definition of quotient Lie algebra of Lie algebra by ideal of Lie algebra
Topics
About: Lie algebra
The table of contents of this article
Starting Context
- The reader knows a definition of Lie algebra.
- The reader knows a definition of ideal of Lie algebra.
- The reader knows a definition of quotient vectors space of vectors space by vectors subspace.
Target Context
- The reader will have a definition of quotient Lie algebra of Lie algebra by ideal 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:
\( V'\): \(\in \{\text{ the Lie algebras }\}\)
\( V\): \(\in \{\text{ the ideals of } V'\}\)
\(*V' / V\): \(= \text{ the quotient vectors space }\), with \([\bullet, \bullet]: V' / V \times V' / V \to V' / V\), \(\in \{\text{ the Lie algebras }\}\)
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Conditions:
\(\forall [v'_1], [v'_2] \in V' / V ([[v'_1], [v'_2]] = [[v'_1, v'_2]])\)
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2: Note
"\([[v'_1], [v'_2]] = [[v'_1, v'_2]]\)" may look confusing, but in "\([[v'_1], [v'_2]]\)", the inner brackets denote the equivalence classes in \(V' / V\) and the outer bracket denotes the bracket operation on \(V' / V\), while in "\([[v'_1, v'_2]]\)", the inner bracket denotes the bracket operation on \(V'\) and the outer bracket denotes the class in \(V' / V\): there should not be any other valid interpretation.
Let us see that the bracket on \(V' / V\) is indeed well-defined.
Let \([v'_1] = [v''_1]\) and \([v'_2] = [v''_2]\).
\(v''_1 = v'_1 + v_1\) and \(v''_2 = v'_2 + v_2\) for some \(v_1, v_2 \in V\).
\([[v''_1, v''_2]] = [[v'_1 + v_1, v'_2 + v_2]] = [[v'_1, v'_2] + [v'_1, v_2] + [v_1, v'_2] + [v_1, v_2]] = [[v'_1, v'_2] + [v'_1, v_2] - [v'_2, v_1] + [v_1, v_2]]\), but \([v'_1, v_2], [v'_2, v_1], [v_1, v_2] \in V\) because \(V\) is an ideal, so, \([v'_1, v_2] - [v'_2, v_1] + [v_1, v_2] \in V\), so, \( = [[v'_1, v'_2]]\).
Let us see that the bracket on \(V' / V\) satisfies the conditions for \(V' / V\) to be a Lie algebra.
Let \([v'_1], [v'_2], [v'_3] \in V' / V\) and \(r_1, r_2 \in F\) be any, where \(F\) is the field over which \(V'\) is a vectors space.
1) \([r_1 [v'_1] + r_2 [v'_2], [v'_3]] = [[r_1 v'_1 + r_2 v'_2], [v'_3]] = [[r_1 v'_1 + r_2 v'_2, v'_3]] = [r_1 [v'_1, v'_3] + r_2 [v'_2, v'_3]] = r_1 [[v'_1, v'_3]] + r_2 [[v'_2, v'_3]] = r_1 [[v'_1], [v'_3]] + r_2 [[v'_2], [v'_3]]\); \([[v'_3], r_1 [v'_1] + r_2 [v'_2]] = [[v'_3], [r_1 v'_1 + r_2 v'_2]] = [[v'_3, r_1 v'_1 + r_2 v'_2]] = [r_1 [v'_3, v'_1] + r_2 [v'_3, v'_2]] = r_1 [[v'_3, v'_1]] + r_2 [[v'_3, v'_2]] = r_1 [[v'_3], [v'_1]] + r_2 [[v'_3], [v'_2]]\).
2) \([[v'_2], [v'_1]] = [[v'_2, v'_1]] = [- [v'_1, v'_2]] = - [[v'_1, v'_2]] = - [[v'_1], [v'_2]]\).
3) \(\sum_{cyclic} [[v'_1], [[v'_2], [v'_3]]] = [[v'_1], [[v'_2], [v'_3]]] + [[v'_2], [[v'_3], [v'_1]]] + [[v'_3], [[v'_1], [v'_2]]] = [[v'_1], [[v'_2, v'_3]]] + [[v'_2], [[v'_3, v'_1]]] + [[v'_3], [[v'_1, v'_2]]] = [[v'_1, [v'_2, v'_3]]] + [[v'_2, [v'_3, v'_1]]] + [[v'_3, [v'_1, v'_2]]] = [[v'_1, [v'_2, v'_3]] + [v'_2, [v'_3, v'_1]] + [v'_3, [v'_1, v'_2]]] = [\sum_{cyclic} [v'_1, [v'_2, v'_3]]] = [0]\).
So, \(V' / V\) is s Lie algebra.
