# What is Lie algebras: Definition and 86 Discussions

In mathematics, a Lie algebra (pronounced "Lee") is a vector space

g

{\displaystyle {\mathfrak {g}}}
together with an operation called the Lie bracket, an alternating bilinear map

g

×

g

g

,

(
x
,
y
)

[
x
,
y
]

{\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}},\ (x,y)\mapsto [x,y]}
, that satisfies the Jacobi identity. The vector space

g

{\displaystyle {\mathfrak {g}}}
together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras.
In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
An elementary example is the space of three dimensional vectors

g

=

R

3

{\displaystyle {\mathfrak {g}}=\mathbb {R} ^{3}}
with the bracket operation defined by the cross product

[
x
,
y
]
=
x
×
y
.

{\displaystyle [x,y]=x\times y.}
This is skew-symmetric since

x
×
y
=

y
×
x

{\displaystyle x\times y=-y\times x}
, and instead of associativity it satisfies the Jacobi identity:

x
×
(
y
×
z
)

=

(
x
×
y
)
×
z

+

y
×
(
x
×
z
)
.

{\displaystyle x\times (y\times z)\ =\ (x\times y)\times z\ +\ y\times (x\times z).}
This is the Lie algebra of the Lie group of rotations of space, and each vector

v

R

3

{\displaystyle v\in \mathbb {R} ^{3}}
may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property

[
x
,
x
]
=
x
×
x
=
0

{\displaystyle [x,x]=x\times x=0}
.

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1. ### I About Classical Lie algebra

Dear @fresh_42 , Hope you are well. Please, I have a question if you do not mind, about Lie Algebra, In page 2 in the book of Lie algebra, written by Humphreys, Classical Lie algebras, ##A, B, C## and ##D##, I did not get it well, especially, symplectic and orthogonal.. Could you please...
2. ### A Differential structure of the group of automorphism of a Lie group

I am working on this I am having trouble with b and c: b) Suppose ##(f_n)_{n=1}^{\infty}## is a sequence in ##Aut(G)##, such that ##(T_e(f_n))_{n=1}^{\infty} \to \psi## converges in ##Aut(\mathfrak g)## I want to show that ## f := \lim_{n\to \infty} f_n## exists as an continuous...
3. ### About representations of Lie groups

I am not quite sure of how this works, i.e. of what exactly I need to do with the hint. Any explanantion would be helpful!
4. ### About lie algebras, vector fields and derivations

-Verify that the space ##Vect(M)## of vector fields on a manifold ##M## is a Lie algebra with respect to the bracket. -More generally, verify that the set of derivations of any algebra ##A## is a Lie algebra with respect to the bracket defined as ##[δ_1,δ_2] = δ _1◦δ_2− δ_2◦δ_1##. In the first...

6. ### A About Schur's lemma in lie algebra

Please, I have a question about Schur's Lemma ; Let $\phi: L \rightarrow g I((V)$ be irreducible. Then the only endomorphisms of $V$ commuting with all $\phi(x)(x \in L)$ are the scalars. Could you explain it, and please, how we can apply this lemma on lie algebra ##L=\mathfrak{s l}(2)##thanks...
7. ### A About Proof of Engel's Theorem

in the Proof of Engel's Theorem. (3.3), p. 13: please, how we get this step: ##L / Z(L)## evidently consists of ad-nilpotent elements and has smaller dimension than ##L##. Using induction on ##\operatorname{dim} L##, we find that ##L / Z(L)## is nilpotent. Thanks in advance,
8. ### A Proving Cartan Subalgebra $\mathbb{K} H$ is Self-Normalizer

Please, How we can solve this: $$\mathfrak{h}=\mathbb{K} H \text { is a Cartan subalgebra of } \mathfrak{s l}_2 \text {. }$$ it is abelian, but how we can prove it is self-normalizer, please:Dear @fresh_42 , if you could help, :heart: 🥹
9. ### A How do we prove that a nonzero nilpotent Lie algebra has a nontrivial center?

Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.12, I have a question: Proposition. (3.2). Let ##L## be a Lie algebra. (c) If ##L## is nilpotent and nonzero, then ##Z(L) \neq 0##. how we prove this, Thanks in advance,
10. ### A Questions about solvable Lie algebras

Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.11, I have a question: Proposition. Let ##L## be a Lie algebra. (a) If ##L## is solvable, then so are all subalgebras and homomorphic images of ##L##. (b) If ##I## is a solvable ideal of ##L## such...
11. ### A What is the definition of quotient Lie algebra?

