Algebraic games—playing with groups and rings

Cross-sections, Quotients, and Representation Rings of Semisimple Algebraic Groups

LetG be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny τ : b G → G is bijective. In particular, f...

Rings as Groups with Operators

(11) KM s 0 (mod 2*--9). Conversely (11) implies (9). Since (9) holds for the modulus 2-9ikf, it follows similarly that (11) holds for the modulus 2~-9 with ikf = 2ikfi. Hence (11) will be true for the given modulus if M = 2~M\. This supplies a proof by induction that (8) is a universal form for every w ^ 4 . If, in addition,* ikf is divisible by every prime p where 3<p^n, we satisfy the necess...

Algebraic Groups and Arithmetic Groups

Embedding Differential Algebraic Groups in Algebraic Groups

Quantifier Elimination We work in K a large rich differentially closed field. All other differential fields are assumed to be small subfields of K. Let L = {+,−, ·, δ, 0, 1} be the language of differential rings. We let L− = {+,−, ·, 0, 1}, the language of rings. If k is a differential field, we can view k either as an L-structure or an L−-structure. Theorem 1.1 (Quantifier Elimination) For any...

Geometric Langlands Duality and Representations of Algebraic Groups over Commutative Rings

In this paper we give a geometric version of the Satake isomorphism [Sat]. As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the coroots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a red...