What is the difference between roots and weights in Lie algebra theory?

In summary, the conversation discusses the basics of Lie algebra theory and how elements of Lie groups can be represented by a set of generators. These generators form a Lie algebra, with a subset of them known as the Cartan subalgebra (CSA) that commute and are diagonalizable. The CSA generators are used to represent physical states, while the remaining generators are referred to as step or ladder operators and transform the eigenvectors of the CSA generators. The algebra can be defined in terms of the integer eigenvalues of the CSA generators, also known as roots or weights. The specific problem mentioned involves the sl(3) algebra and its representations, where the root vectors are calculated differently depending on the representation. The conversation also touches on the adj
  • #1
Andy_X
2
0
I need some help with understanding the basics of Lie algebra theory. I suspect my problems are due to a fundamental misunderstanding somewhere so apologies in advance for the naivety of the questions and for the restatement of elementary mathematics.
As I understand it elements of Lie groups can be represented in terms of a set of generators by exponentiation. The set of generators form a Lie algebra whose properties are determined by the commutators of the generators. A subset of the generators (the Cartan subalgebra or CSA ) commute and are simultaneously diagonalisable. The eigenvectors of the CSA generators are used to represent physical states. The remaining generators, referred to as step or ladder operators, when applied to the eigenvectors of the CSA generators transform them into other eigenvectors with corresponding eigenvalues that differ by 2 from the eigenvalue corresponding to the original eigenvector. The algebra can be largely defined in terms of the eigenvalues of the CSA generators which must be integer and are referred to as roots or weights. Is this correct and is there any difference between a root and a weight?
The specific problem I have involves sl(3). I have seen a representation of this in terms of 8 3*3 real matrices (related to the Gell-Mann matrices) of which 2 are diagonal with diagonals (1, -1,0) and (0,1,-1). I’ve also been told that the root vectors of sl(3) are α(1)=(2,1) α(2)=(-1,2) and θ=(2,1) apparently regardless of the representation used. However the eigenvalues of real diagonal matrices are the diagonal elements themselves so why are the elements of the root vectors different, why do they not differ by 2 and why are there three of them? Clearly I have misunderstood something basic here but despite referring to several texts I am unable to resolve. Any help appreciated.
 
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  • #2
Root is employed (instead of weight) when you are in the adjoint representation of the algebra.
Where have you found eigenvalues differ by 2? do not forget that the states in the CSA are eigenvectors with nul eigenvalues (commutation in the CSA).
Ie for su(3) you have 8 eigenvectors (2 in the CSA) and 3 pairs of ladders
 

1. What are root vectors of Lie algebras?

Root vectors of Lie algebras are a set of vectors that form a basis for a Cartan subalgebra of a given Lie algebra. They are used to define a root system, which plays an important role in the classification and study of Lie algebras.

2. How do root vectors relate to root systems?

Root vectors are the building blocks of root systems. They are used to define the roots of a Lie algebra, which are then used to construct the root system. The root system is a fundamental structure that helps to classify Lie algebras and understand their properties.

3. What is the significance of root vectors in Lie algebras?

Root vectors are crucial in the study of Lie algebras because they provide a way to classify and understand the structure of these algebras. They help to define the roots and root systems, which in turn give important information about the algebra, such as its dimension and the types of representations it can have.

4. How are root vectors calculated?

The calculation of root vectors depends on the specific Lie algebra being studied. In general, root vectors are found by solving a system of equations, known as the root vector equations, using techniques such as linear algebra and group theory.

5. What are some applications of root vectors in mathematics and physics?

Root vectors have numerous applications in both mathematics and physics. For example, they are used in the study of symmetry in physics, such as in particle physics and quantum mechanics. In mathematics, they are used to classify Lie algebras, which have applications in areas such as differential geometry, topology, and representation theory.

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