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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.

6730 questions
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How to prove that a subalgebra $\mathfrak p$ in a semisimple lie algebra $\mathfrak g$ is parabolic if $\mathfrak p^\perp=\mathfrak{rn}(\mathfrak p)$?

Let $\mathfrak g$ be a semissimple Lie algebra over an algebraically closed field of characteristic zero. Suppose $\mathfrak p$ is a subalgebra of $\mathfrak g$ such that $\mathfrak p^\perp=\mathfrak{rn}(\mathfrak p)$, where $\mathfrak p^\perp$…
Júlio César
  • 672
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Existence of product $\ast$ in a Lie algebra so that $[X,Y]=X*Y-Y*X$

I've been studying particle physics, and studying Lie algebra using physics text book doesn't give me enough information, so I'm asking my question here. Given a Lie algeba $\mathcal{A}$ where product is $[\,\cdot\, , \,\cdot\,]$, is there always a…
Henry
  • 3,125
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The Weyl Group of $F_4$

The Weyl Group of $F_4$ is of order $1152=2^{7} \cdot 3^{2}$. By Burnside's theorem the group is solvable. Is there a way to see solvability from the root system? Is it possible to see the order of the group there?
Destin42
  • 51
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Invariant subalgebra of a Lie algebra under an automorphism of the Dynkin diagram

Of all the automorphisms of a (finite-dimensional, semisimple) Lie algebra which induce a particular automorphism of its Dynkin diagram, is there a particular one which is "nicer" than the others? The question came about while solving the…
Ted
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Lie algebra associated to a linear form

Let $F$ be a finite dimensional vector space over a field $k$, if $f : F \to k$ is any linear form, I can define on $F$ a Lie algebra bracket by the following rule $$ [x,y]=f(x)y-f(y)x, $$ or in terms of structure constants $$ c_{ij}^k=f_i…
Alex
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On the root systems

Let $\Phi$ be a root system of $E$. $\alpha,\beta\in \Phi$. Let $\lbrace \beta+i\alpha | i\in \mathbb{Z}\rbrace\cap \Phi$, $\alpha$-string through $\beta$, be $\beta-r\alpha,\ldots,\beta+q\alpha$, and let $\lbrace \alpha+i\beta | i\in…
Peter Greene
  • 151
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Jacobi identity in associative algebra as Lie algebra

In page 2 of Hans Samelson's Notes on Lie Algebras, the text gives an example of Lie algebra $A_L$. Let $A$ be an algebra over $\mathbb{F}$ (a vector space with an associative multiplication $X\cdot Y$). We make $A$ into a Lie algebra $A_L$ (also…
PJ Miller
  • 8,193
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Cartan Subalgebra and regular elements

Let $L$ be semisimple Lie algebra, $x\in L$ semisimple. Prove that if $x$ lies in exactly one Cartan subalgebra, then $x$ is regular.
Cristyan Pinheiro
  • 145
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Exercise 12 of Chapter 7 in Brian Hall's Lie groups, Lie algebras and their representation.

I am stuck on the following exercise in Hall's book: Let $\mathfrak{g}$ be a complex simple Lie algebra with complex structure denoted by J. Let $\mathfrak{g}_\mathbb{R}$ denote the Lie algebra $\mathfrak{g}$ viewed as a real Lie algebra with twice…
richardfatman
  • 143
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Solvable Lie algebra with codimension 1 ideal

There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal". The proof I am thinking of is the following. Suppose the Lie algebra $L$ satisfies…
Earthliŋ
  • 2,490
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adjoint representation completely reducible

Let $\mathcal{A}$ be a Lie algebra. Suppose that adjoint representation of $\mathcal{A}$ is completely reducible (or semisimple). Show that $\mathcal{A}$ can be written as a direct sum of semisimple Lie algebra and abelian Lie algebra. I know an…
Vasco
  • 134
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Adjoint representation of Sl(2,C)

What is the adjoint representation of Sl(2,C). Can anyone help explain this please?
ConnectNumbers66
  • 105
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What are the Weyl group of type $E_8$, $F_4$,$G_2$?

This problem is as titled. The textbook states that the order of the Weyl group of type $E_8$, $F_4$ are $2^{14}3^55^27$ and 1152 respectively, but I am wondering how are these groups like, namely, how can they be decomposed into simpler groups, or…
ShinyaSakai
  • 7,846
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Adjoint representation of $\mathfrak{so}(n)$

So I know $\mathfrak{so}(n)$ consists of anti-symmetric matrices, but I don't see why the adjoint representation is isomorphic to the anti-symmetric tensor square of the vector representation. I see that we can associate each matrix with an…
thegamer
  • 797
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Center of a lie Algebra is an ideal

Let g be a lie algebra, show that the center of g is an ideal in g. attempt at proof: Let $\ \mathfrak g$ be a lie algebra. Define $\mathfrak h$ to be the center of $\ \mathfrak g$. Suppose that $\kappa $ is a sub-algebra of the lie algebra $\…
Alexander King
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