Abstract
We propose a novel quantum integrable model for every nonsimply laced simple Lie algebra \({{\mathfrak {g}}}\), which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the Bethe Ansatz equations of the standard integrable model associated with the quantum affine algebra \(U_q(\widehat{{{\mathfrak {g}}}'})\) of the simply laced Lie algebra \({{\mathfrak {g}}}'\) corresponding to \({{\mathfrak {g}}}\). Our construction is motivated by the analysis of the second classical limit of the deformed Walgebra of \({{\mathfrak {g}}}\), which we interpret as a “folding” of the Grothendieck ring of finitedimensional representations of \(U_q(\widehat{{{\mathfrak {g}}}'})\). We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finitedimensional representations of \(U_q({}^L{\widehat{{{\mathfrak {g}}}}})\), where \(^L{\widehat{{{\mathfrak {g}}}}}\) is the (twisted) affine Kac–Moody algebra Langlands dual to \({\widehat{{{\mathfrak {g}}}}}\). We discuss the analogous structures in the Gaudin model which appears in the limit \(q \rightarrow 1\). Finally, we describe a conjectural construction of the simple \({{\mathfrak {g}}}\)crystals in terms of the folded qcharacters.
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Notes
More precisely, there is a homomorphism \({\text {Rep}} U_q({\widehat{{{\mathfrak {g}}}}}) \rightarrow Z_q({\widehat{{{\mathfrak {g}}}}})[[z^{\pm 1}]]\), so that every \(V \in {\text {Rep}} U_q({\widehat{{{\mathfrak {g}}}}})\) gives rise to a formal power series \(T_V(z)\), and the Fourier coefficients of these series topologically generate \(Z_q({\widehat{{{\mathfrak {g}}}}})\), see [25, Section 8.1].
The XXZtype model is already quantum, but here by a deformation we mean a noncommutative deformation of the commutative algebra of quantum Hamiltonians of the XXZtype model. Therefore, it is a kind of “second quantization.”
More generally, we could consider the case when W is a tensor product of simple representations, with the same invariance property for its highest monomial (but not necessarily for each simple factor). However, we will not do so in this paper.
In the case of \({{\mathfrak {g}}}={{\mathfrak {s}}}{{\mathfrak {l}}}_n\), every finitedimensional irreducible representation of \(U_q({{\mathfrak {s}}}{{\mathfrak {l}}}_n)\) can be lifted to \(U_q(\widehat{{\mathfrak {s}}}{{\mathfrak {l}}}_n)\), but the highest monomial of the resulting representation of \(U_q(\widehat{{\mathfrak {s}}}{{\mathfrak {l}}}_n)\) is not \(\sigma \)invariant, see the above example. The representations with \(\sigma \)invariant monomials are generally much larger.
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Acknowledgements
We thank M. Aganagic, P. Koroteev, A. Okounkov, and A. Zeitlin for useful discussions. E.F. and D.H. were partially supported by a grant from the FranceBerkeley Fund of UC Berkeley. N.R. was partially supported by the NSF grant DMS1902226.
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Appendix: Toward constructing a folded version of the qKZ equations for nonsimply laced Lie algebras
Appendix: Toward constructing a folded version of the qKZ equations for nonsimply laced Lie algebras
The qKZ equations [27] for \(U_q({\widehat{{{\mathfrak {g}}}}})\) (with the twist parameter u, which is an element of the Cartan subgroup H as above), can be written in the form
Here, each \(V_i\) denotes an irreducible finitedimensional representation of \(U_q({\widehat{{{\mathfrak {g}}}}})\), \(V_i(z_i)\) is its shift by the spectral parameter \(z_i\), and \(R^{V_jV_i}_{j,i}(z_j/z_i)\) is the Rmatrix acting on \(V_j(z_j) \otimes V_i(z_i)\) normalized so that it acts as the identity on \(v_j \otimes v_i\), where \(v_j\) and \(v_i\) are highest weight vectors in \(V_j(z_j)\) and \(V_i(z_i)\), respectively. Also, \(u_j\) denotes \(u_{V_j}\).
Let us denote the operator on the righthand side of the jth equation by \(K_j(p)\).
The critical level limit corresponds to \(p \rightarrow 1\). In this limit, we have
Recall that for an auxiliary representation V of \(U_q({\widehat{{{\mathfrak {g}}}}})\) we have the transfer matrix \(t_V(z,u)\). These transfermatrices commute with each other for a fixed u and different V and z. In what follows, we will use the same notation \(t_V(z,u)\) for the action of the transfermatrix \(t_V(z,u)\) on the tensor product \(V_N(z_N) \otimes \ldots \otimes V_1(z_1)\). Thus,
where the subscript a indicates the auxiliary representation V(z).
Proposition 10.1
Suppose that \(V_j\) is such that up to a scalar, \(R^{V_jV_j}(1) = P\), the permutation operator on \(V_j(z) \otimes V_j(z)\) sending \(x \otimes y\) to \(y \otimes x\). Then, \(K_j(1) = t_{V_j}(z_j,u)\).
Proof
If \(R^{V_jV_j}(1) = P\), then formula (10.3) with \(V(z) = V_j(z_j)\) becomes
Using the identify \(A_a P_{aj} = P_{aj} A_j\), we rewrite the RHS of (10.4) as
Next, using the cyclic property of the trace, we rewrite (10.5) as:
Using formula \(A_a P_{aj} = P_{aj} A_j\) again, we rewrite (10.6) as:
In the last formula, the only operator depending on the auxiliary space is \(P_{aj}\) and its trace over the auxiliary space is the identity operator on \(V_j(z_j)\). Hence, (10.7) is equal to \(K_j(1)\). \(\square \)
Next, we discuss under what conditions \(R^{VV}(1) = P\).
