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Simply Connected, Adjoint and Universal Groups of Lie Type
Author(s) -
Schmid Peter
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008066
Subject(s) - mathematics , general linear group , simple lie group , schur multiplier , group of lie type , pure mathematics , lie group , group (periodic table) , adjoint representation , representation of a lie group , type (biology) , character table , algebraic group , outer automorphism group , linear algebraic group , covering group , algebra over a field , automorphism , alternating group , algebraic number , character (mathematics) , combinatorics , group theory , symmetric group , automorphism group , topology (electrical circuits) , mathematical analysis , geometry , physics , ecology , biology , quantum mechanics , topological group
The special linear group is the simply connected group and the projective linear group is the adjoint group of Lie type A n . They are distinguished sections of the (reductive) general linear group which certainly is of this type as well (root system). We shall characterize the general linear group as the universal group of type A n . Indeed we shall introduce corresponding algebraic groups and finite groups for each Lie type (to indecomposable root systems). Knowledge of the universal group implies knowledge of the related simply connected and adjoint groups; in certain respects the universal group even appears to be better behaved (automorphisms, Schur multiplier, character table).