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Results tagged with lie-algebras
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user 24386
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.
4
votes
Accepted
Where does the algebraic closure enter into Block's Theorem?
Block's theorem does not require the base field $k$ to be algebraically closed but one has to be careful when $k$ is imperfect. Then $k$ will admit field extensions of the form $K=k(a)$ with $a\not\in …
- 2,624
1
vote
Irreducible quotient of $U\otimes V$
Your notation suggests that that $u^-$ is a lowest weight vector, so I will asssume that this is the case. Then $u^-\otimes v^+$ generates $U\otimes V$. Indeed, let $W$ be the $\mathfrak{sl}_n$-submod …
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4
votes
Accepted
Good even grading and principal Levi type
If $e$ is principal in a proper Levi subalgebra whose Dynkin diagram involves a component of type
$A_k$ with $k$ odd, then e is not even. This is very easy to see by writing down an explicit $sl_2$-tr …
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13
votes
Accepted
Are S(g) and U(g) isomorphic as g-modules for g Lie algebra over F_p ? Are S(g)^g and U(g)^g...
Question 2 has a negative answer. Indeed, let $L =sl(2,k)$ where $k$ is an algebraically closed field of characteristic $p>3$ and let {$e,h,f$} be the standard basis of $L$. Then it is well-known (and …
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7
votes
Can one show the equivalence of the abstract and classical Jordan decompositions for simple ...
This is how I do this in my third year course on Lie algebras: Since we may assume that the Killing form $\kappa$ of $\mathfrak g$ is is non-degenerate, we can make use of the direct sum decomposition …
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9
votes
Accepted
Commutativity and Kostant sections
In some cases the answer to the weaker version of the question (involving the semisimple
part of $X_2$) is YES. This will happen if $C_g(e)$ is self-dual which is the case, for instance, when $g=gl_N$ …
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10
votes
For $\mathfrak g$ A Lie algebra of type $ E_7 $, $\mathfrak h $ a Cartan subalgebra and $\De...
There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does no …
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16
votes
Accepted
Simple Lie algebras and Jordan decomposition
Any finite dimensional simple Lie algebra over an algebraically closed field of characteristic $p>3$
contains a nonzero element $x$ such that $ad(x)$ is semisimple. This is a nontrivial fact, and the …
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10
votes
Semisimplicity of Lie algebra in positive characteristic
If $g\subset M_n(F)$ and $n\le p-2$ then $g$ is semisimple if and only if the Killing form of g is non-degenerate. This statement is clear (and empty) if $p\in\{2,3\}$, while for $p>3$ one can use var …
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5
votes
A strong relationship between $\mathrm{ad}(X)$ and $1-\mathrm{Ad}_g$ when $\mathrm{Ad}_gX=X$
Your conjecture that ${\rm im} (1-{\rm Ad}_g)\subseteq {\rm im}( {\rm ad}\,X)$ is indeed true. Let me sketch the argument.
First we reduce to the case where the regular element $X\in \mathfrak{g}$ i …
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12
votes
finite dimensional irreducible representation of finite dimensional nilpotent Lie algebra
This is VERY far from being true: consider the $3$-dimensional Heisenberg Lie algebra
$L$ with basis $a$, $b$, $c$ and the only nonzero bracket $[a,b]=c$ (so $c$ is central in $L$). Consider the linea …
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2
votes
Cartan involution for finite W-algebras
In the affine case, there is a related discussion in a paper by Kac-Wakimoto;
see pp. 23-25 in arXiv:math-ph/0304011v2.
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2
votes
Borel subgroups of centralisers of Lie algebra elements in bad characteristic
I think the positive answer to this question follows from some results obtained in the paper The Hesselink stratification of nullcones and base change, Invent. Math., 191 (2013), 631-669, by M. Clarke …
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8
votes
Accepted
Jacobson-Morozov theorem
To each unipotent element $u\in G$ one assigns its weighted Dynkin diagram which is basically
a map $\Delta\colon\, \Pi\rightarrow \{0,1,2\}$ where $\Pi$ is a basis of simple roots of the root syste …
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3
votes
Characterisation of parabolic subalgebras: reference sought
If $\mathfrak{p}^\perp$ is a nipotent subalgebra then it must consist of nilpotent elements. Indeed, let $h\in \mathfrak{p}^\perp$ and suppose ${\rm ad}\,h$ is not nilpotent.
Let $\mathfrak{g}^0(h)$ b …
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