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Results tagged with lie-algebras
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user 9417
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
6
votes
Ternary "Lie structure"
One way to define a Lie structure on a vector space $V$, is as a map $\wedge^2V\to V$ such that its natural extension to $d\colon\wedge^k V\to \wedge ^{k-1}V$ satisfies $d^2=0$. This exactly gives the …
- 4,898
2
votes
Accepted
Quasi-Lie algebras in nature?
Tom Goodwillie:
The homotopy groups of a (say, simply connected) space $X$ form a graded Lie algebra under Whitehead product, in which the even-dimensional part (which is actually the $\pi_n$ for …
5
votes
Nice proofs of the Poincaré–Birkhoff–Witt theorem
I'm partial to Dylan Thurston's proof in his Ph.D thesis. He proves the Duflo isomorphism (which is stronger than PBW) in a graphical/knot-theoretic context. The paper is a real pleasure to read.
- 4,898
8
votes
2
answers
1k
views
Quasi-Lie algebras in nature?
A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand …
- 4,898
0
votes
0
answers
286
views
Good and/or standard notation for the abelianization of a Lie algebra
I'd like to solicit good notations for the abelianization of a Lie algebra $\mathfrak g$. One could write $\mathfrak g/[\mathfrak g,\mathfrak g]$, or even $H_1(\mathfrak g)$ but I'd like something tha …
- 4,898