# 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

[
x
,
x
]
=
x
×
x
=
0

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

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15. ### The Lie bracket of fundamental vector fields

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...
16. ### Show Lie Bracket of X & Y is Linear Comb. of Commuting Vector Fields

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...
17. ### Verifying Lie Bracket for Vector Fields on U

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...
18. ### Solving the Lie Bracket Question in Quantum Mechanics

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...
19. ### Pushforward of Lie bracket

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...