Notes on Affine Hecke Algebras I.
(Degenerated Hecke Algebras and Yangians in Mathematical Physics)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 23 April 2014

Notes and References

This is an excerpt of the paper Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics) by Ivan Cherednik.

Yangians

These algebras are the basic example of quantum groups. I think that they (and their $q\text{-analogs)}$ should be more important for mathematics and physics than $q\text{-analogs}$ of universal enveloping algebras being now in common use. You can find some mathematical arguments in favor of Yangians in [Che1989]. Here I will try to demonstrate only that they are physically natural and give us another interpretation of Yang's $S\text{-matrix}$ as an interwiner.

To introduce (explain) quantum groups one can follow Faddeev's ideology (the quantum inverse scattering method - [Fad1984]) or its particular case - Drinfeld's way [Dri1987]. Faddeev's point of view (as far as I understand it) is that a quantum group is more or less equivalent to the corresponding Bethe-ansatz $\text{(}R\text{-matrix's}$ or not). To be more precises, it should be some hidden composition law of the latter. For Drinfeld the main prolem was to extend a given classical $r\text{-matrix}$ to the quantum one. I'll try to explain here that it is quite possible to come to quantum groups without the concept of $R\text{-matrices}$ and the inverse scattering technique.

The very first step for any scheme of quantization of a given Lie group $G$ (or its Lie algebra $g\text{)}$ is to place at each point $z$ of some space-time the generators $\left\{g_{\alpha }\right\}$ of $g$ with the natural commutation relations $$\begin{array}{cc}[g_{\alpha }\left(z\right),g_{\beta }\left(z\prime \right)]=\sum _{\gamma }{c}_{\alpha \beta }^{\gamma }{g}_{\gamma }\left(z\right)\delta (z-z\prime )& \text{(25a)}\end{array}$$ where $[g_{\alpha },g_{\beta }]=\sum _{\gamma }{c}_{\alpha \beta }^{\gamma }{g}_{\gamma }$ in $g\text{.}$ The r.h.s of (25a) can, in principle, have Schwinger and other terms. Let $g=g{\ell }_N$ (i.e. $g$ is $M_N$ considered as a Lie algebra) $\left\{g_{\alpha }\right\}=\{e_{k\ell },1\le l,\ell \le N\},$ where $e_{k\ell }=1^{k\ell }=\left({\delta }_i^k{\delta }_j^{\ell }\right)$ has the only unit at place $(i,j)\text{.}$ The natural way to introduce states and observables is based on some initial representations $V$ of $g\text{.}$ Let $V$ be ${ℂ}^N$ with the standard action of $g{\ell }_N\text{.}$

The first problem is to define the tensor product $𝒱={\otimes }_{z}V\left(z\right)$ over all points of the space-time, where $V\left(z\right)$ is ${ℂ}^N$ at $z:$ $$\begin{array}{cc}{e}_{k\ell }\left(z\prime \right)v\left(z\right)=\left(e_{k\ell }v\right)\left(z\right)\delta (z-z\prime )\hspace{0.17em}\text{for}\hspace{0.17em}v\left(z\right)\in V\left(z\right)\text{.}& \text{(25b)}\end{array}$$ To solve it one should choose some vacuum state and consider only such states, that are "close" to the vacuum (see works on von Neumann factors). The second problem is to introduce an algebra of observables $𝒜$ operating in $𝒱$ (see e.g. [FRS1989,MSc1990]). The pair $\{𝒱,𝒜\}$ is a quantum group by definition.

Elements of $𝒜$ can be expressed in terms of $\left\{e_{k\ell }\left(z\right)\right\}\text{.}$ But one should avoid to include $\left\{e_{k\ell }\left(z\right)\hspace{0.17em}\text{in}\hspace{0.17em}𝒜\right\}\text{.}$ The latter is to be the least to make $𝒱$ irreducible with respect to the action of $𝒜\text{.}$ The last (obscure enough) property and other similar principles give one some intuition. But, in fact, it is impossible to differ good and bad $𝒜$ without dealing with concrete physical problems.