Lemma 10.2
Let V be an irreducible representation of \(U_q({\widehat{{{\mathfrak {g}}}}})\) such that \(V \otimes V\) is also irreducible. Then, \(R^{VV}(1) = P\).
Proof
It follows from the definition that \(P \circ R^{VV}(1)\) is an intertwining operator \(V \otimes V \rightarrow V \otimes V\). If \(V \otimes V\) is irreducible, then by Schur’s lemma it is a scalar operator. Under our normalization, it then has to be the identity operator, and hence \(R^{VV}(1) = P\). \(\square \)
Note that a representation V satisfying the condition of Lemma 10.2 is called real in [34]. Not all irreducible representations of \(U_q({\widehat{{{\mathfrak {g}}}}})\) are real, as shown in [44], see [34, Sect. 13.6]. However, there is a large class of real representations: Kirillov–Reshetikhin modules.
Proposition 10.3
Let V be any Kirillov–Reshetikhin module over \(U_q({\widehat{{{\mathfrak {g}}}}})\). Then, \(V \otimes V\) is irreducible.
Proof
There are several possible arguments. It follows from [5] that \(V\otimes V\) is cyclic and cocyclic, and hence is irreducible. The statement also follows from the fact that the square of the qcharacter of a Kirillov–Reshetikhin module has a unique dominant monomial; namely, its highest monomial. The latter follows from the results of [35, Sect. 3.2.2]. In addition, the statement has been established by a different method in [39]. \(\square \)
Note however that even if \(V \otimes V\) is reducible, we may still have \(R^{VV}(1) = P\). It would be interesting to describe all irreducible representations V of \(U_q({\widehat{{{\mathfrak {g}}}}})\) satisfying the condition \(R^{VV}(1) = P\).
In any case, the above results readily imply that in the case when all \(V_j\)’s are Kirillov–Reshetikhin modules, the operators \(K_j(1), j=1,\ldots ,N\), are commuting Hamiltonians of the XXZtype integrable model associated with \(U_q({\widehat{{{\mathfrak {g}}}}})\). It is in this sense that we say that one recovers this integrable model in the critical level limit of the qKZ equations.
Similarly, under the above condition on \(V_j\), the operator \(K_j(p)\) is the transfermatrix \(t_{V_j}(z_j,u)\) acting on
However, because of the shifts by \(p^{1}\) these operators do not commute with each other if \(p \ne 1\).
Now suppose that \({{\mathfrak {g}}}\) is a nonsimply laced simple Lie algebra. We would like to construct a “folded qKZ system” such that in the critical level limit the operators on the right hand side become the Hamiltonians of the folded quantum integrable model described in this paper. This means, in particular, that they should correspond to transfermatrices of \(U_q(\widehat{{{\mathfrak {g}}}'})\) rather than \(U_q({\widehat{{{\mathfrak {g}}}}})\). Thus, each \(V_i(z_i)\) should be a representation of \(U_q(\widehat{{{\mathfrak {g}}}'})\).
Unfortunately, naive attempts to construct this folded qKZ system appear to fail:

(1)
For each \(V_i(z_i)\), we have its subspace \((V_i(z_i))(u)\) defined as above and we can take the tensor product of these subspaces, \(\otimes _{i=1}^N (V_i(z_i))(u)\). However, it is not clear why this subspace would be preserved by the operators \(K_j\) or their pdeformed versions \(K_j(p)\) appearing on the RHS of (10.1).

(2)
We take the subspace V(u) of the entire tensor product \(V=\otimes _{i=1}^N V_i(z_i)\). According to Conjecture 5.15,(ii), it contains a subspace isomorphic to a representation M(V) of \(U_q(^L\widehat{{{\mathfrak {g}}}})\). Moreover, since the algebra of transfermatrices of \(U_q(\widehat{{{\mathfrak {g}}}'})\) commutes with the Baxter operators \(Q_j(z), j \in I'\), it follows that all transfermatrices of \(U_q(\widehat{{{\mathfrak {g}}}'})\) preserve this subspace V(u). In particular, the operators \(K_j\) given by formula (10.2), being the transfermatrices of \(U_q(\widehat{{{\mathfrak {g}}}'})\), preserve V(u). But the problem is that on the righthand side of these qKZ equations we have the operators \(K_j(p)\) with \(p\ne 1\), which are the transfer matrices acting on the tensor product of representations in some of which (namely, the ones with \(i=1,\ldots ,j1\)) there is a multiplicative shift in the spectral parameter by \(p^{1}\). It is not clear why these operators should preserve V(u) (where V is the tensor product of the representations \(V_i(z_i)\) without any shift by \(p^{1}\)).
Hence, at the moment it is unclear to us how to fold the qKZ equations for \(U_q(\widehat{{{\mathfrak {g}}}'})\) in such a way that in the critical level limit we recover the commuting Hamiltonians of the folded quantum integrable model associated with \(U_q({\widehat{{{\mathfrak {g}}}}})\).
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Frenkel, E., Hernandez, D. & Reshetikhin, N. Folded quantum integrable models and deformed Walgebras. Lett Math Phys 112, 80 (2022). https://doi.org/10.1007/s11005022015658
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DOI: https://doi.org/10.1007/s11005022015658
Keywords
 Quantum affine algebra
 Deformed Walgebra
 Integrable model
 Gaudin model
 Bethe Ansatz
 qcharacter
Mathematics Subject Classification
 Primary: 81R10
 81R50
 82B23
 Secondary: 17B37
 17B80