Classification of classical r-matrices

Th e o r e m 2.13.1. Evei'y complex simple Lie bialgebra is coboundary and specified by

an element r G g 0 g which may be written as r = + kC where r^ is an element of

g A g and k, is a scalar such that r satisfies the classical Yang-Baxter equation [[r, r]] = 0.

Such an element r is then called a classical r-matrix.

We may now reflect that in Example 2.12.2 we ignored the quasitriangularity of U/i(s(2(C)) when we considered its semi-classical limit. If we had taken it into account, then the definition of the co-commutator could have been written as

S(x) = - -A ") ~ (2.13.2)

from which we would have obtained the results,

^(X) = X > r, (2.13.3)

(5(F) = F > r, (2.13.4)

S(H) = JÎ > r, (2.13.5)

where r G 512(C) 0 512(C) appears in the expansion S l— l-^-hr mod h?.

By virtue of the fact that Dl obeys the quantum Yang-Baxter equation, r obeys the classical Yang-Baxter equation. Explicitly, it is given by

r = ^ H ® H + X ® Y , (2.13.6)

and borrowing terminology from the quantum case we say that r defines (512(C), 5) as a quasitriangular Lie bialgebra. Indeed we have just seen that every complex simple Lie bialgebra is quasitriangular which in a sense ‘explains’ the quasitriangularity of the standard QUEAs.

A subset of topological quasitriangular Hopf algebras is the topological triangular Hopf algebras. These have universal i?-matrices which satisfy — 3 where J = 101. At the semi-classical level this is also a useful sub-division. The complex simple triangular Lie bialgebras are precisely those whose r-matrices are given by r =

2.14. Classification of classical r-matrices

We have already seen that there are standard quasitriangular QUEAs associated with all complex simple Lie algebras. The corresponding standard Lie bialgebras are de­ scribed in terms of the so-called Drinfeld-Jimbo classical r-matrices, r^^. These are non-triangular solutions of the classical Yang-Baxter equation. If I) is the Cartan subal­ gebra of g, we denote by Cq the restriction of the Casimir to 6 0 [3. Then we can choose a basis element E^ for each root subspace g^ such that the Drinfeld-Jimbo r-matrices are given by

= iCo + ^ 0 E.„. (2.14.1)

qSA+

These are by no means the only classical r-matrices for complex simple Lie algebras — there are many ‘nonstandard’ examples. The complete classification consists of two distinct classifications for the non-triangular and triangular cases respectively.

In the case of the non-triangular r-matrices the classification is due to Belavin and Drinfeld. To state their result we need a preliminary definition.

De f in it io n 2.14.1. A quadruple (H o,H i,r ,s ) where IIo and IIi are subsets of the

set n of sim ple roots, r : Ho —> IIi a one-to-one m apping and s e b A 1r is said to be

admissible if it satisfies the following three conditions: 1. (r((a),r(/5)) = (a,/?) for all a,/? G Hq.

2. For every ck G Ho, there is an n such that a, r(o;),... , G Ilo but r^{a) ^ Hq. 3. (t(q;) 0 l)(|(7o + s) + (1 0 ci')(^Co + s) = 0 for all œ G fig.

Let us also introduce an ordering of the positive roots such that a < (3 if = r'^(a)

for some m > 0 (r is extended linearly). The result of Belavin and Drinfeld may now be stated:

T h e o r e m 2.14.2. //(IIo , H i, r, s) is an admissible quadruple then a Cartan- Weyl basis of the root subspaces of g can be chosen such that

r = r^"^ -f- s + Ea A Ep (2.14.2)

a , 0 e A + , 0 < a

is a non-triangular classical r-matrix and moreover every non-triangular classical r-matrix is equivalent via a Lie algebra automorphism to one of this form.

For 512(C) we see that the only non-triangular r-matrix is the Drinfeld-Jimbo solution. Somewhat more interesting is the situation for 5(3 (C). In this case there is also a non­ standard non-triangular r-matrix. For reasons which should become clear in Chapter 3 we call it the Cremmer-Gervais r-m atrix and denote it by r^^. Explicitly

= + I hc,AH/} + E ^ A E .h, (2.14.3) where a, are the simple roots and the notation for the Lie algebra elements is that of Chapter 1.

It is worth setting this result beside the following observation. For any quasitriangular Lie bialgebra, (g, r), and given any element / G g 0 g satisfying

[[/21 — /12, /21 - /12I] + [[/21 — /12, f']] + [[a /21 — /12I] = 0 (2.14.4) (g,r 4- /21 — /12) is also a quasitriangular Lie bialgebra. We say that (g,r 4- j'21 — /12) is related to (g, r) through a semiclassical twist by /. Notice that it follows that every quasitriangular Lie bialgebra is related via such a twisting to the trivial quasitriangular Lie bialgebra (g, 0), so that in fact all the quasitriangular Lie bialgebras associated with a given Lie algebra are related amongst themselves through semiclassical twists. In par­ ticular, in the case of the Cremmer-Gervais r-matrix for 5(3(0), we know that r^'^ and

r ^ ^ are related through a semiclassical twist / = ^ H a 0 / / / ? 4 - E g 0 E ^ p . Of course these

observations are rather trivial. We mention them because the quantum counterparts of these twists are far from trivial, and indeed are far from being completely understood. One of the main results of Chapter 3 is actually a construction of a ‘quantisation’ of this twist.

The classification of the triangular solutions of the classical Yang-Baxter equation was achieved some time after the Belavin-Drinfeld result bv Stolin. For details we refer the