Please, in the definition of quotient Lie algebra If ##I## is an ideal of ##\mathfrak{g}##, then the vector space ##\mathfrak{g} / I## with the bracket defined by: $$[x+I, y+I]=[x, y]+I, for all x, y \in \mathfrak{g}$$, is a Lie algebra called the quotient Lie algebra of ##\mathfrak{g}## by...
12. ### I About derivations of lie algebra

Please, I am looking for a simple example of derivation on ##sl_2##, if possible, I try to use identity map, but not work with me, A derivation of the Lie algebra ##\mathfrak{g}## is a linear map ##\delta: \mathfrak{g} \rightarrow \mathfrak{g}## such that ##\delta([x, y])=[\delta(x), y]+[x...

25. ### A Characterizing the adjoint representation

Let U ∈ SU(N) and {ta} be the set of generators of su(N), a = 1, ..., N2 - 1. The action of the adjoint representation of U on some generator ta can be written as Ad(U)ta = Λ(U)abtb I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known...
26. ### A Diagonalization of adjoint representation of a Lie Group

So, we know that if g is a Lie algebra, we can take the cartan subalgebra h ⊂ g and diagonalize the adjoint representation of h, ad(h). This generates the Cartan-Weyl basis for g. Now, let G be the Lie group with Lie algebra g. Is there a way to diagonalize the adjoint representation Ad(T) of...
27. ### A From Lie algebras to Dynkin diagrams and back again

I am looking for a free online-resource sketching i) the way from Lie algebras to root systems and classification via Dynkin diagrams and ii) back to the Lie Algebra via reconstruction based on the information encoded in the Dynkin diagram. I would prefer a short PDF or web page, not a huge...
28. ### I Q about constricting simple lie algebras from simple roots

I'm reading "lie algebras and particle physics" by Georgi and on I'm up top where he is creating the simple algebras from simple roots and there is something I am not getting here. On page 108 he seems to be making the claim that any simple root φ had the property that any lowering operator of...
29. ### A Understanding the Connection Between Lie Algebras and Rotations

The Lie Algebra is equipped with a bracket notation, and this bracket produces skew symmetric matrices. I know that there exists Lie Groups, one of which is SO(3). And I know that by exponentiating a skew symmetric matrix, I obtain a rotation matrix. ----------------- First, can someone edit...
30. ### MHB Lie Algebras and Similarity Transformations

Say we have an adjoint representation (specifically in n x n matrix format) of a Lie algebra. I have noted that we can create another n x n representation using a similarity transformation on the adjoint rep. I know I haven't discovered anything new but none of my sources mention this. Is...
31. ### MHB How Can I Determine if Two Lie Algebras are Isomorphic?

I now know of two 3D Lie algebras: A_1 with brackets \left [ T^0, ~T^{\pm} \right ] = \pm 2 T^{\pm} , \left [ T^+, ~ T^- \right ] = T^0 and one with brackets: \left [ T^0, ~T^{\pm} \right ] = T^{\mp} and \left [ T^+,~ T^- \right ] = T^0 How can I tell if these two are representations...
32. ### MHB Question about the "E" Lie algebras

I'm in chapter 2 of my Lie algebra text. I have an interest in E_8 but I'm only going to work my way up to E_6 as it has fewer operators. Anyway for now I only have a quick question: Is E_6 defined by its Dynkin diagram or is E_6 defined by something else and the Dynkin diagram follows from...
33. ### Finding the members of the Lie algebra of SO (n)

Homework Statement Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression: R.RT= I with respect to the parameter t...
34. ### MHB SU(2) and elementary properites of Lie Algebras

I've been having fun with my new Lie Algebra text and it occurred to me that working out a couple of basic examples of my own would be a good idea. I got rather large surprise. The example I'm working with is SU(2) and I'm going through some basic properties it has. For all its uses in...
35. ### How to compute the norm of a root from the Cartan matrix?

Hello! As far as I understand, the Cartan matrix is associated with a unique semi simple algebra. How can we compute the norm of a root α from it since its components are invariant under rescaling (if all the simple roots are multiplied by the same constant, the Cartan matrix remains unchanged)...
36. ### MHB What is the Definition of a Derivation for a Lie Algebra?

