\rightarrow H$ be a morphism of (linear or not?) algebraic groups. What are, in general, the conditions to assure $$\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)\text{?}$$
4
votes
1answer
109 views

First adjoint cohomology space of simple Lie algebras

Let $L$ be a central extension of a simple Lie algebra $\mathfrak{g}$ such that $L=[L,L]$. It is not difficult to see that if $H^1(\mathfrak{g}, \mathfrak{g})=0$ then $H^1(L,L)=0$. In other words, if ...
2
votes
0answers
74 views

Derivations of algebras graded by a group

Let $A$ be an algebra. A derivation of $A$ is a linear map $d:A \rightarrow A$ such that $d(ab)=d(a)b + a d(b)$ for $a, b \in A$. If $A$ is a $\mathbb{Z}-$graded algebra, where $\mathbb{Z}$ is the ...
2
votes
0answers
72 views

How can I prove that the following function is increasing according to x1?

Suppose that $0 \le {X_1} < {X_2} < {X_3}$ . How is it possible to prove the following function is increasing based on ${X_1}$ in the range of $0 \le {X_1} < {X_2}$ ? $f({X_1},{X_2},{X_3})...
2
votes
1answer
98 views

Continuity of the derivations from semisimple Banach algebras

Let $A$ be a Banach algebra and $X$ a Banach $A$-bimodule. It is known that if $A$ is a $C^*$-algebra, then by Ringrose theorem every derivation $D:A\rightarrow X$ is continuous. Also, a famous ...
1
vote
1answer
110 views

Locally nilpotent derivations on rings with zero divisors

Almost all books that I have found deal with derivation on several types of rings (or algebras) (for instance, commutative, noncommutative, domains, non-domains etc). However, each paper about ...
16
votes
2answers
849 views

“Insanely increasing” $C^\infty$ function with upper bound

Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set $f^{(0)} = f$, ...
2
votes
2answers
471 views

Find formula for recurrence relation with two function and two variables

$f(n,k) = 2g(n-2,k-1)+f(n-1,k)$ $g(n,k) = g(n-1,k-1)+f(n,k)$ when $n\le0$ or $k\le0: \quad f(n,k) = 0$ when $n < k:\quad f(n,k) = 0$ when $n-k<-1:\quad g(n,k) = 0$ when $k=0:\quad g(n,k) = 1$ $...
16
votes
1answer
542 views

Does every real function have this weak derivation property?

After this question : Does every real function have this weak continuity property? Natrualy there are an other (more difficult) : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
11
votes
3answers
1k views

Why should we study derivations of algebras?

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 ...
2
votes
1answer
1k views

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
votes
1answer
93 views

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
votes
0answers
58 views

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

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 ${\rm Aut}_{L_1}(L)$ ...
1
vote
0answers
131 views

Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains

Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[...
1
vote
2answers
156 views

Prove a $C^{\infty}$ multivariable function is lipchitz via Jacobian matrix [closed]

I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$ is it sufficient to prove the norm of ...
2
votes
0answers
184 views

Derivative with multiple summation operators

I have a defined utility function as Eq.(1), and I am seeking the minimized utility subjects to some constraints. The notation used is as following: \linebreak $V$ is the set of nodes, $v_i\in V$; $O$...
6
votes
1answer
560 views

Semantics of derivations as derivatives

My understanding of how derivations on commutative rings are like derivatives is that a derivation on $R$ is differentiation with respect to a vector field on $\text{Spec}(R)$. But derivations are ...
10
votes
1answer
268 views

An identity in Lie algebras over fields of positive characteristic

Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,...
4
votes
0answers
93 views

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 ...
2
votes
1answer
253 views

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 $u$ ...
0
votes
1answer
267 views

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 = \...
3
votes
1answer
242 views

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 ...