# Matrix form of Lie algebra highest weight representation

• vega12
In summary, the speaker is working with Lie algebras and needs matrix representations for each Lie algebra and highest weight. They have been able to solve this problem for cases where all weights have a multiplicity of 1, but encounter difficulties when a node has a multiplicity greater than one. This results in matrices that do not have the correct dimensions, even though they satisfy the commutation relations. The speaker is seeking references on how to properly handle multiplicities in order to obtain the correct matrix representation. They also clarify that they are referring to semisimple or simple Lie algebras, not just Lie algebras in general.
vega12
I am currently working with Lie algebras and my research requires me to have matrix representations for any given Lie algebra and highest weight. I solved this problem with a program for cases where all weights in a representation have multiplicity 1 by finding how $E_\alpha$ acts on each node of this tree. However, when a node has multiplicity greater than one, I run into problems. For instance, the $\omega_2$ highest weight representation of $D_4$ is dimension 28, but I only have 25 unique nodes, and so my corresponding matrices are only 25 dimensional. They properly satisfy the appropriate commutation relations, but are not the right dimension. The same thing happens with, for example, $C_4$ and highest weight $\omega_2$.

How can one handle multiplicities properly to get the matrix representation with the correct dimension? Any references would be very much appreciated. Thank you!

## What is the Matrix form of Lie algebra highest weight representation?

The Matrix form of Lie algebra highest weight representation is a way of representing elements of a Lie algebra using matrices. It is a useful tool for studying the algebraic structure of Lie algebras and their representation theory.

## What is a Lie algebra?

A Lie algebra is a mathematical structure that studies the algebraic properties of continuous symmetry. It is a vector space with a binary operation called the Lie bracket, which measures how two vectors interact with each other.

## What does "highest weight" refer to in this context?

"Highest weight" refers to a special element in the Lie algebra that is used to construct the highest weight representation. It is a weight that is larger than all other weights in the representation.

## Why is the Matrix form of Lie algebra highest weight representation useful?

The Matrix form allows for a more concrete and efficient way of studying Lie algebras and their representations. It also provides a way to compute and manipulate elements of the algebra more easily.

## What are some applications of the Matrix form of Lie algebra highest weight representation?

The Matrix form is used in various fields of mathematics and physics, such as representation theory, differential geometry, and quantum mechanics. It is also used in practical applications, such as in the study of symmetries in physical systems.

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