Last update: 23 April 2014
This is an excerpt of the paper Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics) by Ivan Cherednik.
Let us discuss the basic example of a factorized (a solution of (6)). The following one is the so-called Yang's Here and further we will denote by the unit matrix in and so on. Setting for any we get an unitary satisfying (6) and (14). In particular, corresponds to a world without any scattering. To verify (6) for (or is an easy exercise. Nevertheless, one can wish to prove (6) without any calculations like it was made for (see above). The best way is to find an interpretation of as a transposition of something interchanges the tensor components). I know four ways to do this. Two of them are based, respectively, on some algebraic geometry (see [Che1979]) and on the theory of the so-called Knizhnik-Zamolodchikov equation from the two-dimensional conformal field theory (see [KZa1984,Che1991]). I will explain here and below only other two making use of (degenerated) affine Hecke algebras and the so-called Yangians [Dri1986].
Mathematically, the idea is simple enough. Let us substitute some operators for in where is from (17). We assume to be pairwise commutative and impose the following conditions Formulas (18) are equivalent to the relations Here and further we will identify with and with Let us check e.g. the first of these formulas. It results from (18a) that Then one can divide (20) by We see that (19) (18), but the converse holds true only for in a "general position".
We have arrived at the following object. Let be the group algebra of with the natural multiplication law: (e.g. It is nothing else but the algebra of formal linear combinations of permutations. Its extension by pairwise commutative symbols with relations (19) is called the degenerated affine Hecke algebra (written It is due to Murphy and Drinfeld (see [Dri1986]). Starting with instead of we get where should stay for in (19a). The algebra isomorphic to for (use the substitution But this is important to understand as some quantum object.
To differentiate from (see (13b)) let us denote the latter by: The main point is that (16) is equivalent to the following identity in To prove the equivalence we need some kind of Wick (or Poincaré-Birckhof-Witt) theorem for One can deduce directly from (19) that each element has the unique representation of the following type: where are some polynomials in Let us denote this sum for after the converse substitution by The only thing we need is to show that and coincide, respectively, with the l.h.s and r.h.s of (16). Let us carry all the in (21) over by means of (18) from the left to the right. Then one obtains and instead of in the l.h.s of (21) and the same elements but in the opposite order in the r.h.s. These differences are exactly what we need. By the way, in the natural notations is involutive in i.e. (cf. (14)). The next theorem (see [Che1987], proposition 3.1 and [Rog1985]) results directly from (21) and (18).
Theorem 1. The collection of elements from extends uniquely to the set with the following properties:
Here (a) is in fact (15). This property can be deduced from (b) (or (18)). Formulas (21), (6) are particular cases of this property. Therefore, the Yang-Baxter relation for Yang's is a direct consequence of the definition of Let us discuss this point.
We see that the l.h.s and r.h.s of (a) induce (operating by conjugations) the same permutation of Hence, the product should be equal to modulo multiplications by some elements from the centralizer (commutant) of in It is not difficult to prove that this centralizer coincides with the algebra of polynomials of (see theorem 3). In particular, (6) for Yang's has to be true up to a multiplication by a scalar function in (use the Wick theorem for Then it is easy to get (6) from this weaker statement. Thus, we have verified, in principle, the Yang-Baxter identity without any calculations. Only by means of formula (b), which is the definition of