Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Verify that the commutator of two derivations of an F-algebra is again a derivation, whereas the ordinary product need not be. i have attached a hint in the image
![1.3. Lie algebras of derivations
Some Lie algebras of linear transformations arise most naturally as
derivations of algebras. By an F-algebra (not necessarily associative) we
simply mean a vector space 2 over F endowed with a bilinear operation
A A A. usually denoted by juxtaposition (unless 2 is a Lie algebra, in
which case we always use the bracket). By a derivation of 2 we mean a linear
map 8: A→ A satisfying the familiar product rule 8(ab) = a8(b)+8(a)b. It
is easily checked that the collection Der 2 of all derivations of 2 is a vector
subspace of End 2. The reader should also verify that the commutator
[8. 8'] of two derivations is again a derivation (though the ordinary product
need not be, cf. Exercise 11). So Der A is a subalgebra of gl(2).
Since a Lie algebra L is an F-algebra in the above sense, Der L is defined.
Certain derivations arise quite naturally, as follows. If xe L, y → [xy] is an
endomorphism of L, which we denote ad x. In fact, ad xe Der L, because
we can rewrite the Jacobi identity (using (L2')) in the form: [x[yz]] = [[xy]z]
+[y[xz]]. Derivations of this form are called inner, all others outer. It is of
course perfectly possible to have ad x = 0 even when x 0: this occurs
in any one dimensional Lie algebra, for example. The map L→ Der L
sending x to ad x is called the adjoint representation of L; it plays a decisive
role in all that follows.
Sometimes we have occasion to view x simultaneously as an element of
L and of a subalgebra K of L. To avoid ambiguity, the notation ad₁x or
adkx will be used to indicate that x is acting on L (respectively, K). For
example, if x is a diagonal matrix, then ado(n,F) (x) = 0, whereas adgi(n,F)(x)
need not be zero.](https://content.bartleby.com/qna-images/question/19616a01-3b9f-4ded-9626-43dcd5e68cc7/63f92f88-cf88-46a2-a7bf-e6716f790cd0/gsk2soa_processed.jpeg)
Transcribed Image Text:1.3. Lie algebras of derivations
Some Lie algebras of linear transformations arise most naturally as
derivations of algebras. By an F-algebra (not necessarily associative) we
simply mean a vector space 2 over F endowed with a bilinear operation
A A A. usually denoted by juxtaposition (unless 2 is a Lie algebra, in
which case we always use the bracket). By a derivation of 2 we mean a linear
map 8: A→ A satisfying the familiar product rule 8(ab) = a8(b)+8(a)b. It
is easily checked that the collection Der 2 of all derivations of 2 is a vector
subspace of End 2. The reader should also verify that the commutator
[8. 8'] of two derivations is again a derivation (though the ordinary product
need not be, cf. Exercise 11). So Der A is a subalgebra of gl(2).
Since a Lie algebra L is an F-algebra in the above sense, Der L is defined.
Certain derivations arise quite naturally, as follows. If xe L, y → [xy] is an
endomorphism of L, which we denote ad x. In fact, ad xe Der L, because
we can rewrite the Jacobi identity (using (L2')) in the form: [x[yz]] = [[xy]z]
+[y[xz]]. Derivations of this form are called inner, all others outer. It is of
course perfectly possible to have ad x = 0 even when x 0: this occurs
in any one dimensional Lie algebra, for example. The map L→ Der L
sending x to ad x is called the adjoint representation of L; it plays a decisive
role in all that follows.
Sometimes we have occasion to view x simultaneously as an element of
L and of a subalgebra K of L. To avoid ambiguity, the notation ad₁x or
adkx will be used to indicate that x is acting on L (respectively, K). For
example, if x is a diagonal matrix, then ado(n,F) (x) = 0, whereas adgi(n,F)(x)
need not be zero.
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