Question about Lie Brackets in Group Theory

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  • #1
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What does it mean when a Lie Bracket has a subscript + or - directly after it?

I found this notation in http://en.wikipedia.org/wiki/Special_unitary_group" [Broken] under the fundamental representation heading

Those are Lie Brackets, right? I know Lie Brackets are being used elsewhere in the article.
 
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  • #2
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My knowledge of Lie groups/algebras are very limited, but in quantum physics this notation usually means commutator/anti-commutator (usually for bosons/fermions).
So
[tex][A,B]_{-} = AB-BA[/tex]
[tex][A,B]_{+} = AB+BA[/tex]

So i would guess, this is the same for the Lie Brackets.
 
  • #3
CompuChip
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My knowledge of Lie groups/algebras are very limited, but in quantum physics this notation usually means commutator/anti-commutator (usually for bosons/fermions).
So
[tex][A,B]_{-} = AB-BA[/tex]
[tex][A,B]_{+} = AB+BA[/tex]

So i would guess, this is the same for the Lie Brackets.
Note that Lie-brackets are abstract things, which take two elements x and y and produce a new one [x, y].
In the section you are referring to, A and B are matrices and the square brackets are not Lie-brackets but simply commutators of matrices. For those it is common to use the notation explained by element4.

Note that if we have a set of matrix commutation relations
[tex][T_i, T_j]_- = T_i T_j - T_j T_i = c_{ij}^k T_k[/tex]
then we can "promote" them into a Lie-group whose algebra is defined by
[tex][t_i, t_j] = c_{ij}^k t_k[/tex]
where the brackets in the former refers to the matrix commutator and in the latter expression to the abstract operation called "Lie-bracket". The matrices [itex]T_i[/itex] are one specific representation called the fundamental representation (if you remember that a representation is a function from the Lie-algebra to some subgroup of matrices, the fundamental representation would simply be the map [itex]t_i \to T_i[/itex] linearly extended).
However, for a set of anti-commutation relations this is not generally possible, as the Lie-bracket must satisfy a requirement
[tex][t_i, t_j] = -[t_j, t_i][/tex]
which the matrix anti-commutator does not satisfy.
 

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