Assume that the space-time is finite (written $z=1,\cdots ,n\text{).}$ First of all, it is natural to include in $𝒜$ the elements ${\sum }_{z=1}^n{e}_{k\ell }\left(z\right)$ for any $k,\ell \text{.}$ One can add ${\sum }_{z=1}^{n-1}{\sum }_{k,\ell =1}^N{e}_{k\ell }\left(z\right)e_{\ell k}(z+1)$ to them (the hamiltonian for the Heisenberg ferromagnet or the so-called XXX-model). In the Bardeen, Cooper, Schrieffer (BCS) theory of superconductivity the hamiltonian of the following type (for $N=2,$ $k=\ell \text{)}$ is important: $$u\sum _{z=1}^n{e}_{k\ell }\left(z\right)+\sum _{z,z\prime =1}^n\sum _{m=1}^N{e}_{km}\left(z\right)e_{m\ell }\left(z\prime \right)\text{.}$$

Summarizing, we see that linear combinations of operators $$\sum _z{c}_1\left(z\right)e_{k\ell }\left(z\right),\sum _{z,z\prime }{c}_2(z,z\prime )\sum _m{e}_{km}\left(z\right)e_{m\ell }\left(z\prime \right),\hspace{0.17em}\sum _{z,z\prime ,z″}{c}_3(z,z\prime ,z″)\sum _{m,r}{e}_{km}\left(z\right)e_{mr}\left(z\prime \right)e_{r\ell }\left(z″\right),\cdots$$ for some scalar functions $c_1,c_2,c_3,\cdots$ and every $1\le k,\ell \le N$ are natural candidates to incorporate.

Of course, it is possible to consider analogous elements with two or more matrix free indices in place of $(k,\ell )\text{.}$ But, generally speaking, $𝒜$ is already big enough without them. We will show below that for the simplest $c$ the only above elements form an algebra acting irreducibly on $𝒱\text{.}$ Although such more complicated combinations can be significant for another choice of $c\text{.}$

In our definition below the points $z,z\prime ,z″,\cdots$ will be ordered $\text{(}c\ne 0$ only for $z<z\prime <z″\cdots \text{).}$ If one changes the order he well get another algebra of observables and some other representation isomorphic to the initial pair. The corresponding interwiner will be precisely Yang's $S\text{.}$

In fact, this interpretation of $S$ is dual to the above one (by means of ${\mathscr{H}}_n^{\prime }\text{).}$ We note that some points are in Yang's paper (Phys. Rev. 168 (1968)), which are close to our approach to Yangians.

Case $n=2\text{.}$ Formulas (25) show that we can use the tensor notations from sec. 2: $$𝒱=V\otimes V,\hspace{0.17em}{e}_{k\ell }\left(z\right)={}^z{e}_{k\ell },\hspace{0.17em}V={ℂ}^N,\hspace{0.17em}z=1,2\text{.}$$ Let us consider $𝒱$ as a module under the action of the algebra generated by $e_{k\ell }^0=e_{k\ell }\left(1\right)+e_{k\ell }\left(2\right)=e_{k\ell }\otimes 1+1\otimes e_{k\ell }$ and $e_{k\ell }^1=u_1{e}_{k\ell }\left(1\right)+u_2{e}_{k\ell }\left(2\right)+\sum _{m=1}{e}_{mk}\otimes e_{\ell m}$ for all $1\le k,\ell \le N\text{.}$ Simple calculations give that $V$ is irreducible for $u=u_2-u_1\ne \pm 1\text{.}$

Thus, $𝒜$ is big enough to make $𝒱$ irreducible (for a generic $u\text{).}$ However, $𝒜$ is not very big. Namely, it is not far from $g{\ell }_N$ operating on $𝒱$ by $\left\{e_{k\ell }^0\right\},$ since for special $u=1$ (respectively $u=-1\text{)}$ the symmetric $S^{2}V$ (external ${\mathrm{\Lambda }}^{2}V\text{)}$ square of $V$ is the only $𝒜\text{-submodule}$ of $𝒱\text{.}$ The idea is to define quantum groups (Yangians) like this $𝒜$ but for any initial representations and $n\text{.}$

The aim of the next general definition is to make $V^{\otimes n}$ irreducible (for some generic parameters) but not to loose the classic theory of decomposing of $V^{\otimes n}$ under the diagonal action of $g{\ell }_N\text{.}$ For some special values of parameters we should reproduce in terms of $𝒜$ the classic results like the decomposition $V^{\otimes 2}=S^{2}V\oplus {\mathrm{\Lambda }}^{2}V$ above.

