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Unformatted text preview: tric bilinear form on V . Then o(V, f ) is
called the orthogonal Lie algebra relative to f .
(2) Suppose that f is a nondegenerate skew symmetric form on V : f (v, v ) = 0 for all
v ∈ V . (If char F = 2 this is equivalent to the condition that f (v, w) = −f (w, v )
for all v, w.) In this case dim V = 2n (even) and o(V, f ) is called the symplectic
Lie algebra relative to f .
We will look at these examples in more detail later.
Example 1.2.7. Other easy examples of linear Lie algebras are:
(1) t(n, F ) ⊆ gl(n, F ), the set of upper triangular n × n matrices over F
(2) n(n, F ) ⊆ t(n, F ), the set of strictly upper triangular matrices (with 0 on the
(3) d(n, F ) ⊆ t(n, F ), the set of diagonal n × n matrices wi...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11