Let A be an associative algebra. (a) Show that for any a E A the map da : A A defined by da(b) = ab – ba is a derivation. Such derivations are called inner derivations. (b) Show that inner derivations form an ideal in Der(A). (c) Show that any derviation of the associative algebra A = Mat,(C) of complex n x n matrices is inner. %3D
Let A be an associative algebra. (a) Show that for any a E A the map da : A A defined by da(b) = ab – ba is a derivation. Such derivations are called inner derivations. (b) Show that inner derivations form an ideal in Der(A). (c) Show that any derviation of the associative algebra A = Mat,(C) of complex n x n matrices is inner. %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![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 A over F endowed with a bilinear operation
Ax A →
A, usually denoted by juxtaposition (unless A is a Lie algebra, in
which case we always use the bracket). By a derivation of A 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 A of all derivations of A is a vector
subspace of End A. 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(A).
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 x e L, y [xy] is an
endomorphism of L, which we denote ad x. In fact, ad x e Der L, because
we can rewrite the Jacobi identity (using (L2')) in the form: [x[yz]]
+[y[xz]]. Derivations of this form are called inner, all others outer. It is of
course perfectly possible to have ad x =
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.
= [[xy]z]
0 even when x + 0: this occurs
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 adx or
adgx will be used to indicate that x is acting on L (respectively, K). For
example, if x is a diagonal matrix, then ad»n,F) (x)
need not be zero.
0, whereas ada(n,F)(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ebb9219-e1b9-4a64-9ea9-f4066c39787f%2F230ab44b-8844-495f-b74e-79beed9eb6d6%2Fma4nz3b_processed.jpeg&w=3840&q=75)
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 A over F endowed with a bilinear operation
Ax A →
A, usually denoted by juxtaposition (unless A is a Lie algebra, in
which case we always use the bracket). By a derivation of A 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 A of all derivations of A is a vector
subspace of End A. 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(A).
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 x e L, y [xy] is an
endomorphism of L, which we denote ad x. In fact, ad x e Der L, because
we can rewrite the Jacobi identity (using (L2')) in the form: [x[yz]]
+[y[xz]]. Derivations of this form are called inner, all others outer. It is of
course perfectly possible to have ad x =
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.
= [[xy]z]
0 even when x + 0: this occurs
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 adx or
adgx will be used to indicate that x is acting on L (respectively, K). For
example, if x is a diagonal matrix, then ad»n,F) (x)
need not be zero.
0, whereas ada(n,F)(x)
Transcribed Image Text:Let A be an associative algebra.
(a) Show that for any a E A the map da : A → A defined by da(b) = ab – ba is a
derivation. Such derivations are called inner derivations.
(b) Show that inner derivations form an ideal in Der(A).
(c) Show that any derviation of the associative algebra A
n x n matrices is inner.
= Mat,(C) of complex
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