Geodesic
Construction of eigenfunctions for a~system of quantum minors of the~monodromy matrix for an~$SL(n,\mathbb C)$-invariant spin chain
Teoretičeskaâ i matematičeskaâ fizika, Tome 189 (2016) no. 2, pp. 149-175

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We consider the problem of seeking the eigenvectors for a commuting family of quantum minors of the monodromy matrix for an $SL(n,\mathbb C)$-invariant inhomogeneous spin chain. The algebra generators and elements of the $L$-operator at each site of the chain are implemented as linear differential operators in the space of functions of $n(n{-}1)/2$ variables. In the general case, the representation of the $sl_n(\mathbb C)$ algebra at each site is infinite-dimensional and belongs to the principal unitary series. We solve this problem using a recursive procedure with respect to the rank $n$ of the algebra. We obtain explicit expressions for the eigenvalues and eigenvectors of the commuting family. We consider the particular cases $n=2$ and $n=3$ and also the limit case of the one-site chain in detail.
Keywords: Yang–Baxter equation, intertwining operator, Yangian, separation of variables.
Mots-clés : $R$-matrix 
@article{TMF_2016_189_2_a0,
     author = {P. A. Valinevich and S. \`E. Derkachev and P. P. Kulish and E. M. Uvarov},
     title = {Construction of eigenfunctions for a~system of quantum minors of the~monodromy matrix for an~$SL(n,\mathbb C)$-invariant spin chain},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {149--175},
     publisher = {mathdoc},
     volume = {189},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_189_2_a0/}
}
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P. A. Valinevich; S. È. Derkachev; P. P. Kulish; E. M. Uvarov. Construction of eigenfunctions for a~system of quantum minors of the~monodromy matrix for an~$SL(n,\mathbb C)$-invariant spin chain. Teoretičeskaâ i matematičeskaâ fizika, Tome 189 (2016) no. 2, pp. 149-175. http://geodesic.mathdoc.fr/item/TMF_2016_189_2_a0/