TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
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Contents<br />
Introduction vi<br />
Notations 1<br />
1 Adjoint orbits of sp(2n) and o(m) 3<br />
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
1.2 Nilpotent orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
1.3 Semisimple orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />
1.4 Invertible orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />
1.5 Adjoint orbits in the general case . . . . . . . . . . . . . . . . . . . . . . . . . 16<br />
2 Quadratic Lie algebras 17<br />
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
2.2 Singular quadratic Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 22<br />
2.2.1 Super-Poisson bracket and quadratic Lie algebras . . . . . . . . . . . . 22<br />
2.2.2 The dup number of a quadratic Lie algebra . . . . . . . . . . . . . . . 23<br />
2.2.3 Quadratic Lie algebras of type S1 . . . . . . . . . . . . . . . . . . . . 28<br />
2.2.4 Solvable singular quadratic Lie algebras and double extensions . . . . . 32<br />
2.2.5 Classification singular quadratic Lie algebras . . . . . . . . . . . . . . 36<br />
2.3 Quadratic dimension of quadratic Lie algebras . . . . . . . . . . . . . . . . . . 48<br />
2.3.1 Centromorphisms of a quadratic Lie algebra . . . . . . . . . . . . . . . 48<br />
2.3.2 Quadratic dimension of reduced singular quadratic Lie algebras and the<br />
invariance of dup number . . . . . . . . . . . . . . . . . . . . . . . . . 50<br />
2.3.3 Centromorphisms and extensions of a quadratic Lie algebra . . . . . . 52<br />
2.4 2-step nilpotent quadratic Lie algebras . . . . . . . . . . . . . . . . . . . . . . 54<br />
2.4.1 Some extensions of 2-step nilpotent Lie algebras . . . . . . . . . . . . 54<br />
2.4.2 2-step nilpotent quadratic Lie algebras . . . . . . . . . . . . . . . . . . 56<br />
3 Singular quadratic Lie superalgebras 61<br />
3.1 Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras 61<br />
3.2 The dup-number of a quadratic Lie superalgebra . . . . . . . . . . . . . . . . . 70<br />
3.3 Elementary quadratic Lie superalgebras . . . . . . . . . . . . . . . . . . . . . 73<br />
3.4 Quadratic Lie superalgebras with 2-dimensional even part . . . . . . . . . . . . 76<br />
3.4.1 Double extension of a symplectic vector space . . . . . . . . . . . . . 78<br />
iv