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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.

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras 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.

Concretely, a Lie algebra $\mathfrak{g}$ over a field $\mathbf{k}$ is a $\mathbf{k}$-vector space equipped with an alternating bilinear multiplication $[{-}\,{-}]\colon \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}$ called the Lie bracket that satisfies the Jacobi identity:

$$\big[x\,[y\,z]\big] + \big[z\,[x\,y]\big] + \big[y\,[z\,x]\big] = 0$$

Examples

  • $\mathbb{R}^3$ endowed with the cross product forms a Lie algebra.

  • For any any associative algebra $A$ with multiplication $\cdot$, you can define a Lie bracket on $A$ as a literal commutator between two elements, $[v\,w]= v\cdot w-w\cdot v\,,$ making $A$ into a Lie algebra.

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Is Lie bracket always $[x, y]=xy-yx$?

I’m reading Stillwell’s Naive Lie Theory, where he defined Lie bracket on page 80 as $$[x, y]=xy-yx$$ However, on page 82 he said In general , a Lie algebra is a vector space with a bilinear operation $[,]$ , satisfying $$[x,y]=-[y,x]$$ $$[x,[y,z]]…
athos
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totally antisymmetric structure constants and metric of a Lie Algebra

Given a (semisimple) Lie Algebra, and choosing a base $\{T^a\}$, the structure constants $f^{ab}_c$ are defined by $ [ T^a, T^b ] = f^{ab}_cT^c $ The metric tensor is matrix obtained from the Killing form between two elements of the…
francesco
  • 133
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Do a complex Lie algebra and its realification have the same radical?

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. The radical of a Lie algebra $\mathfrak{g}$, denoted as $\mathrm{rad}(\mathfrak{g})$, is the unique maximal solvable ideal of $\mathfrak{g}$. Let $\mathfrak{g}$ be a finite-dimensional complex…
Jianing Song
  • 1,707
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Lie algebras containing $ so_3(\mathbb{R}) $

Every subalgebra of a solvable Lie algebra is solvable. So a Lie algebra $ \mathfrak{g} $ containing a subalgebra isomorphic to $ \mathfrak{so}_3(\mathbb{R}) $ cannot be solvable. Is the converse true? In other words, is it true that a Lie algebra $…
Ian Gershon Teixeira
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ad-hoc basis in Frobenius Lie algebras

Let $\mathfrak{g}$ be a Frobenius Lie algebra, that is, there exist $f \in \mathfrak{g}^{*}$ such that the bilinear form defined by $b(x,y)=f([x,y])$ is non-degenerate. My question is : Why can we assure that there exist some basis $\lbrace e_1,…
ferolimen
  • 618
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Ideals in Frobenius Lie Algebras

Let $\mathfrak{g}$ be a Frobenius Lie algebra, that is, there exist $f \in \mathfrak{g}^{*}$ such that the bilinear form defined by $b(x,y)=f([x,y])$ is non-degenerate. Since $b$ is non-degenerate there exist a unique $x_p \in \mathfrak{g}$ called…
ferolimen
  • 618
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Producing non-abelian Lie algebra from abelian one

Let $\mathfrak{g}$ be a finite-dimensional abelian Lie algebra over a field $k$ of characteristic zero. I was wondering how many constructions do we have to produce a non-abelian Lie algebra $\overline{\mathfrak{g}}$ out of $\mathfrak{g}$ i.e. using…
eightc
  • 686
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Induced Lie algebra homomorphism not clear

Let $f: M \longrightarrow M'$ be a diffeomorphism between two smooth manifolds. We denote by $Vect(M)$ the space of smooth vector fields (ie. derivations) on $M$. Then, the map $f_*: Vect(M) \longrightarrow Vect(M')$ defined by…
Spida
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A question on a four dimensional nilpotent Lie Algebra

Let $L$ be a four-dimensional nilpotent Lie Algebra. Suppose $\text{dim}(Z(L))=1$ where $Z(L)$ is the center of the Lie Algebra. There is a claim that $\text{dim}([L,L])<3$, which I don't understand. It is clear that $\text{dim}([L,L])\leq 3$ since…
Riju
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Lie bracket of ideals is ideal

I am asked to prove that given two ideals $I,J\subset \mathfrak{g}$ in a Lie algebra, then the set $$ [I,J]=\{[x,y]:x\in I,y\in J\} $$ is another ideal of $\mathfrak{g}$. It is somewhat easy to see that due to the Jacobi identity, for any $a\in…
topolosaurus
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Root system of a semisimple Lie algebra spans the dual.

Let $L$ be a semisimple Lie algebra and $H$ a Cartan subalgebra. The roots of $L$ with respect to $H$ are the elements of $$ \Phi = \{\alpha \in H^*\setminus 0 : L_\alpha \neq 0\} $$ where $L_\alpha = \{x \in L : [h,x] = \alpha(h)x \text{ for all }…
Fibonacci Cube K
  • 712
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Understanding $G_2$ as a particular subgroup of $SO(7)$

The Wikipedia article on the exceptional Lie group $G_2$ has the following definition of the group: $$G_2=\{g\in SO(7):g^*\varphi=\varphi, \varphi = \omega^{123} + \omega^{145} + \omega^{167} + \omega^{246} - \omega^{257} - \omega^{347} -…
G. Smith
  • 597
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How to compute the adjoint action?

I have some matrix lie algebra $\mathfrak{g}$ and I want to compute the matrix of the adjoint action of $x\in\mathfrak{g}$, $\mathrm{ad}_x:\mathfrak{g}\rightarrow\mathfrak{g}$ given by $y\mapsto[x,y]$. How can I proceed? I have been told that it…
Daniel Robert-Nicoud
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Which of $\{J_0,J_1,J_2\}$ and $\{J_0,J_+,J_-\}$ is a generator for the Lie algebra $\mathfrak{su}(2)$?

I have just started learning about Lie algebra in the context of quantum mechanics and got confused with this: Some sources say the generators are $J_0,J_1$ and $J_2$ and some use $J_0,J_+$ and $J_−$. Which set is correct? Or if both are correct…
Saurabh Shringarpure
  • 797
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Inner automorphisms of a lie algebra

I am having trouble under standing why something of the form $\exp(ad x)$ where $ad x$ is nilpotent is a an automorphism of a lie algebra. As far as I understand the exponential map is mapping from the Lie Algebra to a Lie group. But my…
samlanader
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