Lie algebras Definition and 87 Threads

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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