THESIS
2021

1 online resource (xi, 85 pages) : illustrations (some color)

**Abstract**
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

^{Xs }.

We make a review about root systems and representations of sl(3,C) and complex exceptional Lie
algebra g

_{2}. Our calculations of orthogonal projections and Dynkin indices are based on sl(3,C) and g

_{2}.
We study the orthogonal projections of irreducible root systems to the plan...[

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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

^{Xs }.

We make a review about root systems and representations of sl(3,C) and complex exceptional Lie
algebra g

_{2}. Our calculations of orthogonal projections and Dynkin indices are based on sl(3,C) and g

_{2}.
We study the orthogonal projections of irreducible root systems to the plane spanned by the root system
of type A

_{2}. Ignoring the zero vector, these projections form the type g

_{2} 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 g

_{2}.

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 g

_{2}-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 A

_{2} and type g

_{2} 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 g

_{2} in e

_{8} 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 e

_{8}. After calculating the relevant branching rules with SageMath, we study the Dynkin indices for e

_{7ᐧ1/2}.

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