What is Lie algebra: Definition and 167 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. H

    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. S

    I Fundamental representation and adjoint representation

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  3. cianfa72

    I ##SU(2, \mathbb C)## parametrization using Euler angles

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  4. H

    A About Universal enveloping algebra

    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..
  5. H

    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...
  6. H

    A About universal enveloping algebra

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  7. H

    A About Proof of Engel's Theorem

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  8. H

    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. H

    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. H

    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. H

    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. H

    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...
  13. H

    A Understanding the Second Direction in Semi Simple Lie Algebra: A Guide

    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,
  14. H

    About semidirect product of Lie algebra

    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...
  15. H

    How to compute the Casimir element of Lie algebra sl(2)?

    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...
  16. bella987

    Deriving the commutation relations of the Lie algebra of Lorentz group

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  17. redtree

    I Why are discontinuous Lorentz transformations excluded from the Poincare group?

    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...
  18. L

    I Jacobi identity of Lie algebra intuition

    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...
  19. P

    Simple lie algebra that holds just four generators?

    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...
  20. J

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  21. fresh_42

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  22. J

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  23. F

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  24. V

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  25. Y

    I Why isn't this a Lie group?

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  26. S

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  27. M

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  28. P

    A Questions about representation theory of Lie algebra

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  29. RicardoMP

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  30. L

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  31. L

    Finding Cartan Subalgebras for Matrix Algebras

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  32. Jason Bennett

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  33. Jason Bennett

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  34. D

    Other Textbooks for tensors and group theory

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  35. G

    A Lie Algebra and Lie Group

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  36. P

    A Highest weight of representations of Lie Algebras

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  37. M

    I Spin matrices and Field transformations

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  38. arivero

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  39. D

    I Relation Between Cross Product and Infinitesimal Rotations

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  40. fresh_42

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    Greg Bernhardt submitted a new blog post Lie Algebras: A Walkthrough The Representations Continue reading the Original Blog Post.
  41. fresh_42

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  42. fresh_42

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  43. N

    Left invariant vector field under a gauge transformation

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  44. N

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  45. S

    I What is the definition of a Semi-simple Lie algebra?

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  46. B

    I Can we construct a Lie algebra from the squares of SU(1,1)

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  47. JTC

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  48. A

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  49. A

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  50. A

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