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Results tagged with lie-algebras
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user 5740
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.
5
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Lie algebra with cyclic structure constants
Endow your algebra with the symmetric bilinear form making $\{e_i\}$ an orthornormal basis (the Gram matrix is the identity matrix). Since $c_{ij}^k=([e_i,e_j],e_k)$, the cyclicity condition is equiva …
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12
votes
Accepted
Is every G-invariant function on a Lie algebra a trace?
The answer to the general question is "no":
If $\mathfrak{g}$ is solvable, by Lie's theorem its commutant $\mathfrak{g}^{\prime}=[\mathfrak{g},\mathfrak{g}]$ is represented by strictly upper triang …
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2
votes
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Adjoint orbits of small subspaces in Lie algebras
I think that answers to your questions can be obtained along the following lines.
Fix the dimension of $V$ to be $k\leq 1/2\dim V$ and let $n=\dim\mathfrak{g}.$ Condiser the real algebraic Grassmania …
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4
votes
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Nilpotent matrices related to Lie algebras of special orthogonal groups in characteristic 0
Let $\lambda$ be a partition of $n.$ Then there exists a skew-symmetric nilpotent matrix whose Jordan blocks sizes are $\lambda_i$ if and only if every even parts has even multiplicity. This follows e …
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2
votes
Is there an analog of Clifford Theorem in the setting of Lie algebras?
If the module $M$ is finite-dimensional then the answer seems to be affirmative.
Let $Soc_I(M)$ be the maximal semisimple $I$-submodule of $M,$ which is non-zero by the finite-dimensio-nality of $M. …
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4
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On the full reducibility of representations of reductive Lie algebras
To complement Jim's answer, there is a thorough discussion of complete reducibility for reductive Lie algebras (with proofs, but only in char=0) in Sections 1.6 and 1.7 of Dixmier's "Enveloping algebr …
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6
votes
Applications of Chevalley Restriction Theorem
Note that both algebras are polynomial rings, i.e. free commutative algebras. Thus knowing that one of the two sides of the isomorphism is a polynomial ring implies that the other is, too. For $G$ cla …
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12
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Regular nilpotent element in complex simple Lie algebra
Regular elements need not be semisimple! For example, in the Lie algebra $\frak{sl}_2$, every non-zero element is regular, with the centralizer spanned by the element itself. Among the elements of the …
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2
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A question on involutions on the Lie algebra of vector fields
Extending Ian's comment, any involution on $M$ induces an involution on the Lie algebra of the vector fields on $M.$ There should be plenty of those.
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4
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Confusion in some notations in Lie sub-algebras of exceptional Lie algebra
${\bf Q1}$ From the general principles [the following claim is false, as pointed out by Dave Witte Morris in the comments], the subsystem $R_{\alpha}^{\perp}=\{\gamma\in R: (\gamma, \alpha)=0\}$ of a …
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3
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Relation between Lie Algebra Cohomology and Number of Relations of a Cyclic Module?
This is FALSE as stated, even for the augmentation ideal. For example, consider the 3-dimensional Heisenberg Lie algebra g and the trivial module k. Then it's easy to see that the third (=top) cohomol …
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Cyclic vectors in irreducible representations of simple Lie algebras
Summary The answer to the first question is affirmative and to the second question is negative, but for rather mundane reasons. In the simple Lie algebra case, cyclicity of ${\rm ad}\, a$ for some …
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4
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Infinite dimensional unitary representations of SU(2) for non-half-integer j?
You have to distinguish between representations of a compact group $SU(2)$ and of its Lie algebra $su(2)$, which can be complexified to $sl(2,\mathbb{C}).$ By Peter-Weyl theorem, every irreducible rep …
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Weight spaces of representations of finite dimensional simple Lie algebras
The requisite property follows from the following key proposition:
$U_{\lambda}$ is a finitely generated right $U_0$-module.
Notation The subscripts denote the grading of the universal envelopi …
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12
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How to decompose a composition of representations?
As David remarked, this is nearly a hopeless task in general, but some special cases can be computed explicitly.
If $\mathfrak{g}$ is a member of a skew reductive dual pair $(\mathfrak{g},\mathfrak{ …
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