For a symmetrizable generalized Cartan matrix (GCM) AA, the Gabber–Kac theorem identifies the Kac–Moody algebra g(A)\mathfrak{g}(A) with the algebra given by Chevalley generators and Serre relations, which is what makes generators-and-relations computations possible throughout Kac–Moody theory. For a non-symmetrizable AA the Kac–Moody algebra is defined only as a quotient of the Serre algebra by an implicit radical r(A)\mathfrak{r}(A), and nothing is known about that radical: it is open whether r(A)=0\mathfrak{r}(A)=0 for every AA, and no explicit complete set of defining relations beyond the Serre relations is known for any non-symmetrizable AA. The problem asks for a complete, explicit presentation of g(A)\mathfrak{g}(A) for arbitrary non-symmetrizable AA, equivalently a computation of r(A)\mathfrak{r}(A). The quantized setting motivates the question but does not enlarge it: Fang's construction shows that the naive specialization q→1q\to 1 of the quantum-double construction need not be well-defined when the Cartan matrix is not averaged, while his averaged specialization is well-defined and surjective, so the missing presentation theory is precisely what obstructs a non-averaged quantized treatment.

Background

A generalized Cartan matrix (GCM) is an integral square matrix A=(aij)A=(a_{ij}) with aii=2a_{ii}=2, off-diagonal entries aij≤0a_{ij}\le 0, and aij=0a_{ij}=0 exactly when aji=0a_{ji}=0. A GCM is called symmetrizable if it can be rescaled by positive numbers to a symmetric matrix (formally, DADA is symmetric for some diagonal matrix DD with positive entries); every 2×22\times 2 GCM is symmetrizable, and non-symmetrizable matrices first occur in rank 3 — Fang's 3×33\times 3 example below is one. To any GCM one attaches a Lie algebra by generators and relations: start from the free Lie algebra on Chevalley generators ei,fie_i,f_i and a Cartan subalgebra h\mathfrak{h}, impose the Chevalley relations [ei,fj]=δijhi[e_i,f_j]=\delta_{ij}h_i, [h,ei]=⟨h,αi⟩ei[h,e_i]=\langle h,\alpha_i\rangle e_i, [h,fi]=−⟨h,αi⟩fi[h,f_i]=-\langle h,\alpha_i\rangle f_i (where αi∈h∗\alpha_i\in\mathfrak{h}^* are the simple roots and hi∈hh_i\in\mathfrak{h} the corresponding simple coroots) and the Serre relations (ad ei)1−aij(ej)=0(\mathrm{ad}\,e_i)^{1-a_{ij}}(e_j)=0, (ad fi)1−aij(fj)=0(\mathrm{ad}\,f_i)^{1-a_{ij}}(f_j)=0 for i≠ji\ne j; call the result the Chevalley–Serre (or Serre) algebra g~(A)\tilde{\mathfrak{g}}(A). Kac's definition of the Kac–Moody algebra is g(A)=g~(A)/r(A)\mathfrak{g}(A)=\tilde{\mathfrak{g}}(A)/\mathfrak{r}(A), where r(A)\mathfrak{r}(A) is the maximal ideal intersecting h\mathfrak{h} trivially; the quotient is needed because for a non-symmetrizable AA no argument is known that rules out extra relations. The Gabber–Kac theorem (Gabber–Kac 1981; Kac's book, Theorem 9.11) states that for symmetrizable AA the radical r(A)\mathfrak{r}(A) vanishes, so the Serre relations are a complete set of defining relations; its proof uses the invariant symmetric bilinear form attached to a symmetrizable GCM, an object that provably does not exist on the derived algebra when AA is non-symmetrizable. For non-symmetrizable AA the Serre algebra and the Kac–Moody algebra g(A)\mathfrak{g}(A) itself — the contragredient construction, so called because h\mathfrak{h} and its dual act dually on the Chevalley generators — therefore differ a priori, and the literature even carries two conventions for the derived algebra g′(A)\mathfrak{g}'(A): Gao (1994, 2000), studying central extensions and Leibniz homology (the non-antisymmetric analogue of Lie algebra homology introduced by Loday) for non-symmetrizable data, worked with the Serre-style object, while Kac's notation denotes its quotient; Hu–Pei–Liu (2008) noted explicitly that Kac's g′(A)\mathfrak{g}'(A) is the quotient of Gao's and re-derived their cohomological results only for the quotient object. The object asked for here is Kac's quotient g(A)=g~(A)/r(A)\mathfrak{g}(A)=\tilde{\mathfrak{g}}(A)/\mathfrak{r}(A). On the quantized side, the enveloping algebra of a symmetrizable Kac–Moody algebra has a presentation by quantum Serre relations and a specialization map sending the deformation parameter qq to 11; Fang (2014) constructed the candidate qq-deformation for arbitrary GCMs through Nichols algebras (graded braided Hopf algebras B(V)=T(V)/I(V)\mathfrak{B}(V)=T(V)/\mathfrak{I}(V), where I(V)\mathfrak{I}(V) is the defining ideal of a braided vector space VV, i.e. a space whose tensor powers carry an action of the braid groups) of diagonal type — meaning the braid operators act as scalars, vi⊗vj↦qijvj⊗viv_i\otimes v_j\mapsto q_{ij}v_j\otimes v_i — with braiding matrix (qcij)(q^{c_{ij}}). He showed by an explicit counterexample — the GCM (2−2−1−12−1−3−12)\begin{pmatrix}2&-2&-1\\-1&2&-1\\-3&-1&2\end{pmatrix} — that the naive specialization need not be well-defined, and proved that after replacing CC by its averaged (symmetric) matrix Cˉ\bar C with entries cˉij=(cij+cji)/2\bar c_{ij}=(c_{ij}+c_{ji})/2 one does get a well-defined surjective averaged specialization σ:UA1(C)→U(g(Cˉ))\sigma:U_{\mathcal{A}_1}(C)\to U(\mathfrak{g}(\bar C)) onto the enveloping algebra of the averaged matrix's Kac–Moody algebra. Here A=K[q1/2,q−1/2]\mathcal{A}=\mathbb{K}[q^{1/2},q^{-1/2}] over a characteristic-zero field K\mathbb{K} and A1\mathcal{A}_1 is the localization of K[q1/2]\mathbb{K}[q^{1/2}] at (q1/2−1)(q^{1/2}-1), so that half-integer averaged entries cˉij\bar c_{ij} are absorbed by q1/2q^{1/2} in the base ring; Uq(C):=Dq(Cˉ)U_q(C):=D_q(\bar C) is the quantum-double Nichols construction attached to the averaged braiding matrix (qcˉij)(q^{\bar c_{ij}}), its A1\mathcal{A}_1-form UA1(C)U_{\mathcal{A}_1}(C) is generated by Ei,Fi,Ki±1,[Ki;0]E_i,F_i,K_i^{\pm 1},[K_i;0], and g(Cˉ)\mathfrak{g}(\bar C) is defined for an arbitrary matrix via a Kac realization (h,Π,Π∨)(\mathfrak{h},\Pi,\Pi^\vee). The averaging, however, is guaranteed to preserve only a linear isomorphism class of the Nichols algebra, not its algebra structure. Adjacent results that do not touch this core: Joseph–Lamprou (2009) built the crystal B(∞)B(\infty) — the combinatorial skeleton encoding the negative part of the quantized enveloping algebra — for non-symmetrizable data via Littelmann path models (a realization of crystals as piecewise-linear paths in the weight lattice; no algebra presentation), Heckenberger–Zheng (2018) computed root multiplicities of rank-two diagonal Nichols algebras, and Marquis (2019) produced the first failures of Gabber–Kac-type simplicity — but only for complete Kac–Moody groups (the maximal, completed groups attached to a GCM by the Tits functor) over finite fields of positive characteristic, leaving the characteristic-zero Lie algebra problem open.

