definition of Lie algebra
Topics
About: Lie algebra
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of field.
Target Context
- The reader will have a definition of Lie algebra.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(F\): \(\in \{\text{ the fields }\}\)
\(V\): \(\in \{\text{ the vectors spaces over } F\}\), with \([\bullet, \bullet]: V \times V \to V\)
//
Conditions:
\(\forall v_1, v_2, v_3 \in V, \forall r_1, r_2 \in F\)
(
1) \([r_1 v_1 + r_2 v_2, v_3] = r_1 [v_1, v_3] + r_2 [v_2, v_3]\) \(\land\) \([v_3, r_1 v_1 + r_2 v_2] = r_1 [v_3, v_1] + r_2 [v_3, v_2]\)
\(\land\)
2) \([v_2, v_1] = - [v_1, v_2]\)
\(\land\)
3) \(\sum_{cyclic} [v_1, [v_2, v_3]] = 0\)
)
//
2: Natural Language Description
Any vectors space, \(V\), over any field, \(F\), with any bracket, \([\bullet, \bullet]: V \times V \to V\), such that for any \(v_1, v_2, v_3 \in V\) and any \(r_1, r_2 \in F\), 1) \([r_1 v_1 + r_2 v_2, v_3] = r_1 [v_1, v_3] + r_2 [v_2, v_3]\) and \([v_3, r_1 v_1 + r_2 v_2] = r_1 [v_3, v_1] + r_2 [v_3, v_2]\); 2) \([v_2, v_1] = - [v_1, v_2]\) 3) \(\sum_{cyclic} [v_1, [v_2, v_3]] = 0\)
3: Note
Inevitably, for each \(v \in V\), \([v, 0] = [0, v] = 0\): \([v, 0] = [v, 0 v + 0 v] = 0 [v, v] + 0 [v, v] = 0 + 0 = 0\); \([0, v] = [0 v + 0 v, v] = 0 [v, v] + 0 [v, v] = 0 + 0 = 0\).
Lie algebra is a not-necessarily-associative algebra: \([r_1 v_1 + r_2 v_2, r'_1 v'_1 + r'_2 v'_2] = r_1 [v_1, r'_1 v'_1 + r'_2 v'_2] + r_2 [v_2, r'_1 v'_1 + r'_2 v'_2] = r_1 (r'_1 [v_1, v'_1] + r'_2 [v_1, v'_2]) + r_2 (r'_1 [v_2, v'_1] + r'_2 [v_2, v'_2]) = (r_1 r'_1) [v_1, v'_1] + (r_1 r'_2) [v_1, v'_2]) + (r_2 r'_1) [v_2, v'_1] + (r_2 r'_2) [v_2, v'_2])\), but the associativity, \([ [v_1, v_2], v_3] = [v_1, [v_2, v_3]]\), is not guaranteed to hold.
