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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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)
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)
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
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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