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Results tagged with lie-algebras
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user 113258
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.
2
votes
1
answer
176
views
Prove: Lie algebra generated by two $n\times n$ shift matrices is $\mathfrak{so}(n,\mathbb{C...
I wish to have a proof for the following result:
Let $U_n$ be an $n\times n$ upper shift matrix, and $L_n = U_n^T$ be a lower shift matrix. For example,
$$
U_5 = \begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
0 …
- 593
13
votes
1
answer
494
views
What's the dimension of the Lie algebra generated by transpositions on $n$ objects?
Define a Lie bracket on the group algebra of the permutation group $S_n$ in the following way:
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
where $\sigma, \tau \in S_n$, and the multiplicat …
- 593
7
votes
0
answers
124
views
Softwares to determine semi-simple types of Lie algebras generated over $\mathbb{R}$ or $\ma...
I wish to determine the type of a Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. For example,
\begin{align}
n^+ =
\begin{pmatrix}
…
- 593
10
votes
1
answer
338
views
What is the Lie superalgebra generated by permutations?
Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
wher …
- 593