Loosely speaking a derivation D is defined as a function on an algebra A that has the property D(ab) = (Da)b + a(Db). Now, if we define the map ad_x: y \mapsto [x,y] and apply this to the Jacobi identity we get ad_x[y,z] = [ ad_x(y),z ] + [ y, ad_x(z) ] . This does not look quite like the...
37. ### Explicit construction of isomorphisms of low-rank Lie algebras.

I've worked that out in my Mathematica notebook "Lie-Algebra Matrices.nb" in directory "Extra Mma notebooks" in SemisimpleLieAlgebras.zip, but I must concede that some of my derivations seem rather inelegant, not using features of the algebras very well. The last comment in How should I show...
38. ### Why do several Lie groups have the same Lie algebra when acting on a space?

Why is it that several lie groups can have the same Lie algebra? could it have to do with the space where they act transitively? Could two different Lie groups acting transitively on the same space have the same Lie algebra?
39. ### Interesting question regarding Lie algebras

Is there any finite dimensional Lie algebra that is not isomorphic to any of the subalgebras contained in GL(n) ?
40. ### Lie groups & Lie Algebras in Nuclear & Particle Physics

Hi, I'm a student of Nuclear Engineering (MS level) at University of Dhaka, Bangladesh. I completed my Honours and Master Degree with Mathematics. I have chosen to complete a thesis paper on "Application of Lie groups & Lie Algebras in Nuclear & Particle Physics." I need some guideline...
41. ### Geometry Lie Groups, Lie Algebras, and Representations by Hall

Author: Brian Hall Title: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction Amazon link https://www.amazon.com/dp/1441923136/?tag=pfamazon01-20 Level: Grad Table of Contents: General Theory Matrix Lie Groups Definition of a Matrix Lie Group Counterexamples...
42. ### Lie algebras contained in Universal Enveloping Algebra

I've got a general question about Lie algebras, which is basically this: Q: What is there to be said about the Lie algebras that can be identified in the universal enveloping algebra of a particular Lie algebra? E.g. if I have a Lie algebra of the form [A,B] = ηB, then I would like to...
43. ### Are Matching Commutation Relations Enough to Prove Lie Algebra Isomorphism?

Hi! I just need a "yes" or "no" answer. If i have to show that two lie algebras are isomorphic, is it sufficient to show that their generators fulfill the same commutation relations?
44. ### Regarding Lie algebras

Hello! I thought I had a good picture of this stuff but now I suspect that I've mixed up things completely.. For su(2) we have: f_{ab}\,^c = \epsilon_{ab}\,^c. Which means that, for the basis of our tangentspace we get: [\partial_a, \partial_b] = i \epsilon_{ab}\,^c\, \partial_c Where...
45. ### Riemannian symmetric spaces and Lie algebras

I'm interested in the crossover of Lie groups/differential geometry and I'm a bit confused about the relation of Lie algebras with symmetric spaces. Take for instance the Lie group G=SL(2,R), we take the quotient by K=SO(2) as isotropic group(maximal compact subgroup) and get the symmetric...
46. ### Understanding Lie Algebras: Structure Constants and Commutation Relations

the commutator in the Baker-Campbell-Hausdorff formula must be proportional to some linear combination of the generators of the group (because of closure) The constants of proportionality are called the Structure Constants of the group, and if they are completely known, the commutation relations...
47. ### Dimension formulas for Lie algebras

Hi. 1. Can anyone definitively tell me what the dimension formula for the classical Lie algebras? For example, I know for SO(2n) or D_n, the dimension formula is SO(N)--> (N*(N-1))/2 E.g. SO(8) is 8*7/2 = 28. Ok, so what about SU(N+1) i.e. A_n, SO(2n+1) i.e. B_N and Sp(n) i.e...
48. ### Lie Algebras: [H1, H2] = 0 => h1h2 = h2h1

Hi! I have the following problem: I tried to solve this by using the adjoint represantation and although it looks very promising, I can't really get it to work. If I define C_x(y) := xyx^{-1} as the conjugation, then what needs to be shown is: C_{h_1}h_2 = h_2. I'm using these symbols...
49. ### Length of roots in su(2) su(3) and other Lie algebras.

Hi all I found these equalities from Gordon Brown (1963). He uses the killing form to measure the length of the roots in a semi simple algebra. First and second equalities are quite obvious and come from the definition. Could you help me for the last one which prove that we have a...
50. ### Two approaches to lie algebras

Hello! I've read some lie algebra in both group theory books and differential geometry books, and is confused about the different perspectives. The group theory approach is usually that the author introduce the generators and show that these fulfil the algebra. In differential geometry our...