Sweedler. Hopf algebras (1969)(T)(350s).djvu: eKnigu

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Sweedler. Hopf algebras (1969)(T)(350s).djvu

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Date Sep 25, 2004

Proof, a) A simple subcoalgebra is irreducible;
so by the theorem each simple subcoalgebra lies in
an irreducible component...
Replacing C by Y,
E by f(Y) and F by Fflf(Y) shows that we may
assume C (and E) are finite dimensional...
M is a coalgebra map, so
by (9-1.4) M((C O C)i+J) c c1+J, hence
C^j = M(Ci ® Cj) c M((C ® CI+J) c C1+J...
These will be of interest in
studying graded bialgebras and in the construction
of universal pointed irreducible coalgebras...
We leave as an exercise the easy verifications
that ttF|c = f and that the uniqueness of F follows
from the uniqueness of the F^'s...
The map given in d) from B(V) ® B(w) to B(u) is
a graded bialgebra map since B(u) is commutative...
Define
- hVii and E)=
for all i if the characteristic is zero and for i<(p
if the characteristic is p > 0...
By the remarks
above a D -module is rational and completely
reducible only if it is rational and completely
reducible over C ...
The pair (UL, i) -
or simply UL - is called a universal enveloping
algebra for L if for any associative algebra A
the map f ¦> fi gives a 1-1 correspondence between
the algebra maps f from UL to A and the Lie maps
from L to A~...