3:16 AM
A new tag was created by shrinklemma.
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There are at least two ways of defining the inner automorphisms of a real Lie algebra $\mathfrak{g}$. One is the algebraic definition: an inner automorphism is $\exp (\text{ad} X)$, where $X$ is an nilpotent element of $\mathfrak{g}$. The other is the analytic definition: the automorphism group $... In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras g {\displaystyle {\mathfrak {g}}} whose only ideals are {0} and g {\displaystyle {\mathfrak {g}}} itself. It is important to emphasize that a one-dimensional Lie algebra (which is necessarily abelian) is by definition not considered a simple Lie algebra, even though such an algebra certainly has no nontrivial... 3 hours later… 6:14 AM It seems that Saad is fond of deleting newly created tags and rejecting edits about tags (e.g. improve usage, retag... ) @MartinSleziak And tag on this question was deleted by him, too. 1 hour later… 7:21 AM I improved usage and wiki for , still pending now 7:32 AM @Andrews FYI If there is no question using that tag, the tag will be deleted by the system in 24 hours. @ArcticChar There was, but deleted by Saad. And there're a lot of questions relevant to right now. If I mannually add this tag to relevant questions, I'm afraid they'll be deleted soon 2 hours later… 9:27 AM Just to have it saved - in case this instance of the tag is deleted - I will add here links to the revisions for the tag-excerpt and the tag-wiki. @Andrews Well, there are people who are much stricter about creating new tags. However, I think I have mentioned this link before: Should every new tag be discussed on meta before creation? 2 hours later… 11:05 AM I added to 2 questions in case of the tag being deleted. And since we already have tag is suitable to me. 4 Let$\mathfrak{g} \subset gl(V)$be a semisimple Lie algebra. I already know that symmetric bilinear form$f(x,y)=\mathbf{Trace}(XY)$is nonsingular on$\mathfrak{g}$. And I've read that any ideal$\mathfrak{g_1}$of$\mathfrak{g}$is also semisimple, from which it also follows that$\mathfrak{g...

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We know that each complex semisimple lie algebra $L$ is a direct sum of a chosen Cartan subalgebra $H$ and finitely many weight spaces, each of which is associated with an element in $H^*=\operatorname{Hom}(H,\mathbb{C})$, also known as a root. The set of roots of $L$ forms a root system which c...

11:51 AM
@MartinSleziak This tag was added again (actually rolled back)

Actually, I was wondering whether the is too specialized, as it's just the 1-dimensional version of Grassmann numbers, which is a more general concept. But anyway, I decided to leave it there and add a tag excerpt and tag wiki.

12:15 PM
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[EDIT: I know what the notation means, and I can easily show that $x=\epsilon \otimes 1 + 1\otimes \epsilon$ satisfies $x^3=0$ but $x^2 \neq 0$. That's not what this question is about. It might be better to restrict the question to Synthetic Differential Geometry where it makes more sense. There ...

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of n dimensions leads to the Grassmann numbers. The algebra of dual numbers is a ring that is a local ring since the principal ideal...
We'll see whether the tag survives. In any case, it now has tag-excerpt and tag-wiki.

4 hours later…
4:24 PM
@MartinSleziak Well, the tag was again removed: math.stackexchange.com/posts/3179368/revisions

4:34 PM
@MartinSleziak The tag was edited away too.

4 hours later…
8:19 PM