# Question about Lie Brackets in Group Theory

• Savant13
In summary, the Lie Brackets with subscripts + or - are used to represent commutator/anti-commutator operations in quantum physics, where [A,B]_- represents AB-BA and [A,B]_+ represents AB+BA. These brackets are also used in Lie groups and algebras, with the square brackets representing matrix commutators and the curly brackets representing abstract Lie-brackets. The matrices T_i in the fundamental representation are one specific example of this, where the Lie-brackets satisfy a specific requirement.
Savant13
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" 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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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
$$[A,B]_{-} = AB-BA$$
$$[A,B]_{+} = AB+BA$$

So i would guess, this is the same for the Lie Brackets.

element4 said:
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
$$[A,B]_{-} = AB-BA$$
$$[A,B]_{+} = AB+BA$$

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
$$[T_i, T_j]_- = T_i T_j - T_j T_i = c_{ij}^k T_k$$
then we can "promote" them into a Lie-group whose algebra is defined by
$$[t_i, t_j] = c_{ij}^k t_k$$
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 $T_i$ 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 $t_i \to T_i$ linearly extended).
However, for a set of anti-commutation relations this is not generally possible, as the Lie-bracket must satisfy a requirement
$$[t_i, t_j] = -[t_j, t_i]$$
which the matrix anti-commutator does not satisfy.

## What are Lie brackets in Group Theory?

Lie brackets, also known as commutators, are a mathematical operation used to measure the failure of two elements in a group to commute. In other words, they are used to determine how much two elements do not behave like they should in a group.

## How are Lie brackets calculated?

Lie brackets are calculated by taking the difference between the product of the two elements and the product of the elements in the opposite order. For example, if we have two elements a and b, the Lie bracket [a,b] is calculated as ab - ba.

## Why are Lie brackets important in Group Theory?

Lie brackets are important in Group Theory because they allow us to study the structure and behavior of groups in a more detailed way. They help us identify how elements in a group interact with each other and how they can be used to create new elements.

## What are some applications of Lie brackets?

Lie brackets have many applications in mathematics, physics, and engineering. They are used in the study of differential equations, Lie algebras, and symplectic geometry. In physics, they are used to describe the behavior of quantum systems and in engineering, they are used in the design of control systems.

## Can Lie brackets be generalized to other algebraic structures?

Yes, Lie brackets are not limited to just groups. They can be generalized to other algebraic structures such as Lie algebras, Lie superalgebras, and Lie groups. In fact, the concept of a Lie bracket has been extended to other mathematical fields, such as topology and category theory.

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