What is Lie algebra: Definition and 168 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
Consider ##\mathbb R^2## as the Euclidean plane. Since it is maximally symmetric it has a 3-parameter group of Killing vector fields (KVFs).
Pick orthogonal cartesian coordinates centered at point P. Then the 3 KVFs are given by: $$K_1=\partial_x, K_2=\partial_y, K_3=-y\partial_x + x...
Dear @fresh_42 , Hope you are well.
Please, I have a question if you do not mind, about Lie Algebra,
In page 2 in the book of Lie algebra, written by Humphreys,
Classical Lie algebras, ##A, B, C## and ##D##, I did not get it well, especially, symplectic and orthogonal..
Could you please...
I have some clarifications on the discussion of adjoint representation in Group Theory by A. Zee, specifically section IV.1 (beware of some minor typos like negative signs).
An antisymmetric tensor ##T^{ij}## with indices ##i,j = 1, \ldots,N## in the fundamental representation is...
Hi,
I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements
\begin{pmatrix}
e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\
ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}...
Please, I have a question about this:
The Universal enveloping algebra of a finite dimensional Lie algebra is Noetherian.
How we can prove it? Please..
Please, I have a question about Schur's Lemma ;
Let $\phi: L \rightarrow g I((V)$ be irreducible. Then the only endomorphisms of $V$ commuting with all $\phi(x)(x \in L)$ are the scalars.
Could you explain it, and please, how we can apply this lemma on lie algebra ##L=\mathfrak{s l}(2)##thanks...
Please, I have a question about universal enveloping algebra: Let ##U=U(\mathfrak{g})## be the quotient of the free associative algebra ##\mathcal{F}## with generators ##\left\{a_i: i \in I\right\}## by the ideal ##\mathcal{I}## generated by all elements of the form ##a_i a_j-a_j a_i-\sum_{k \in...
in the Proof of Engel's Theorem. (3.3), p. 13:
please, how we get this step:
##L / Z(L)## evidently consists of ad-nilpotent elements and has smaller dimension than ##L##.
Using induction on ##\operatorname{dim} L##, we find that ##L / Z(L)## is nilpotent.
Thanks in advance,
Please, How we can solve this:
$$
\mathfrak{h}=\mathbb{K} H \text { is a Cartan subalgebra of } \mathfrak{s l}_2 \text {. }
$$
it is abelian, but how we can prove it is self-normalizer, please:Dear @fresh_42 , if you could help, :heart:
Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.12, I have a question:
Proposition. (3.2). Let ##L## be a Lie algebra.
(c) If ##L## is nilpotent and nonzero, then ##Z(L) \neq 0##.
how we prove this,
Thanks in advance,
Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.11, I have a question:
Proposition. Let ##L## be a Lie algebra.
(a) If ##L## is solvable, then so are all subalgebras and homomorphic images of ##L##.
(b) If ##I## is a solvable ideal of ##L## such...
Please, in the definition of quotient Lie algebra
If ##I## is an ideal of ##\mathfrak{g}##, then the vector space ##\mathfrak{g} / I## with the bracket defined by:
$$[x+I, y+I]=[x, y]+I, for all x, y \in \mathfrak{g}$$,
is a Lie algebra called the quotient Lie algebra of ##\mathfrak{g}## by...
Please, I am looking for a simple example of derivation on ##sl_2##, if possible, I try to use identity map, but not work with me,
A derivation of the Lie algebra ##\mathfrak{g}## is a linear map ##\delta: \mathfrak{g} \rightarrow \mathfrak{g}## such that ##\delta([x, y])=[\delta(x), y]+[x...
Please, I need some clarifications about second direction, in the file attached,
$$
\text { Then ad } x \text { ad } y \text { maps } L \rightarrow L \rightarrow I \text {, and }(\text { ad } x \text { ad } y)^2 \text { maps } L \text { into }[I I]=0 \text {. }
$$Thank you in advance,
Homework Statement: About semidirect product of Lie algebra
Relevant Equations: ##\mathfrak{s l}_2=## ##\mathbb{K} F \oplus \mathbb{K} H \oplus \mathbb{K} E##
Hi,
Please, I have a question about the module of special lie algebra:
Let ##\mathbb{K}## be a field. Let the Lie algebra...
Homework Statement: please, could you help me to know hoe I compute the Casimir element of lie algebra sl(2), I know the basis and their relations, but i could not find the book explain in details how we get the Casimir element.. I think it is related to killing form, but also I could not find...
This is the defining generator of the Lorentz group
which is then divided into subgroups for rotations and boosts
And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps:
especially...
The full Lorentz group includes discontinuous transformations, i.e., time inversion and space inversion, which characterize the non-orthochronous and improper Lorentz groups, respectively. However, these groups are excluded from the Poincare group, in which only the proper, orthochronous...
My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article...
I’m reading Weinberg’s QFT books, and stacking how to solve problem 15.4.
Weinberg says there is no simple lie algebra with just four generators, but I have no idea how to approach this problem. If the number of generators are only one or two, it can easy to say there is not such a simple lie...
1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##.
The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...
Let ##\mathfrak{A}:=\operatorname{span}\left\{D_n:=x^n\dfrac{d}{dx}\, : \,n\in \mathbb{Z}\right\}## and ##\mathfrak{B}:=\operatorname{span}\left\{E_n:=x^n\dfrac{d}{dx}\, : \,n\in \mathbb{N}_0\right\}## with the usual commutation rule.
