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Preprint Number 1761
1761. Jan DobrowolskiSets, groups, and fields definable in vector spaces with a bilinear form E-mail: Submission date: 15 April 2020 Abstract: We study definable sets, groups, and fields in the theory T_∞ of
infinite-dimensional vector spaces over an algebraically closed field equipped
with a nondegenerate symmetric (or alternating) bilinear form. First, we define
an (ℕ × ℤ,≤_{lex})-valued dimension on definable
sets in T_∞ enjoying many properties of Morley rank in strongly minimal
theories. Then, using this dimension notion as the main tool, we prove that all
groups definable in T_∞ are (algebraic-by-abelian)-by-algebraic, which,
in particular, answers a question of Granger. We conclude that every infinite
field definable in T_∞ is definably isomorphic to the field of scalars
of the vector space. We derive some other consequences of good behaviour of the
dimension in T_∞, e.g. every generic type in any definable set is a
definable type; every set is an extension base; every definable group has a
definable connected component. Mathematics Subject Classification: Keywords and phrases: |

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