Kassel. Quantum groups (Springer, 1995)(T)(539s).djvu: eKnigu

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Kassel. Quantum groups (Springer, 1995)(T)(539s).djvu

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This means that, for any triple (U, V, W) of objects of C, there exists an
isomorphism
auvw ;(U ®V)®W ^U ®{V <&W) B.4)
such that the square
(U®V)®W ^X^ U ®{V ® W)
(ftg>g)*g)h f<3(g®h) B.5)
commutes whenever f,g,h are morphisms in the category
282 Chapter XI...
The first and last equalities follow from E.3), the second and fifth ones
from the naturality of the associativity constraint, i.e., from Relation B.5),
the third from the Pentagon Axiom B.6), and the fourth one from the
induction hypothesis...
We shall apply this method in Section 5 to
exhibit explicit isotopy invariants that, will allow us to complete the proof
of Theorem X.4.2 asserting the existence of the Jbnes-Conway polynomial...
Therefore their images under F obtained after per-
performing the substitutions indicated above are identical...
Proof, (a) It suffices to prove the first statement for the generators <7j,...,
<Tn_\ of B,,...
D.7)
Relation D.7) is equivalent to
(id,, ®(^)-1)(c±rViV)x (id, fc/OOv.vC*)* = id,,,, D8)
Taking transposes and using Lemma 11.3.3, we see that D.8) is equivalent
to D.1.c)...
A.8)
This implies that the natural isomorphism cvv is a solution of the Yang-
Baxter equation for any object V of a braided tensor category...
Henceforth, we shall consider the set N as the
set of objects of the strict tensor category B...
We now prove that the commutativity of (Cn 2) and of (Crl m) implies
the commutativiiy of (Cn m + 1)...
D,2)
The naturality in Definition 4.1 means that the square
X®V -^* V ® X
|/«idv idv®/ D.3)
Y®V ^^ V®Y
commutes for any morphism / : X -> Y in C...
From now on the symbol = displayed in the figures means
that the corresponding morphisms are equal in C...
One can check using Reidemeister
Transformations (I ) and (II) that if L is a ribbon with s(L) = b(L), then
its quantum trace tiq(L) is the closure of L drawn in Figure 5.2...
A)(F23)(id ® id 0 A) ((id 0 A)(F)) (id 0 id ® A)($)
(A ® id <8> id)($)(A <8) id 0 id) ((A 0 id)(F))
(A (8> id (8>id)(F1^1)F1^1
= F34(id<8> id <8> A)(F23)$234(id® A » id) ((id ® A)(F))
$2~34(id(8) id ® A)($)(A @id ®id)($)$^23
(id <8>A(8> id)((A ® id)(F-1) J $123(A 0 id
= F34(id <8> id (8) A)(F23)$234(id 0 A <g) id) ((id ®
(id <8> A (8>id)($)(id<8>A <8> id) ((A <8>id)(F~
$123(A®id®id)(F121)Ff21
= F34(id(8) id (8) A)(F23)$234(id® A ®id)(F231)F2-31
F23(id® A ® id) (F23(id®A)(F)$(A(g)id)(F-1)F1-21JF2-31
F23(id<8) A (8)id)(F12)*123(A @id ®id)(F1V)Ff21
which proves C.4)...
We must check that we have
(P2 o Cy v = Cy y o (P2 ¦ The latter is equivalent to F~1(RFJl = (RF~lJi,
which follows from C.9)...
If f = Y2rt>0 a'i'1™ *s another formal series, then
the sum/+f and the product //' of/ and f'in...
Therefore if
it
A is separated and complete, it defines an element, still denoted f(ha), in
A= A...
It is a left K-module
with a K-linear map fiM : A®M —> M such that
g) = Pu o (idA®fiM) and P M o (ry®idM) = id,...
The last part of Definition 5.1 means
that the unit r\ of A is given by
V(f) = n E.6)
for all /GC[[/i]]...
As an immediate consequence of Proposition 2.1 and of Definition 2.2,
we see that the space of morphisms of K-algebras from A to a separated
complete K-algebra A is in bijectioii with the set of maps f:X—>A such
that / vanishes on R...
Since wean be expressed in such a way that
the generators X, killing the highest weight vector appear to the right of
Kj, we see that the actions of S(u)u and of 0 are the same as the actions
of the elements obtained from the part of Rfi corresponding to ? = 0 in
Formula B.13)...