We study nilpotent orbits and Dynkin indices of Lie algebras. After a brief review of nilpotent orbit theory, we study the subregular orbit. We find that the set of subregular elements is stable under the action of the adjoint group. For any element X in a complex reductive Lie algebra g, denote the Jordan decomposition of X by X = X_s + X_n, where X_s (respectively, X_n) is the semisimple (respectively, nilpotent) part of X. Then X is a subregular element of g if and only if X_n is a subregular nilpotent element of g^X_s .

We make a review about root systems and representations of sl(3,C) and complex exceptional Lie algebra g₂. Our calculations of orthogonal projections and Dynkin indices are based on sl(3,C) and g₂. We study the orthogonal projections of irreducible root systems to the plan...[ Read more ]

We study nilpotent orbits and Dynkin indices of Lie algebras. After a brief review of nilpotent orbit theory, we study the subregular orbit. We find that the set of subregular elements is stable under the action of the adjoint group. For any element X in a complex reductive Lie algebra g, denote the Jordan decomposition of X by X = X_s + X_n, where X_s (respectively, X_n) is the semisimple (respectively, nilpotent) part of X. Then X is a subregular element of g if and only if X_n is a subregular nilpotent element of g^X_s .

We make a review about root systems and representations of sl(3,C) and complex exceptional Lie algebra g₂. Our calculations of orthogonal projections and Dynkin indices are based on sl(3,C) and g₂. We study the orthogonal projections of irreducible root systems to the plane spanned by the root system of type A₂. Ignoring the zero vector, these projections form the type g₂ root system for all the root systems except type C. Through these orthogonal projections, we can see the close relation between sl(3,C) and g₂.

We also study the Dynkin indices of Lie algebras. Dynkin indices describe the ways of embedding a simple subalgebra into a complex simple Lie algebra. We calculate the Dynkin indices of the finite-dimensional sl(3,C)-, so(5,C)-, and g₂-modules. We also derive a general formula for the Dynkin index of finite-dimensional sl(l+1,C)-module.

As an application of computational Lie theory, we recalculate the Dynkin indices of subalgebras of type A₂ and type g₂ in the five exceptional simple Lie algebras. We use the computer algebra system SageMath to compute the branching rules and find maximal subgroups (subalgebras) of the exceptional Lie groups (algebras). In this process, we find a new Dynkin index of g₂ in e₈ with value 4.

Lastly, we study the Lie algebra e_7ᐧ1/2. This Lie algebra comes from P. Deligne’s work on the exceptional series of Lie groups. We study the structure of e_7ᐧ1/2 by realizing it as the derived subalgebra of a certain Heisenberg type parabolic subalgebra of e₈. After calculating the relevant branching rules with SageMath, we study the Dynkin indices for e_7ᐧ1/2.