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Essential dimension of algebraic groups Meyer, Aurel Nathan


We study the essential dimension of linear algebraic groups. For a group G, essential dimension is a measure for the complexity of G-torsors or, more generally, the complexity of any algebraic or geometric structure with automorphism group G. This makes essential dimension a powerful invariant with many interesting and surprising connections to problems in algebra and geometry. We show that for various classes of groups, including finite (algebraic) groups and algebraic tori, the essential dimension is related to minimal faithful representations. In many cases this renders the exact value of the essential dimension computable and we explore several of its consequences. An important open problem is the essential dimension of the projective linear group PGLn. This topic is closely related to the structure theory of central simple algebras, which may be viewed as twisted forms of the algebra of n x n matrices. We study central simple algebras with additional structure such as a distinguished Galois subfield. We prove new bounds on the essential dimension of these algebras and, as a corollary, of the group PGLn.

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