definition of general linear Lie algebra, \(\mathfrak{gl} (V)\)
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 %field name% vectors space.
- The reader knows a definition of %structure kind name% endomorphism.
Target Context
- The reader will have a definition of general linear Lie algebra, \(\mathfrak{gl} (V)\).
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\(*\mathfrak{gl} (V)\): \(= \{f: V \to V \in \{\text{ the linear maps }\}\}\), \(\in \{\text{ the Lie algebras }\}\) with the \(F\) vectors spaces structure and the bracket specified below
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Conditions:
\(\forall f_1, f_2 \in \mathfrak{gl} (V), \forall v \in V, \forall r \in F\)
(
\((f_1 + f_2) (v) = f_1 (v) + f_2 (v)\)
\((r f) (v) = r f (v)\)
\([f_1, f_2] (v) = f_1 \circ f_2 (v) - f_2 \circ f_1 (v)\)
)
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2: Note
Let us see that \(\mathfrak{gl} (V)\) is indeed an \(F\) vectors space.
1) \(\forall f_1, f_2 \in \mathfrak{gl} (V) (f_1 + f_2 \in \mathfrak{gl} (V))\) (closed-ness under addition): \((f_1 + f_2) (r_1 v_1 + r_2 v_2) = f_1 (r_1 v_1 + r_2 v_2) + f_2 (r_1 v_1 + r_2 v_2) = r_1 f_1 (v_1) + r_2 f_1 (v_2) + r_1 f_2 (v_1) + r_2 f_2 (v_2) = r_1 (f_1 (v_1) + f_2 (v_1)) + r_2 (f_1 (v_2) + f_2 (v_2)) = r_1 (f_1 + f_2) (v_1) + r_2 (f_1 + f_2) (v_2)\), so, \(f_1 + f_2 \in \mathfrak{gl} (V)\).
2) \(\forall f_1, f_2 \in \mathfrak{gl} (V) (f_1 + f_2 = f_2 + f_1)\) (commutativity of addition): \((f_1 + f_2) (v) = f_1 (v) + f_2 (v) = f_2 (v) + f_1 (v) = (f_2 + f_1) (v)\).
3) \(\forall f_1, f_2, f_3 \in \mathfrak{gl} (V) ((f_1 + f_2) + f_3 = f_1 + (f_2 + f_3))\) (associativity of additions): \(((f_1 + f_2) + f_3) (v) = (f_1 + f_2) (v) + f_3 (v) = f_1 (v) + f_2 (v) + f_3 (v) = f_1 (v) + (f_2 (v) + f_3 (v)) = f_1 (v) + (f_2 + f_3) (v) = (f_1 + (f_2 + f_3)) (v)\).
4) \(\exists 0 \in \mathfrak{gl} (V) (\forall f \in \mathfrak{gl} (V) (f + 0 = f))\) (existence of 0 element): let \(0\) be the zero map; \(0 \in \mathfrak{gl} (V)\); \((f + 0) (v) = f (v) + 0 (v) = f (v)\).
5) \(\forall f \in \mathfrak{gl} (V) (\exists f' \in \mathfrak{gl} (V) (f' + f = 0))\) (existence of inverse element): let \(f' = - 1 f\); \(f' \in \mathfrak{gl} (V)\); \((f + f') (v) = f (v) + f' (v) = f (v) - 1 f (v) = 0 (v)\).
6) \(\forall f \in \mathfrak{gl} (V), \forall r \in F (r . f \in \mathfrak{gl} (V))\) (closed-ness under scalar multiplication): \((r f) (r_1 v_1 + r_2 v_2) = r f (r_1 v_1 + r_2 v_2) = r (r_1 f (v_1) + r_2 f (v_2) = r_1 r f (v_1) + r_2 r f (v_2) = r_1 (r f) (v_1) + r_2 (r f) (v_2)\).
7) \(\forall f \in \mathfrak{gl} (V), \forall r_1, r_2 \in F ((r_1 + r_2) . f = r_1 . f + r_2 . f)\) (scalar multiplication distributability for scalars addition): \(((r_1 + r_2) f) (v) = (r_1 + r_2) f (v) = r_1 f (v) + r_2 f (v) = (r_1 f) (v) + (r_2 f) (v) = (r_1 f + r_2 f) (v)\).
