Answer: Lie Algebras: Adjoint Representations of Same Dimension as Basis

In summary, the conversation discusses the concept of Lie algebras and their representations, specifically the basic and adjoint representations. It is mentioned that for some Lie algebras, such as so(3), the dimensions of the basic and adjoint representations are the same. The conversation then asks if there are other Lie groups/algebras that have this property and requests recommendations for further reading on the topic.
  • #1
yorik
7
0
Hello,

I hope it's not the wrong forum for my question which is the following:

Is there some list of Lie algebras, whose adjoint representations have the same dimension as their basic representation (like, e.g., this is the case for so(3))? How can one find such Lie algebras? Could you recommend some literature to me?

Thanks in advance!
 
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  • #2


I'm not very well-founded in Lie-algebra's, but the adjoint of an element x is the map
[tex]\operatorname{ad}_x: y \mapsto [x, y][/tex]
isn't it?
So the adjoint is linear, i.e.
[tex]\operatorname{ad}_x + \operatorname{ad}_y = \operatorname{ad}_{x + y}, \operatorname{ad}_x(y) + \operatorname{ad}_x(z) = \operatorname{ad}(y + z)[/tex]
etc. - then isn't the adjoint representation always of the same dimension.
I.e. the basic representation provides the generators [itex]g_i[/itex] of the Lie-algebra, and then [itex]\operatorname{ad}_{g_i}[/itex] generate the adjoint representation?
 
  • #3


Well, the basic idea of the Lie groups and lie algebras is the following. A Lie group G is a group, whose elements can be written as exp(i * ak Xk), where k runs from 1 to N and over k is summed (Einstein summation convention), ak are some real-valued parameters and Xk are some linearly independent hermitian operators (so called generators). They form an N-dimensional vector space (however, the dimension of the Hilbert space on which they act is not specified).

Now, the generators satisfy [Xk, Xl] = i fklmXm, where the f's are the so called structure constants. They define the Lie algebra of the Lie group G. It turns out, you can find infinitely many generator spaces (of different), which satisfy the algebra. The smallest irreducible generator space gives us the basic (fundamental) representation of the Lie Algebra.

We can also define the so called adjoint representation. We define
(Tk)lm = - i fklm. It has the same dimension as the Lie group. But, in generally, its dimension is not equal to the dimension of the fundamental representation. However, it's the case for so(3), the Lie Algebra of SO(3): http://math.ucr.edu/home/baez/lie/node5.html.

So my question is: Are there any other Lie groups / algebras for which the dimension of the two representations are equal like it is for SO(3) / so(3)?
 

Related to Answer: Lie Algebras: Adjoint Representations of Same Dimension as Basis

1. What is a Lie algebra?

A Lie algebra is a mathematical structure that describes the algebraic properties of a set of elements, typically represented by matrices or vectors. It is a fundamental tool in the study of abstract algebra and its applications in fields such as physics and engineering.

2. What is the adjoint representation of a Lie algebra?

The adjoint representation of a Lie algebra is a way of associating each element of the algebra with a linear transformation on the algebra itself. It provides a way of studying the properties of the algebra through its own structure, rather than relying on external structures like vector spaces.

3. What is the dimension of the adjoint representation?

The dimension of the adjoint representation is equal to the dimension of the basis of the Lie algebra. This means that for a Lie algebra with n elements, the adjoint representation will also have n dimensions.

4. How is the adjoint representation related to the basis of a Lie algebra?

The adjoint representation is closely related to the basis of a Lie algebra, as it is defined in terms of the basis elements and their corresponding linear transformations. The basis elements serve as a basis for the adjoint representation, and the linear transformations are defined by the structure constants of the algebra.

5. What are some applications of the adjoint representation of Lie algebras?

The adjoint representation of Lie algebras has many applications in mathematics and physics. It is used in the study of differential equations, Lie groups, and symmetries in physics. It also has applications in areas such as algebraic geometry and algebraic topology.

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