Let us use the rational function in $\lambda \in ℂ$ $$\begin{array}{cc}L\left(\lambda \right)=1+\sum _{r,k,\ell }{\lambda }^{-r}{E}_{k\ell }^{r-1}{1}^{\ell k},& \text{(26)}\end{array}$$ where $1\le r\le n,$ $1\le k,\ell \le N,$ $1^{\ell k}$ is $e_{\ell k}$ considered as $N\times N\text{-matrix}$ (see above). Letters $E_{k\ell }^{r-1}$ are assumed to be pairwise non-commuting. E.g. for $n=1,$ $N=2$ $$L\left(\lambda \right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+{\lambda }^{-1}\left(\begin{array}{cc}{E}_{11}^0& E_{21}^0\\ E_{12}^0& E_{22}^0\end{array}\right)\text{.}$$ Although $L$ is a matrix with non-commutative matrix elements we can use multi-index notations from sec. 2. In particular, ${}^{1}L=L\otimes 1,$ ${}^{2}L=1\otimes L\text{.}$ Let us impose on $\left\{E_{k\ell }^{r-1}\right\}$ the Yang-Baxter-Faddeev relation $$\begin{array}{cc}R({\lambda }_1-{\lambda }_2){}^{1}L\left({\lambda }_1\right){}^{2}L\left({\lambda }_2\right)={}^{2}L\left({\lambda }_2\right){}^{1}L\left({\lambda }_1\right)R({\lambda }_1-{\lambda }_2)& \text{(27)}\end{array}$$ for any ${\lambda }_1,{\lambda }_2\in ℂ,$ where (see (8)) $$\begin{array}{cc}R\left(\lambda \right)=PS\left(\lambda \right)=\lambda +P& \text{(28)}\end{array}$$ One can show directly that (27) for Yang's $R$ is equivalent to the system of the following relations: $$\begin{array}{cc}\begin{array}{ccc}[E_{ij}^r,E_{k\ell }^s]& =& E_i^{r+s}{\delta }_j^k-E_{kj}^{r+s}{\delta }_i^{\ell }\\ & +& \sum _{a+b=r+s-1}^{a<r\le b}(E_{kj}^a{E}_{i\ell }^b-E_{kj}^b{E}_{i\ell }^a)\text{.}\end{array}& \text{(29)}\end{array}$$ The quotient-algebra of the algebra of non-commutatitve polynomials in $\left\{E\right\}$ by relations (29) is called the yangian of level $n$ for $g{\ell }_N$ (written ${𝒴}_N^n\text{).}$ See [Dri1986,Che1986].

Given a set $u=(u_1,\cdots ,u_n)$ consider $e_{k\ell }\left(r\right)={}^r{e}_{k\ell }$ acting on the corresponding components of $𝒱=V^{\otimes n}(V={ℂ}^N)$ and put $$\begin{array}{ccc}{\stackrel{\sim }{L}}_u\left(\lambda \right)& =& (1+\frac{1}{\lambda -u_n}\sum _{k,\ell }{e}_{k\ell }\left(n\right)1^{\ell k})(1+\frac{1}{\lambda -u_{n-1}}\sum _{k,\ell }{e}_{k\ell }(n-1)1^{\ell k})\cdots \\ & \cdots & (1+\frac{1}{\lambda -u_1}\sum _{k,\ell }{e}_{k\ell }\left(1\right)1^{\ell k})\prod _{r=1}^n\left(\frac{\lambda -u_r}{\lambda }\right)\text{.}\end{array}$$ Here $\left\{1^{\ell k}\right\}$ commute with $\left\{e\right\}$ and determine "the position" of $e_{\ell k}\left(r\right)$ in the corresponding $N\times N\text{-matrix.}$ This $L$ is a function in $\lambda$ having its values in $N\times N\text{-matrices}$ with the matrix elements from the algebra $M_N^{\text{<mo>⊗</mo><mi>n</mi>}}$ generated by $\{e\left(r\right),1\le r\le n\}\text{.}$ It is convenient for the sake of more invariant writings to denote $1^{\ell k}$ by $e^{\ell k}\left(0\right)$ or ${}^0{e}_{\ell k}\text{.}$ Then $${\stackrel{\sim }{L}}_u\left(\lambda \right)={}^{0n}R(\lambda -u_n){}^{0\hspace{0.17em}n-1}R(\lambda -u_{n-1})\cdots {}^{01}R(\lambda -u_1){\lambda }^{-n},$$ where $R$ is from (28): $R\left(\lambda \right)=\lambda 1+\sum _{k,\ell }{e}_{k\ell }\otimes e_{\ell k}\text{.}$ It results directly from (8) that ${\stackrel{\sim }{L}}_u\left(\lambda \right)$ is a solution of equation (27), Hence, the corresponding ${\stackrel{\sim }{E}}_{k\ell }^{r-1}$ $(1\le r\le n)$ from the decomposition of ${\stackrel{\sim }{L}}_u\left(\lambda \right)$ (see (26)) give us the representation ${𝒴}_N^n\ni E_{k\ell }^{r-1}\to {\stackrel{\sim }{E}}_{k\ell }^{r-1}\in M_N^{\text{<mo>⊗</mo><mi>n</mi>}}=\text{End}\left(𝒱\right)$ of ${𝒴}_N^w$ in $𝒱$ (written $𝒱\left(u\right)\text{).}$ Two simple examples:

a)	${\stackrel{\sim }{E}}_{k\ell }^0=e_{k\ell }\left(1\right)+\cdots e_{k\ell }\left(n\right)-(u_1+\cdots +u_n)1,$
b)	the operators $e_{k\ell }^0,e_{k\ell }^1$ for $n=2$ (see above) are some linear combinations of ${\stackrel{\sim }{E}}_{k\ell }^0,{\stackrel{\sim }{E}}_{k\ell }^1$ modulo $1\text{.}$

