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...
@ArcticChar There was, but deleted by Saad. And there're a lot of questions relevant to semisimple-lie-algebras right now. If I mannually add this tag to relevant questions, I'm afraid they'll be deleted soon
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...
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...
Actually, I was wondering whether the dual-numbers 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.
[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...