Twisted Affine Lie Algebras: Principal Subspaces and New Rogers–Ramanujan Identities — E8 Intelligence Research

FINDING: Principal subspaces of basic modules for twisted affine Lie algebras \(A_{2n}^{(2)}\) admit exact sequences and \(q\)-series multisum identities, including new Rogers–Ramanujan-type families and Nandi's identities. | MATH: The key structures are: (1) Principal specialization of characters: \(\chi_{\Lambda_0}(q) = \prod_{i=1}^n (1-q^{2i-1})^{-1} \prod_{i=1}^n (1-q^{2i})^{-1}\) for \(A_{2n}^{(2)}\) at level 1 (up to grading shifts); (2) Exact sequences among principal subspaces yield recursion relations for \(q\)-series coefficients; (3) New quadruple-sum identities of the form \(\sum_{k_1,\dots,k_4 \ge 0} \frac{q^{Q(k)}}{\prod_{j=1}^4 (q;q)_{k_j}}\) where \(Q(k)\) is a quadratic form derived from the Cartan matrix of \(A_{2n}^{(2)}\); (4) Rogers–Ramanujan-type products: \(\prod_{j\ge 1} (1-q^j)^{-a_j}\) with exponents \(a_j\) periodic mod \(2n+1\). | CONNECTION: The root system of \(A_{2n}^{(2)}\) is \(B_n\) (crystallographic, with long roots ratio \(\sqrt{2}\) to short roots). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com