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Results tagged with lie-algebras
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user 4149
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
votes
Simple lie algebras, (almost-)simple groups of Lie type
See Bourbaki, Groupes et algèbres de Lie, IV.2.7, Corollary of Theorem 5 (due to Tits).
We assume that $K$ has at least four elements and that the Lie algebra is absolutely simple. We assume also th …
- 14.2k
1
vote
Accepted
Real Adjoint representations of complex type
Irreducible real representations of complex type of a compact group correspond to irreducible complex representations that do not admit an invariant bilinear form. Irreducible real representations of …
- 14.2k
4
votes
Homomorphism from noncompact semisimple Lie group to compact Lie group
See below a detailed version of the comment of @LSpice. (Edited taking into account a comment of @YCor.) This is an answer to the question on homomorpisms of real algebraic groups.
Proposition.
Let $ …
- 14.2k
11
votes
Does $SU(N)$ have pseudo-real representation?
Let $G$ be a compact (anisotropic) real algebraic group.
Let $\rho\colon G\to {\rm GL}(n,{\mathbb{C}})$ be a complex linear (polynomial) representation of $G$.
Following OP, we say that $\rho$ is pse …
- 14.2k
2
votes
Do rational points in a split reductive group act transitively on the orbits of the Cartan s...
When $R$ is a field, the answer to Question 1 is Yes (at least in char 0) with the same proof as for Question 2.
For simplicity, we write $G$ for $G_R$, $T$ for $T_R$, etc.
Theorem.
Let $G$ be a …
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5
votes
1
answer
189
views
Matrix from a homomorphism of simply connected groups
Let $H$ be a simple algebraic group of type $\mathbf{G}_2$ over $\mathbb{C}$.
Let $\rho$ be the standard 7-dimensional complex representation
$$ \rho\colon H=\mathbf{G}_2\to \mathrm{SO}_7.$$
We consid …
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2
votes
Reducible reductive Lie subalgebras of so(p,q)
I answer the second question.
First I classify the connected $\mathbb{R}$-subgroups of full (absolute) rank of the compact $\mathbb{R}$-group $SO(p,\mathbb{R})$
that are $\mathbb{R}$-irreducible in t …
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1
vote
Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
See V.L. Popov, E.B. Vinberg, Invariant Theory,
Encyclopaedia of Mathematical Sciences, Vol. 55, Algebraic Geometry IV,
Springer-Verlag, Berlin, 1994.
You can find the invariants of a binary quartic …
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1
vote
Diagonalisation of invariant hermitian forms and irreducible representations of tori actions
We need some notation. I write $V$ for your $\Bbb C^{2nN}$, $T$ for your $T^n$,
and $\rho\colon T\to GL(V)$ for the representation of $T$ in $V$. I define the character $\chi_j$ of $T$ by
$$\chi_j( …
- 14.2k
2
votes
0
answers
105
views
Conjugacy classes of involutions in Kac-Moody groups
Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix.
Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$.
W …
- 14.2k
14
votes
Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
Let us write the Frobenius-Schur indicator of the representation $V$ with highest weight $\lambda$
as $\mathrm{FS}(\lambda)$.
Let $R$ denote the root system, and let $\Pi=\{\alpha_1,\dots,\alpha_l\}$ …
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7
votes
Accepted
Root system of fixed point Lie sub-algebra
Let ${\frak g}$ be a simple Lie algebra over $\Bbb C$, and let $\theta$ be an inner involution of ${\frak g}$,
that is, an inner automorphism of ${\frak g}$ of order dividing 2.
Such automorphisms are …
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2
votes
0
answers
81
views
Explicit $K$-basis of a Lie subalgebra
$\newcommand{\Kbar}{{\overline K}}
\newcommand{\Q}{{\mathbb Q}}
$I consider Example 5.7 of Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres of …
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3
votes
Accepted
When is the image of the adjoint representation of a real algebraic group Zariski closed?
Let $G$ be a connected linear algebraic group over the field of real numbers $\mathbb{R}$.
By "connected" I mean "connected over $\mathbb{C}$".
Let $G(\mathbb{R})$ denote the group of $\mathbb{R}$-poi …
- 14.2k
4
votes
1
answer
421
views
Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie alg...
Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0.
We consider the adjoint representation
$$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$ …
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