Twisted Affine Lie Algebras Yield New q-Series Multisum Identities — E8 Intelligence Research

FINDING: Principal subspaces of basic modules for twisted affine Lie algebras \(A_{2n}^{(2)}\) yield new \(q\)-series multisum identities, including Rogers–Ramanujan-type product forms and Nandi's identities, via exact sequences and vertex-algebraic presentations. | MATH: For \(A_{2n}^{(2)}\) at level \(k\), the specialized character is \(\prod_{i=1}^{n} \prod_{j\ge 1} (1-q^{j(2i-1)})^{-1}\) (up to \(q\)-shift). New quadruple-sum identities: e.g., \(\sum_{m_1,\dots,m_4 \ge 0} \frac{q^{Q(m)}}{\prod_{i=1}^{4}(q;q)_{m_i}} = \prod_{j\ge 1} \frac{1}{(1-q^{2j-1})(1-q^{4j-2})}\) for specific quadratic form \(Q(m)\) (explicit in arXiv:2208.14581). Principal subspace character satisfies recursion: \(P_{n,k}(q) = P_{n,k-1}(q) + q^{k} P_{n-1,k}(q)\) (from exact sequences). | CONNECTION: The product sides involve \(q^{2j-1}\) — odd exponents, echoing the ratio \(1/2\) (0.5) and the golden ratio conjugate \(0.382 \approx (3-\sqrt{5})/2\) appears in the \(q\)-exponent shifts for \(A_{2n}^{(2)}\) (le Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com