Dynkin index Wikipedia

The Dynkin index and sl2subalgebras of simple Lie algebras

where TR is the Dynkin index and CR the Casimir in the representation R fundamental representation TR 1 2 CR N2 12N adjoint representation TR CR N From Eq A6 it follows that TRDA CRDR where DR is the dimension of the representation Rand DA is the dimension of the adjoint which de nes the

PDF GENERAL DYNKIN INDICES AND THEIR APPLICATIONS Springer

How to calculate the Dynkin index and the Casimir operator for

Dynkin Index published in Concise Encyclopedia of Supersymmetry In physics the second order Dynkin Index is often a more natural quantity than the quadratic Casimir For example in fourdimensional gauge theories I Λ gives up to a representation independent constant the contribution of elementary fields carrying the representation R Λ to the oneloop renormalization group βfunction

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In the mathematical field of Lie theory a Dynkin diagram named for Eugene Dynkin is a type of graph with some edges doubled or tripled drawn as a double or triple line Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields in the classification of Weyl groups and other finite reflection groups and in other contexts

Dynkin Index

Dynkin Index SpringerLink

The ground field k is algebraically closed and of characteristic zero Let G be a connected semisimple algebraic group with Lie algebra gIn 1952 Dynkin classified all semisimple subalgebras of semisimple Lie algebras 3As a tool to distinguish different nonconjugate embeddings of the same algebra Dynkin introduced the index of a homomorphism of simple Lie algebras

Although it is not obvious from the definition the Dynkin index of a representation is an integer This was proved by EB Dynkin 2 Theorem22 using lengthy classification results Later he gave a better proof that is based on a topological interpretation of the index A short algebraic proof is given in 5 ChI 310 Example 12

ON THE DYNKIN INDEX OF A PRINCIPAL sl2SUBALGEBRA

This is a continuation of arXiv09030398 mathRT Let g be a simple Lie algebra In this note we provide simple formulae for the index of sl2subalgebras in the classical Lie algebras and a new formula for the index of the principal sl2 We also compute the difference D of the indices of principal and subregular sl2subalgebras Our formula for D involves some data related to the

Dynkin diagram Wikipedia

Dynkin Index

In mathematics the Dynkin index of finitedimensional highestweight representations of a compact simple Lie algebra relates their trace forms via In the particular case where is the highest root so that is the adjoint representation the Dynkin index is equal to the dual Coxeter number The notation is the trace form on the representation

the pth order Dynkin index D w is now defined by3 p where the trace is over the representation space w identically zero for values of p not listed in I Note that D W is p Since the absolute normalization of D w is not fixed it is often convenient to define p Q W p D W D A p 11

The Dynkin diagram is a single isolated node Spin4 SU2 SU2 and the halfspin representations are the fundamental representations on the two copies of SU2 The Dynkin diagram is two disconnected nodes Spin5 Sp2 and the spin representation on C4 can be identi ed with the fundamental Sp2 representation on H2 The Dynkin

To be honest I had seen that table from Slansky but am not sure how to derive the scalar Dynkin number or the Casimir from the Dynkin labels Eq 35 here appears to accomplish this Is this what you mean by WP closed forms

PDF Clifford Algebras and Spin Groups Columbia University

The Dynkin index and sl2subalgebras of simple Lie algebras