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This chapter deals with central extensions and groups locally of minimal type. It begins with a discussion of the general lemma on the behavior of the scheme-theoretic center with respect to the formation of central quotient maps between pseudo-reductive groups; this lemma generalizes a familiar fact in the connected reductive case. The chapter then considers four phenomena that go beyond the quadratic case, along with a pseudo-reductive group of minimal type that is locally of minimal type. It shows that the pseudo-split absolutely pseudo-simple k-groups of minimal type with a non-reduced root system are classified over any imperfect field of characteristic 2. In this classification there is no effect if the “minimal type” hypothesis is relaxed to “locally of minimal type.”

This chapter describes constructions when Φ‎ has a double bond. In particular, it considers a construction that goes beyond SO(q)'s and provides the right generalization of the basic exotic construction for type-Bn when n is not equal to 2. A type-B generalized basic exotic k-group is the universal smooth k-tame central extension of a type-B adjoint generalized basic exotic k-group. The root-field hypothesis in the rank-1 adjoint type-B case is automatic in the higher-rank case. The chapter also builds a large class of absolutely pseudo-simple k-groups of type C via fiber products using type-B generalized basic exotic groups. Finally, it discusses exceptional construction for rank-2 and introduces auxiliary Weil restrictions to explain how the generalized exotic groups of types B and C, as well as the rank-2 basic exceptional groups, underlie a construction beyond the standard case that is exhaustive under a locally of minimal type hypothesis.

This chapter deals with field-theoretic and linear-algebraic invariants. It first presents a construction of non-standard pseudo-split absolutely pseudosimple k-groups with root system A1 over any imperfect field k of characteristic 2. It then considers an absolutely pseudo-simple group over a field k, along with a pseudo-split pseudo-reductive group over an arbitrary field k. It also establishes the equality over k of minimal fields of definition for projection onto maximal geometric adjoint semisimple quotients. This is followed by two examples that illustrate the root field in A1-cases. The chapter concludes with a discussion of a classification of the isomorphism classes of pseudo-split pseudo-simple groups G over an imperfect field k of characteristic p subject to the hypothesis that G is of minimal type. The associated irreducible root datum, which is sufficient to classify isomorphism classes in the semisimple case, is supplemented with additional field-theoretic and linear-algebraic data.

This chapter considers some preliminary notions, starting with standard pseudo-reductive groups, Levi subgroups, and root systems. It reviews the “standard construction” of pseudo-reductive k-groups and shows that any connected reductive group equipped with a chosen split maximal torus is generated by that maximal torus and its root groups for the simple positive and negative roots relative to a choice of positive system of roots in the root system. It also discusses the basic exotic construction, noting that the only nontrivial multiplicities that occur for the edges of Dynkin diagrams of reduced irreducible root systems are 2 and 3. Finally, it explains the minimal type pseudo-reductive k-group G, along with quotient homomorphism between pseudo-reductive groups.

This chapter describes the construction of canonical central extensions that are analogues for perfect smooth connected affine k-groups of the simply connected central cover of a connected semisimple k-group. A commutative affine k-group scheme of finite type is k-tame if it does not contain a nontrivial unipotent k-subgroup scheme. The chapter establishes good properties of the universal smooth k-tame central extension, noting that the property “locally of minimal type” is inherited by pseudo-reductive central quotients of pseudo-reductive groups. Although inseparable Weil restriction does not generally preserve perfectness, the chapter shows that the formation of the universal smooth k-tame central extension interacts with derived groups of Weil restrictions.

This chapter deals with “generalized standard” pseudo-reductive group over a field k. Because the notion of root field has only been defined for absolutely pseudo-simple G whose root system over ks is reduced, the first step is to remove that restriction. Thus, in the non-reduced case the focus is on the longest (equivalently, divisible) roots when defining the root field of G. For absolutely pseudo-simple k-groups of minimal type with a non-reduced root system over ks the condition that the root field coincides with the ground field is a reasonable one to impose for the purpose of a structure theorem. The chapter proves a rigidity property of generalized standard presentations involving the notion of “pseudo-isogeny.” It concludes with a detailed discussion of the structure theorem.