Problem Statement

Determine the defining relations of the Kac–Moody algebra g(A)\mathfrak{g}(A) attached to an arbitrary non-symmetrizable generalized Cartan matrix AA. Concretely, the problem — the non-symmetrizable Gabber–Kac problem, open since Kac's construction and GCM theory in the 1980s — asks, with g~(A)\tilde{\mathfrak{g}}(A) the Chevalley–Serre algebra and r(A)\mathfrak{r}(A) its maximal ideal intersecting the Cartan subalgebra trivially, to compute r(A)\mathfrak{r}(A) explicitly: decide whether r(A)=0\mathfrak{r}(A)=0 for every GCM AA (in which case the Gabber–Kac theorem extends and Serre relations are always complete), or, where r(A)≠0\mathfrak{r}(A)\ne 0 for a non-symmetrizable AA, give a complete explicit generating set for r(A)\mathfrak{r}(A), thereby furnishing a full presentation of g(A)\mathfrak{g}(A) beyond the Serre relations. In either case the target is the presentation problem for arbitrary non-symmetrizable AA: exhibiting one non-symmetrizable matrix together with a generating set for that matrix alone does not settle the problem, since what is asked is a computation of r(A)\mathfrak{r}(A) — equivalently a complete explicit presentation of g(A)\mathfrak{g}(A) — that settles the general non-symmetrizable case. The quantized material of Background (Fang's counterexample to well-definedness of the naive specialization, his averaged specialization, and the Nichols-ideal questions he cites) motivates the problem and records known obstructions, but it is not part of the required completion boundary.

References

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  2. Victor G. Kac, Infinite-Dimensional Lie Algebras, 3rd edition, Cambridge University Press, Cambridge, 1990. ISBN 9780521466936; online ISBN 9780511626234, https://doi.org/10.1017/CBO9780511626234
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