Affine Difference Algebraic Groups

The central objects of study in this thesis are affine difference algebraic groups. Similar to the case of affine algebraic groups, these groups can all be realized as subgroups of some general linear group defined by algebraic difference equations. However, the defining equations here are not simply polynomials in the matrix entries but difference polynomials, i.e., the defining equations involve a difference operator σ which has to be interpreted as a ring endomorphism. For example, if G is the difference algebraic subgroup of GL n defined by the algebraic difference equations Xσ(X) T = σ(X) T X = I n , where σ : C → C is the complex conjugation map, then G(C) is the group of all complex unitary n × n-matrices. A more classical example of a difference field, i.e., a field equipped with an endomorphism, is C(x) with σ(f (x)) = f (x + 1) or σ(f (x)) = f (qx) for q ∈ C ×. Difference algebra, i.e., the systematic study of solutions of difference equations from an algebraic point of view, grew out of the study of difference equations over C(x) in much the same way as algebraic geometry grew out of the study of polynomial equations over Q or C. Alternatively, affine difference algebraic groups may be described as affine group schemes with a certain additional structure (the difference structure). As schemes they are typically not of finite type, but they enjoy a certain finiteness property with respect to the difference structure; they are " of finite σ-type ". From an algebraic point of view, an affine difference algebraic group G over a difference field k corresponds to a Hopf algebra k{G} over k together with a ring endomorphism σ : k{G} → k{G} that extends σ : k → k. The structure maps of the Hopf algebra are required to commute with σ, and k{G} is required to be finitely σ-generated over k, i.e., there exists a finite set B ⊂ k{G} such that B, σ(B), σ 2 (B),. .. generates k{G} as a k-algebra. Difference algebraic groups are the discrete analog of differential algebraic groups, i.e., groups defined by algebraic differential equations. Differential algebraic groups have always played an important role in differential algebra (see, e. See also [Mal10] for a more geometric approach to differential algebraic groups and [Bui98] for an arithmetic analog of difference/differential algebraic groups. Over the last …