8) \(\forall f_1, f_2 \in \mathfrak{gl} (V), \forall r \in F (r . (f_1 + f_2) = r . f_1 + r . f_2)\) (scalar multiplication distributability for vectors addition): \((r (f_1 + f_2)) (v) = r (f_1 + f_2) (v) = r (f_1 (v) + f_2 (v)) = r f_1 (v) + r f_2 (v) = (r f_1) (v) + (r f_2) (v) = (r f_1 + r f_2) (v)\).
9) \(\forall f \in \mathfrak{gl} (V), \forall r_1, r_2 \in F ((r_1 r_2) . f = r_1 . (r_2 . f))\) (associativity of scalar multiplications): \(((r_1 r_2) . f) (v) = (r_1 r_2) f (v) = r_1 (r_2 f (v)) = r_1 (r_2 f) (v) = (r_1 (r_2 f)) (v)\).
10) \(\forall f \in \mathfrak{gl} (V) (1 . f = f)\) (identity of 1 multiplication): \((1 f) (v) = 1 f (v) = f (v)\).
Let us see that \([]\) satisfies the conditions to be a bracket.
1) \([r_1 f_1 + r_2 f_2, f_3] = r_1 [f_1, f_3] + r_2 [f_2, f_3]\) \(\land\) \([f_3, r_1 f_1 + r_2 f_2] = r_1 [f_3, f_1] + r_2 [f_3, f_2]\): \([r_1 f_1 + r_2 f_2, f_3] = (r_1 f_1 + r_2 f_2) \circ f_3 - f_3 \circ (r_1 f_1 + r_2 f_2) = r_1 f_1 \circ f_3 + r_2 f_2 \circ f_3 - r_1 f_3 \circ f_1 + r_2 f_3 \circ f_2 = r_1 (f_1 \circ f_3 - f_3 \circ f_1) + r_2 (f_2 \circ f_3 - f_3 \circ f_2) = r_1 [f_1, f_3] + r_2 [f_2, f_3]\); \([f_3, r_1 f_1 + r_2 f_2] = f_3 \circ (r_1 f_1 + r_2 f_2) - (r_1 f_1 + r_2 f_2) \circ f_3 = r_1 f_3 \circ f_1 + r_2 f_3 \circ f_2 - r_1 f_1 \circ f_3 - r_2 f_2 \circ f_3 = r_1 (f_3 \circ f_1 - f_1 \circ f_3) + r_2 (f_3 \circ f_2 - f_2 \circ f_3) = r_1 [f_3, f_1] + r_2 [f_3, f_2]\).
2) \([f_2, f_1] = - [f_1, f_2]\): \([f_2, f_1] = f_2 \circ f_1 - f_1 \circ f_2 = - (f_1 \circ f_2 - f_2 \circ f_1) = - [f_1, f_2]\).
3) \(\sum_{cyclic} [f_1, [f_2, f_3]] = 0\): \([f_1, [f_2, f_3]] + [f_2, [f_3, f_1]] + [f_3, [f_1, f_2]] = [f_1, (f_2 \circ f_3 - f_3 \circ f_2)] + [f_2, (f_3 \circ f_1 - f_1 \circ f_3)] + [f_3, (f_1 \circ f_2 - f_2 \circ f_1)] = f_1 \circ (f_2 \circ f_3 - f_3 \circ f_2) - (f_2 \circ f_3 - f_3 \circ f_2) \circ f_1 + f_2 \circ (f_3 \circ f_1 - f_1 \circ f_3) - (f_3 \circ f_1 - f_1 \circ f_3) \circ f_2 + f_3 \circ (f_1 \circ f_2 - f_2 \circ f_1) - (f_1 \circ f_2 - f_2 \circ f_1) \circ f_3 = f_1 \circ f_2 \circ f_3 - f_1 \circ f_3 \circ f_2 - f_2 \circ f_3 \circ f_1 + f_3 \circ f_2 \circ f_1 + f_2 \circ f_3 \circ f_1 - f_2 \circ f_1 \circ f_3 - f_3 \circ f_1 \circ f_2 + f_1 \circ f_3 \circ f_2 + f_3 \circ f_1 \circ f_2 - f_3 \circ f_2 \circ f_1 - f_1 \circ f_2 \circ f_3 + f_2 \circ f_1 \circ f_3 = 0\).
