What is Lie bracket: Definition and 19 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
Hi, a doubt related to the calculation done in this old thread.
$$\left(L_{\mathbf{X}} \dfrac{\partial}{\partial x^i} \right)^j = -\dfrac{\partial X^j}{\partial x^i}$$
$$L_{\mathbf{X}} {T^a}_b = {(L_{\mathbf{X}} \mathbf{T})^a}_b + {T^{i}}_b \langle L_{\mathbf{X}} \mathbf{e}^a, \mathbf{e}_i...
We had a thread long time ago concerning the Lie dragging of a vector field ##X## along a given vector field ##V## compared to the Fermi-Walker transport of ##X## along a curve ##C## through a point ##P## that is the integral curve of the vector field ##V## passing through that point.
We said...
-Verify that the space ##Vect(M)## of vector fields on a manifold ##M## is a Lie algebra with respect to the bracket.
-More generally, verify that the set of derivations of any algebra ##A## is a Lie algebra with respect to the bracket defined as ##[δ_1,δ_2] = δ _1◦δ_2− δ_2◦δ_1##.
In the first...
Please, I need some clarifications about second direction, in the file attached,
$$
\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 {. }
$$Thank you in advance,
Hi,
I've been doing a course on Tensor calculus by Eigenchris and I've come across this problem where depending on the way I compute/expand the Lie bracket the Torsion tensor always goes to zero. If you have any suggestions please reply, I've had this problem for months and I'm desperate to...
As an aside, fresh_42 commented and I made an error in my post that is now fixed. His comment, below, is not valid (my fault), in that THIS post is now fixed.Assume s and w are components of vectors, both in the same frame
Assume S and W are skew symmetric matrices formed from the vector...
Prove that for a 2 sphere in R3 the Lie bracket is the same as the cross product using the vector: X = (y,-x,0); Y = (0,z-y)
[X,Y] = JYX - JXY where the J's are the Jacobean matrices.
I computed JYX - JXY to get (-z,0,x). I computed (y,-x,0) ^ (0,z,-y) and obtained (xy,y2,yz) = (z,0,x)...
I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups.
Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e,
Lgh = gh where g,h ∈ G
Which corresponds to a translation by g.
Is this an example of a module over a ring...
Homework Statement
Determine the Lie bracket for 2 elements of SU(3).
Homework Equations
[X,Y] = JXY - JYX where J are the Jacobean matrices
The Attempt at a Solution
I exponentiated λ1 and λ2 to get X and Y which are 3 x 3 matrices.. If the group elements are interpreted as vector...
Hello! So I have 2 vector fields on a manifold ##X=X^\mu\frac{\partial}{\partial x^\mu}## and ##Y=Y^\mu\frac{\partial}{\partial x^\mu}## and this statement: "Neither XY nor YX is a vector field since they are second-order derivatives, however ##[X, Y]## is a vector field". Intuitively makes...
I am trying to prove the following:
$$3d\sigma (X,Y,Z)=-\sigma ([X,Y],Z)$$
where ##X,Y,Z\in\mathscr{X}(M)## with M as a smooth manifold. I can start by stating what I know so it is easier to see what I do wrong for you guys.
I know that a general 2-form has the form...
I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t
but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...
Hello
In textbook by Kobayashi and Nomizu derivation of rank k in space of all differential forms on a manifold is defined to be operator that is linear, Leibnitz and maps r-forms into r+k-forms. By Leinbitz I mean, of course: D(\omega \wedge \eta)=(D \omega) \wedge \eta + \omega \wedge (D...
The Maurer-Cartan one-form ##\Theta = g^{-1} dg## is though of as a lie algebra valued form.
It arises in connection with Yang-Mill's theory where the gauge potential transforms as
$$A \mapsto g Ag^{-1} - g^{-1} dg.$$
However, one also defines for lie-algebra valued differential forms...
Homework Statement
The Lie bracket of the fundamental vector fields of two Lie algebra elements is the fundamental vector field of the Lie bracket of the two elements:
[\sigma(X),\sigma(Y)]=\sigma([X,Y])
Homework Equations
Let \mathcal{G} a Lie algebra, the fundamental vector field of an...
Homework Statement
Show that if the vector fields X and Y are linear combinations (not necessarily with constant coefficients) of m vector fields that all commute with one another, then the lie bracket of X and Y is a linear combination of the same m vector fields.
The Attempt at a Solution...
If we have vect (u) which denotes an infinite-dimensional vector space of all vector fields on u. As infinitesimal elements of the continuous group of Diff(u) they form a Lie Algebra. We then can define the bracket of two vector fields in v and w. If in coordinates:
v = \sum_{i}V i...
Hi!
I was doing an assignment in quantum mechanics and came upon the following fact I cannot explain to me.
I hope someone of you can and will be willing to :)
Consider the creation and annihilation operators: a^+ and a and also the momentum and position operators p and x...
one elementary result that you see when you first learn differential geometry is that the pushforward of the Lie bracket of two vector fields is the Lie bracket of the pushforward of the two vector fields, i.e.
let \phi be a diffeomorphism from manifold M to N, and let v, w be two vector...