My question is: How can we prove or disprove the Lie algebra...
I want to show that ##[C, a_{r}] = 0##. This means that:
$$ Ca_{r} - a_{r}C = \sum_{i,j} g_{ij}a_{i}a_{j}a_{r} - a_{r}\sum_{i,j} g_{ij}a_{i}a_{j} = 0$$
I don't understand what manipulating I can do here. I have tried to rewrite ##g_{ij}## in terms of the structure...
Hello everyone, I am new here, so please let me know if I am doing something wrong regarding the formatting or the way I am asking for help.
I did not really know how to start off, so first I tried to just write out all the ##\mu \nu \rho \sigma## combinations for which ##\epsilon \neq 0## and...
I'm following the lecture notes by https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf.
On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM.
The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the...
"The group given by ## H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0...
Posting for my son (who does not have an account here):
He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants...
I am just starting a QM course. I hope these are reasonable questions. I have been given my first assignment. I can answer the questions so far but I do not really understand what's going on. These questions are all about so(N) groups, Pauli matrices, Lie brackets, generators and their Lie...
I have confusions about representation theory. In the following questions, I will try to express it as best as possible.
For this thread say representation is given as
ρ: L → GL(V)
where L is the Lie group(or symmetry group for a physicist)
GL(V) is the general linear...
I'm solving these problems concerning the SU(4) group and I've reached the point where I have determined the Cartan matrix of SU(4), its inverse and the weight schemes for (1 0 0) and (0 1 0) highest weight states.
How do I decompose the (1 0 0) and (0 1 0) into irreps of SU(3) x U(1) using...
In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...
This is one problem from Robin Ticciati's Quantum Field Theory for Mathematicians essentially asking us to find Cartan subalgebras for the matrix algebras ##\mathfrak{u}(n), \mathfrak{su}(n),\mathfrak{so}(n)## and ##\mathfrak{so}(1,3)##. The only thing he gives is the definition of a Cartan...
3) Taylor expansion question in the context of Lie algebra elements:
Consider some n-dimensional Lie group whose elements depend on a set of parameters \alpha =(\alpha_1 ... \alpha_n) such that g(0) = e with e as the identity, and that had a d-dimensional representation D(\alpha)=D(g(
\alpha)...
1) Likely an Einstein summation confusion.
Consider Lorentz transformation's defined in the following matter:
Please see image [2] below.
I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...
Hello,
I am an undergraduate who has taken basic linear algebra and ODE. As for physics, I have taken an online edX quantum mechanics course.
I am looking at studying some of the necessary math and physics needed for QFT and particle physics. It looks like I need tensors and group theory...
Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...
Hello there,
Given a Lie Algebra ##\mathfrak{g}##, its Cartan Matrix ##A## and a finite representation ##R##, is there a way of determining its highest weight ##\Lambda## in a simple way?
In my course, we consider ##\mathfrak{g}=A_2= \mathfrak{L}_{\mathbb{C}}(SU(3))##. It is stated that the...
Let us for a moment look a field transformations of the type
$$\phi(x)\longmapsto \exp\left(\frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}\right)\phi(x),$$
where ##\omega## is anti-symmetric and ##S^{\mu\nu}## satisfy the commutation relations of the Lorentz group, namely
$$\left[S_{\mu \nu}, S_{\rho...
Given a representation of a Lie Group, is there a equivalence between possible electric charges and projections of the roots? For instance, in the standard model Q is a sum of hypercharge Y plus SU(2) charge T, but both Y and T are projectors in root space, and so a linear combination is. But I...
Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example:
$$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...
Homework Statement
For a left invariant vector field γ(t) = exp(tv). For a gauge transformation t -> t(xμ). Intuitively, what happens to the LIVF in the latter case? Is it just displaced to a different point in spacetime or something else?
Homework EquationsThe Attempt at a Solution
I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups.
Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e,
Lgh = gh where g,h ∈ G
Which corresponds to a translation by g.
Is this an example of a module over a ring...
Hello! I am a bit confused by some definitions. We have that a Lie algebra is abelian if ##[a,b]=0## for all ##a,b \in L## and ##L'## is an invariant subalgebra of ##L## if ##[a,b]=0## for all ##a \in L'## and ##b \in L##. From here I understand that ##L'## is abelian. Then they define a...
I am trying to decompose some exponential operators in quantum optics. The interesting thing is that the operators includes operators from Su(1,1) algebra $$ [K_+,K_-]=-2K_z \quad,\quad [K_z,K_\pm]=\pm K_\pm.$$
For example this one: $$ (K_++K_-)^2.$$ But as you can see they are squares of it.
I...
Good Day
I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc.
But I am making a lot of progress.
There is, however, one issue that continues to elude me.
I often read how Lie developed Lie Groups to study symmetries of PDE's
May I ask if someone...
Homework Statement
How can you find all inequivalent (non-isomorphic) 2D Lie algebras just by an analysis of the commutator?
Homework Equations
$$[X,Y] = \alpha X + \beta Y$$
The Attempt at a Solution
I considered three cases: ##\alpha = \beta \neq 0, \alpha = 0## or ##\beta = 0, \alpha =...
Homework Statement
Consider the contractions of the 3D Euclidean symmetry while preserving the SO(2) subgroup. In the physics point of view, explain the resulting symmetries G(2) (Galilean symmetry group) and H(3) (Heisenberg-Weyl group for quantum mechanics) and give their Lie algebras...