PlanetMath: Jacobi identity interpretations

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Jacobi identity interpretations

(Definition)

The Jacobi identity in a Lie algebra $\mathfrak{g}$ has various interpretations that are more transparent, whence easier to remember, than the usual form

$\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.$

One is the fact that the adjoint representation ¹ $\operatorname{ad}:\mathfrak{g} \rightarrow \mathfrak{gl}(\mathfrak{g})$ really is a representation. Yet another way to formulate the identity is

$\displaystyle \operatorname{ad}(x)[y,z]=[\operatorname{ad}(x)y,z]+[y,\operatorname{ad}(x)z],$

i.e., $\operatorname{ad}(x)$ is a derivation on $\mathfrak{g}$ for all $x \in \mathfrak{g}$ .

Footnotes

...¹: Here, “ $\mathfrak{gl}(\mathfrak{g})$ ” means the space o endomorphisms of $\mathfrak{g}$ , viewed as a vector space, with Lie bracket on $\mathfrak{gl}(\mathfrak{g})$ being commutator.

"Jacobi identity interpretations" is owned by rspuzio. [ full author list (3) | owner history (3) ]

(view preamble)

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: derivation, identity, representation, commutator, Lie bracket, vector space, endomorphisms, adjoint representation, interpretations, Lie algebra, Jacobi identity

This is version 5 of Jacobi identity interpretations, born on 2002-09-20, modified 2007-04-01.
Object id is 3468, canonical name is JacobiIdentityInterpretations.
Accessed 3265 times total.

Classification:

AMS MSC:

17B99 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 2 ]

Discussion

forum policy

No messages.

Interact