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.
I shall try to interpret this mathematical trick as some quantization procedure. We will look for observables which correspond to (I shall remind that is the momentum of the particle. Therefore, a quantization of angles is, in fact, a quantization of impulses).
Yang's (17) is of a very symmetric type. Any in-state for (see (1,2)) can be expressed in terms of where for permutations Hence, it is natural to diminish the space of states. We fix and some initial set of indices Let Here and play different roles. The orderings of are indexed by They are in one-one correspondence with the "sectors" — the connected components of Therefore, we will use the visual name "sector" in place of "ordering". The states (for every given sector) are numbered by By definition,
The natural (but wrong) idea is to introduce the quantum angles by relations If were independent it might be possible. But they are linearly dependent. However, one can try the following: Given we define the action of only on the "vacuum" states for each sector. Let us assume that and all in are pairwise distinct (fox example Then the number of sectors (i.e. orderings of is equal to the number of states for each of them. Therefore, the definition (23) is, in principle, consistent. If the set is not "generic" we should be more precise (we will not consider this case here).
All the sectors are glued together by the (see (15)). In particular, Identifying with with and with we obtain the basis of eigenvectors for in the group algebra All these are true for being in a general position only. Simple calculations show that and satisfy the relations (18-19), where acts on by left multiplications (cf. [Che1986-2]). Deduce this statement from (23').
We have collected the vacuum states (they linearly generate all the states) together in the space of states for the initial sector Starting with other sectors we will obtain some isomorphic representations of The will be interwiners between these "sector" representations. In fact, this interpretation of is very close to the ideology of superselection sectors (see [FRS1989,MSc1990]). We will not discuss here the latter, but formulate the corresponding mathematical theorem. As a matter of fact it has been partially proven.
Theorem 2 (see [Rog1985,Che1986-2]). Given let us denote by the space with the natural left (regular) action of and the action of which can be uniquely determined by means of the following relations Then is an irreducible for being in a general position for any The operator gives an isomorphism which appears to be a
This theorem is in fact equivalent to theorem 1. Indeed, any considered as a function of with its values in is equal to (coincides with after the substitution where all the should be collected on the right). Therefore, the isomorphism above is a direct corollary of statement (b) of theorem 1. The irreducibility of is clear, because form a basis in of eigenvectors with respect to with pairwise distinct eigenvalues (for in a general position).
It is worth mentioning that is more natural than from some other physical point of view. Let us summarize its "quantum" properties.
|(a)||The subalgebras is a maximal commutative in i.e. the commutant (centralizer) of coincides with|
|(b)||The centre (commutant) of consists of all symmetric polynomials in (due to I. Bernstein).|
|(c)||Haag-duality. The commutant of where is generated by and is equal to generated by (see e.g. [Che1987]).|
Compare (c) with the corresponding axiom from [FRS1989]. As for the commutant of modulo the centre is more than complimentary
Let us consider (see above) with the relations in place of (19a). Here plays the role of the Planck constant For we get the algebra which is the "quasi-classical" limit of and especially simple. For example, it is evident that any element can be represented in the form for appropriate polynomials Moreover, if for some then for any such that Indeed, The latter is impossible. In particular, the subalgebra coincides with its commutant in
There is a nice mathematical trick to extend the above statement to any sufficiently close to To calculate this commutant for any one should solve linear equations for coefficients of the polynomials in the decomposition If the commutant contains with for some then certain determinants of minors are to be equal to zero (and vice versa). To be more precise, given the rank of the above system for polynomials of degree for such is less than the corresponding rank for The determinants are scalar polynomial functions in Some of them do not equal zero at for Hence, they have no common zeroes not only at but in a neighbourhood of Therefore, and in this neighbourhood. We have proved the required statement for small But any for is isomorphic to (see above). Hence, the coincidence of and its commutant holds true for arbitrary as well.
The best way to prove (a, b, c) is to use the following statement. each element has the unique representation: where are some rational function in are from theorem 1.