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Results tagged with lie-algebras
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user 3545
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.
7
votes
Accepted
relate parabolic subalgebras to gradings?
Recall that when $\mathfrak{g}$ is a Lie algebra over a field $k$, a derivation of $\mathfrak{g}$ is a $k$-linear map $D: \mathfrak{g} \rightarrow \mathfrak{g}$ such that
$$D( [X,Y] ) = [DX, Y] + [X, …
- 13.3k
13
votes
Figure out the roots from the Dynkin diagram
This question is answered, probably in many textbooks on Lie algebras, Chevalley groups, representation theory, etc.. I always think that Bourbaki's treatment of Lie groups and algebras is a great pl …
- 13.3k
5
votes
Classification of real forms up to inner automorphisms
I'd recommend looking at the article "Strong real forms and the Kac classification", by Jeffrey Adams -- it's an expository paper from 2005 which answers your question.
For more details, Adams is car …
- 13.3k
5
votes
The exceptional Lie algebra $\mathfrak{g}_2$ and binary cubics
Well, I don't know about "deep", but here is another perspective. My point of view is much different than the classical invariant point of view -- I use the word "syzygy" about as often as I use the …
- 13.3k
6
votes
About the definition of E8, and Rosenfeld's "Geometry of Lie groups"
Here's one of my favorite definitions, that gives a construction of (many forms) of $E_8$ as well as a plethora of other exceptional groups. I hope it's ok to construct the Lie algebras here, and let …
- 13.3k
12
votes
Accepted
Type of 26-dimensional representation of different real forms of the complex simple Lie alge...
I think the best way to see the signature of these quadratic forms is by using the formula from "A Classification Theorem for Albert Algebras" by R. Parimala, R. Sridharan, and Maneesh L. Thakur, Tran …
- 13.3k
23
votes
Matrix representation for $F_4$
As I understand the question, the OP would be happy to see a description of the lowest-dimension fundamental representation of $F_4$ (and perhaps $E_6$, $E_7$, $E_8$), and is happy with the descriptio …
- 13.3k
5
votes
Uniform setting for computing orders of algebraic groups over finite quotients of the integers?
The places to look are:
Steinberg, "Endomorphisms of linear algebraic groups." Memoir AMS 80, (1968), and
Gross, "The motive of a reductive group" Invent. math. 130, 287 ± 313 (1997).
(I learne …
- 13.3k
11
votes
Definition of $\textrm{GSpin}_{2n}$ and its root datum
A good concise reference is Deligne's article on the Weil conjectures for K3 surfaces. See "La conjecture de Weil pour les surfaces K3" in Inventiones 15 (1971/72): 206-226. An English version is ea …
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