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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.
3
votes
Accepted
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 …
- 14.5k
5
votes
Accepted
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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8
votes
Non-faithful irreducible representations of simple Lie groups
By the highest weight theory, finite-dimensional irreducible representations of a finite-dimensional simple Lie algebra $\mathfrak{g}$ are parametrized by the dominant weights $\lambda$ in the weight …
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6
votes
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Simple modules for direct sum of simple Lie algebras
This is true more generally for any two finite-dimensional Lie algebras over a field. Recall that a $\frak{g}$-module is the same as a module over $U (\frak{g})$, the universal enveloping algebra of $ …
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12
votes
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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4
votes
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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5
votes
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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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6
votes
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What is known about the morphism $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$ induced by $L\hookright...
The symmetrization mapping $\sigma$ from the symmetric algebra $S(L)$ to the universal enveloping algebra $U(L)$ is an isomorphism of $L$-modules. Since $L$ is a direct summand of $S(L)$, its isomorph …
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2
votes
Accepted
Centralizer of hermitian matrices with zero trace
As you have pointed out, one can work in the Lie algebra ${\frak{su}}(N)$. You want to classify the centralizers of various subspaces of this Lie algebra, and without loss of generality you can assume …
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2
votes
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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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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2
votes
Accepted
Copies of ax+b inside the AN part of an Iwasawa decomposition?
Here is an elementary argument addressing Q1. The adjoint action of the subalgebra $\mathfrak{a}$ on $\mathfrak{g}$ has the following two key properties:
It is diagonalizable (the operators $ad(H)$ …
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6
votes
Structure of $[S(\mathfrak{g})\otimes S(\mathfrak{g})]^G$ for semisimple $\mathfrak{g}$
When $G=GL(n),$ this is the "invariants of matrices", as Alexander Chervov has pointed out. The full description by generators and relations is only known for small $n.$ So it's not "textbook material …
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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
votes
Accepted
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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