[Paper Review] Semisimple Algebraic Groups in Characteristic Zero

J. S. Milne|arXiv (Cornell University)|May 9, 2007

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper provides a concise, conceptually clean derivation of the classification of split semisimple algebraic groups over fields of characteristic zero by leveraging Tannaka duality and tensor category theory, showing that such groups are classified by the same root data as their associated semisimple Lie algebras. The key contribution is a natural, representation-theoretic bridge between Lie algebras and algebraic groups via tensor categories.

ABSTRACT

It is shown that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. This article is extracted from Milne 2007.

Motivation & Objective

• To provide a simple and natural derivation of the classification of split semisimple algebraic groups in characteristic zero.
• To establish a direct link between the classification of semisimple Lie algebras and that of semisimple algebraic groups via tensor category theory.
• To show that the representation theory of these groups, particularly their finite-dimensional representations, is fully captured by their root data.
• To clarify the structural relationship between semisimple Lie algebras, semisimple algebraic groups, and tensor categories in characteristic zero.

Proposed method

• Utilizes Tannaka duality to reconstruct an algebraic group from its category of finite-dimensional representations.
• Applies the theory of M-gradations on tensor categories to associate group homomorphisms from diagonalizable groups to the Tannaka dual of a category.
• Uses the universal tensor map from the set of isomorphism classes of simple objects to the free abelian group modulo tensor relations to define the weight lattice.
• Constructs the algebraic group as the Tannaka dual of a full subcategory of representations of a Lie algebra with highest weights in a given lattice X.
• Leverages the fact that the Tannaka dual of the category of representations of a split semisimple Lie algebra with highest weights in X is a split semisimple algebraic group with root system (V,R) and weight lattice X.
• Applies the adjoint representation and the action of exponentials of adjoint endomorphisms to prove conjugacy of maximal tori.

Experimental results

Research questions

• RQ1 How can the classification of split semisimple algebraic groups in characteristic zero be derived directly from the classification of semisimple Lie algebras?
• RQ2 What is the precise role of tensor categories and Tannaka duality in connecting Lie algebras and algebraic groups?
• RQ3 How do the weight lattices Q(R) and P(R) of a root system relate to the classification of algebraic groups?
• RQ4 Can the isogeny class of a semisimple algebraic group be recovered from a lattice embedding between the root system's weight lattices?
• RQ5 What is the geometric meaning of the Tannaka dual of a subcategory of representations with fixed highest weights?

Key findings

• Every diagram (V,R,X), where (V,R) is a reduced root system and X is a lattice with Q(R) ⊂ X ⊂ P(R), arises from a unique split semisimple algebraic group over k.
• The Tannaka dual of the full subcategory of representations of a split semisimple Lie algebra with highest weights in X is a split semisimple algebraic group with root system (V,R) and weight lattice X.
• Isogenies between split semisimple algebraic groups correspond exactly to lattice embeddings X → X' that preserve the root system.
• The center of a split semisimple algebraic group G is isomorphic to the quotient of the weight lattice X by the root lattice Q(R), and is recovered as the kernel of the adjoint representation.
• Maximal tori in the algebraic group G(𝔤) are conjugate via elements in G(𝔤)(k), as shown by exponentiating nilpotent elements of the Lie algebra.
• The category of finite-dimensional representations of a split semisimple algebraic group is equivalent to the Tannaka dual of the category of representations of its Lie algebra with highest weights in the weight lattice.