# Questions tagged [lie-algebras]

For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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• 101
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### Complexification of complex Lie algebras like $\mathfrak{su}(2)$

I'm reading Brian Hall's book on Lie theory. He defines the complexification $V_{\mathbb{C}}$ of a real vector space $V$ as the linear combinations $v_1+iv_2$, with $v_1,v_2\in V$. Next, he proceeds ...
1 vote
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### Velocity vector vs Matrix Differential

I am having trouble understanding the equivalence of taking the derivative of the matrix and taking the velocity vector. I came across a proof of the Lie Algebra of $O(n)$ as follows: Let $\gamma(t)$ ...
1 vote
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### uniqueness of generators of Lie groups

Are the generators of any particular Lie group always unique? Let's take $SU(3)$ group as an example. It does have 8 generators which are explicitly written as eight $3\times3$ matrices in the ...
• 111
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### Real Commutant Algebra of a Set of Matrices

Suppose I have a collection of $N\times N$ real, symmetric matrices $R_1, R_2, \dots$ and I want to find their orthogonal commutant---that is, the group of real, orthogonal matrices that commute with ...
1 vote
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### Simple Lie subalgebra of a semisimple Lie algebra

For a semisimple Lie algebra with a nondegenrate trace form, it's well-known that it can be decomposed as the direct sum of simple Lie ideals, and hence has a simple Lie ideal. But in general, how to ...
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### Invariant Solutions of PDEs-Linear fokker-planck equation with an odd drift

I am currently writing my master thesis on Lie group analysis and recently I came across this infinitesimal generator: I am trying to obtain group invariant solutions in their implicit form and so ...
1 vote
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### The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.

SETUP. It is a standard result that $\text{GL}(n,\Bbb{R})/O(n)$ is isomorphic to the set $P'$ of positive definite $n\times n$ matrices, as manifolds: the basic idea is that $\text{GL}(n,\Bbb{R})$ ...
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### product of structure constants [closed]

I'm not that familiar with Lie Algebra, however for a basis $F_{i} \in \mathfrak{su}(N)$ such that $$[F_{i}, F_{j} ] = \sum_{k}f_{ijk}F_{k}$$ I have seen it written that $f_{ikl}f_{jkl} = \delta_{ij}$....
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### Grothendieck ring of $Rep(\mathfrak{sl}_2)$

The Grothendieck ring of the abelian category $Rep(\mathfrak{sl}_2)$ of finite-dimensional representations of $\mathfrak{sl}_2$ is, according to Bakalov-Kirillov's Lecture notes on tensor categories ...
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1 vote
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### Can't "finite" be removed from the definition of a root system? [duplicate]

In every definition of finite root systems I could find (reduced or not-necessarily-reduced), the root system is always assumed to be finite in the definition. It's a very minor point, but isn't this ...
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### Nilpotent Lie-Algebra $g$: $g^{i+1} ⊆ g^i$ ideal in $g$?

Assume $g$ to be a nilpotent Lie-Algebra. Nilpotency means that we can find an index $n$ such that: $g^n = \{0\}$ for the series defined as: $g^0 = g$ $g^{i+1} = \operatorname{span}\{[g,g^i]\}$ Why is ...
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• 821
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### Finding inverse to inner automorphism (Humphreys' Lie Algebra book)

This is basically an algebra question. In Humphreys' book on Lie algebras he states that one can find the inverse to $\exp \delta$, where $\delta$ is a nilpotent derivation - say $\delta^k=0$, by ...
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### Why $\omega|_S\in\Omega^1(S,\mathfrak{g})$ iff $\omega(\nabla,e)\in\Omega^1(U,\mathfrak{g})$?

Let $M$ be a smooth manifold, let $\mathcal{S}$ be a $G$-strucutre on $M$ and let $\nabla$ be a connection on $TM$. Let $\omega\in\Omega^1(\text{Fr}(TM),\mathfrak{gl}_r)$ be the connection 1-form ...
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### Does isometry on PSD matrices preserve eigenvalues?

Let $S$ be the set of symmetric matrices and $T: S\rightarrow S$ be a linear isometry. Moreover, let $T$ be a bijection from the space of PSD matrices to the set of PSD matrices. Must $T$ preserve ...
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### Lie algebra question about the highest weight of a root system

I'm reading the book James E. Humpreys's book Introduction to Lie algebras and Representation Theory . In this book the section 13.4 said in case $\Phi$ is irreducible, there is a unique highest root (...
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• 1,711
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### Relation between linear representation and "induced adjoint representation" of Lie algebra?

Consider a representation $\rho \colon \mathfrak{g} \mapsto \mathrm{End}(V)$ of a Lie algebra $\mathfrak{g}$ on a vector space $V$. What can we say about the induced representation on the space of ...
• 303
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### Vector field of rotation around an axis in spherical coordinates

This is a qualifying exam question in Differential Geometry. I'm new to the subject and am reading up from John Lee's Introduction to Smooth Manifolds. Let $M=S^2\subset\mathbb{R}^3$ be the unit ...
1 vote
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### Binomial theorem for ideals

I was proving the statement that if $I$ and $J$ solvable ideals of Lie algebra $L$, then $I + J$ is a solvable ideal of $L$. The proof is we know $$(I+J)/J\cong I/I\cap J.$$ Since $I,J$ are solvable ...
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1 vote
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### Is my proof valid for Exercise 7.3 of book "Introduction to Lie Algebras" by Karin Erdmann and Mark J. Wildon

Exercise 7.3 of book "Introduction to Lie Algebras" by Karin and Mark: Show that $V$ is irreducible if and only if for any non-zero $v \in V$ the submodule generated by $v$ contains all ...
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