Thereom 4 a) The space $𝒱\left(u\right)$ is an irreducible ${𝒴}_N^n\text{-module}$ if and only if $u_i-u_j\ne 1$ for every $1\le i,j\le n\text{.}$ For $w\in S_n$ one has $$\begin{array}{cc}{R}_w\left(u\right)L_u\left(\lambda \right)=L_{w\left(u\right)}\left(\lambda \right)R_w\left(u\right),& \text{(30)}\end{array}$$ where $R_w\left(u\right)=P_w{S}_w\left(u\right),$ $S_w$ is from (15). In particular, if $𝒱\left(u\right)$ is irreducible then the mapping $$𝒱\left(u\right)\ni x\to R_w\left(u\right)x,{\stackrel{\sim }{E}}_{k\ell }^{r-1}\to R_w\left(u\right){\stackrel{\sim }{E}}_{k\ell }^{r-1}{R}_w^{-1}\left(u\right)$$ is an isomorphism from $𝒱\left(u\right)$ onto $𝒱\left(w\left(u\right)\right)\text{.}$

Let us prove identity (30). Consider fig. 5. Let us calculate the corresponding $S\text{-matrices}$ (see (13) and fig. 4). One has $$\begin{array}{ccc}& & {}^{12}S\left(u_{13}\right){}^{01}S\left(u_{12}\right){}^{23}S(\lambda -u_3){}^{12}S(\lambda -u_2){}^{01}S(\lambda -u_1)\\ & =& {}^{23}S(\lambda -u_1){}^{12}S(\lambda -u_3){}^{01}S(\lambda -u_2){}^{23}S\left(u_{13}\right){}^{12}S\left(u_{12}\right)\text{.}\end{array}$$ The simple rule of turning the latter into its $R\text{-matrix}$ version is as follows. The upper left indices should be changed to coincide with the indices of the arguments. We obtain the identity $${}^{13}R{}^{12}R\left({}^{03}R{}^{02}R{}^{01}R\right)=\left({}^{01}R{}^{03}R{}^{02}R\right){}^{13}R{}^{12}R,$$ where the arguments are omitted. Here ${}^{13}R\left(u_{13}\right){}^{12}R\left(u_{12}\right)={}^{13}{P}^{13}S\left(u_{13}\right){}^{12}P{}^{12}S\left(u_{12}\right)=P_w{}^{23}S\left(u_{13}\right){}^{12}S\left(u_{12}\right)=R_w\left(u\right)$ for $w=(3,1,2)=s_2{s}_1\text{.}$ The products in brackets are $L_u\left(\lambda \right)$ and $L_{w\left(u\right)}\left(\lambda \right)\text{.}$

Let us compare the corresponding mappings of theorem 4 and theorem 2. The latter is the right multiplication by $S_w\left(\mathrm{\Theta }\right)\text{.}$ The first is the conjugation by $R_w\left(u\right)=P_w{S}_w\left(u\right)\text{.}$ The identification of $\mathrm{\Theta }$ and $u$ makes it evident that these two should be very closely connected. In particular, they are degenerated (i.e. $S_w\left(\mathrm{\Theta }\right),$ $R_w\left(u\right)$ are non-invertible) for the same values of $\mathrm{\Theta }=u\text{.}$ Moreover, $𝒱\left(u\right)$ and $M\left(\mathrm{\Theta }\right)$ are simultaneously irreducible.

We will not discuss here the precise mathematical statements (see [Dri1986,Che1987]). Roughly speaking, the ${𝒴}_N^n\text{-submodules}$ of $𝒱\left(u\right)$ are in one-to-one correspondence with ${\mathscr{H}}_n^{\prime }\text{-submodules}$ of $M\left(\mathrm{\Theta }\right)$ for $N>n\text{.}$ The degeneration of $S_w\left(\mathrm{\Theta }\right)$ and $R_w\left(u\right)$ for the same parameters is the particular case of this correspondence. Practically, if one can describe the submodules of $M\left(\mathrm{\Theta }\right),$ he can construct all the submodules of $𝒱\left(u\right)\text{.}$ Of course, the first problem is more convenient to settle.

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