Quantum Algebra
Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
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For translation-invariant prime-qudit codes the Levin-Wen formula now gives exact topological entropy without boundary overestimation.
· “Resolving spurious topological entanglement entropy in stabilizer codes”
The maps extend the known derivation property to general parameters and determine the automorphism group of the related superalgebra.
· “Derivations on the triplet W-algebras with mathfrak{sl}₂-symmetry”
The result unifies known real gauges and predicts them in self-dual Chern-Simons theories.
· “Reality and Complexity of F-symbols in 2+1d Topological Phases”
No Hamiltonian or categorical data needed: six axioms make lattice wave functions string-net ground states.
Inverse level matrices flow to mirror SCFTs; the duality wall reproduces the E8_1 character.
· “3d mathcal N=4 rank-zero mirror symmetry, TQFT interfaces, and Zagier duality of Nahm sums”
Every q-real converges at least as fast as the golden one, proving a q-analogue of Hurwitz's theorem.
· “Coefficients of q-real numbers: their combinatorial meaning and growth”
A tableau-to-web recipe gives a non-recursive basis for quantum sl_n invariants; for sl_3, face depths pick out the leading term.
The bar involution plus quasi-R-matrix gives real F-matrices for every simple Lie algebra and level, but the braiding does not descend.
The recursion over weight-diagram arrows yields explicit matrices for all four polarizations of the quantum supergroup.
A full classification for affine gl and osp superalgebras, with matrix invariants as precursors of supersymmetric K-matrices.
New superfield formulas embed N=1 SUSY affine vertex algebras in free fields; at level zero the map fails.
· “Free field realizations of N=1 SUSY affine vertex algebras”
A cocycle twist keeps a property-(T) projection 1/2 away from every irreducible character.
· “A Counterexample To Universal Character Density For Compact Quantum Groups”
A chain-locality condition rules out non-product endpoint symmetries, so unknown unitaries are identifiable up to fewer gauges.
· “Markov Constraints Enhance Identifiability in Quantum Shadow Inversion”
For N=4,5 the hexagon web is a direct sum of indecomposable foam objects; general N is conjectured.
· “6-valent vertex in the mathfrak{gl}_N web category and its categorification”
Extends the known fusion-2-category state sum and distinguishes homotopy classes invisible to homotopy groups.
· “Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds”
Finite length is proved, so the Grothendieck group is an equivariant copy of the dual subregular left-cell module.
· “Grothendieck Group of the Level -1 Type D: Gaps in Jin's Proof and a Complete Proof”
The center is Sym(L3)-modules; the paper derives the Cartan matrix, cohomology, quiver, and ten p=5 objects.
Exact quaternionic error correction, in the same picture, reduces to real compression coefficients.
The refined quantum Hikita conjecture is proved for ADE quivers with minuscule framings and for the Jordan quiver.
The limit ties quantum state sums to classical geometry for knot and cusp complements.
· “mathrm {U}_(qtilde q)mathfrak{sl}(2;mathbb R) Turaev-Viro invariants for cusped 3-manifolds”
A colimit of equivariant coherent sheaves has Grothendieck ring equal to the integral quantum Cartan algebra, with its coproduct…
· “A categorification of the integral form of the Cartan subalgebra for quantum sl(2)”
A mutation-free seed recipe places C_{w,v}'s Grothendieck ring inside cluster-algebra bounds.
· “Monoidal seeds of the categories mathcal{C}_(wv) over quiver Hecke algebras”
The new divisibility implies the exponent conjecture for every group-theoretical semisimple Hopf algebra over any field.
· “Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras”
The largest quantum symmetry group erases same-size differences, while finer quantum equivalences still separate matrices.
· “Complex Hadamard Matrices - Quantum Symmetries, Equivalence and Non-Local Games”
A 2x2 determinant, checked by exact rational arithmetic, forces a^38=0 and puts the algebra in the strongly rational c=1 family.
· “C₂-Cofiniteness and Rationality of the Icosahedral Orbifold V_(L₂)^(A₅)”
Multiplet counts 3, 10, 35 come from a Weyl-group ratio; each quiver arrow is a differential operator.
· “Langlands Duality and Invariant Differential Operators: the Case SL(2n+1)”
Representation-theoretic functors reproduce the operators and prove their eigenfunctions are orthogonal.
In higher Hilbert spaces, every 'linear functional' is represented by an object, uniquely up to isometry.
No odd Jacobi diagram survives AS and IHX through four loops, so knot inversion remains undetectable.
A commutative Frobenius algebra End(B) accounts for the vacuum degeneracy and rescales torus partition functions by its dimension.
· “A Comment On Topological Degeneracy In Gauged WZW Models”
For S3 in characteristic 3, torus block spaces in degrees 1 and 2 mod 4 cannot be built from degree zero.
· “Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups II”
A probability textbook argues the semicircle law is the natural first continuous distribution for students heading into quantum physics.
State spaces are derived Homs into powers of the canonical coend; degree zero recovers the non-semisimple Reshetikhin–Turaev theory.
· “3-dimensional TQFTs from derived categories of quantum group representations”
The q-binomial multiplication rule becomes an actual direct-sum decomposition of equivariant vector bundles.
Twisted differential forms compose without transversality; middle cohomology matches Kähler holomorphic quantization.
The new invariant also yields a compact move-generating set and a Vassiliev invariant of order one.
An explicit induction functor makes Hopf Koszul duality a monoidal equivalence on the localizing subcategory.
· “Monoidal structures arising from B_infty-algebras with applications to Hopf algebras”
Every candidate algebra has a zero-character; the true skein algebra has none, so no isomorphism exists.
· “A Disproof of Santharoubane's Conjecture on Presentations of Generic Skein Algebras”
A spanning-tree trick controls red loops, and skein modules bound the TFT's state spaces.
Cyclotomic q-web categories realize quantum affine gl_p, so their decomposition matrices come from canonical bases.
For every dg manifold, Hochschild and Cartan cohomology carry the same operations, uniting two classical Duflo-type theorems.
· “Duflo--Kontsevich-type isomorphisms for Tamarkin--Tsygan calculi of dg manifolds”
If the theory is reflection positive, the defect bicategory is nearly a 3-Hilbert space with a spherical weight.
· “Topological defects in reflection positive topological field theories”
Resolves a conjecture: compact quantum groups with all scaling automorphisms inner must have tracial Haar measure.
The boundary Casimir monodromy is the iota-quantum Weyl group action for $(so_m⊂sl_m)$, confirming Conjecture 1.1.
A root-of-unity category now powers TQFTs that can distinguish lens spaces.
· “Relative modular categories from mathfrak{osp}(2 vert 2n)”
Pure braids become signed reflection-subword models for Catalan, parking, and z=0 HOMFLYPT counts.
· “Noncrossing Combinatorics, the Full Twist, and Decategorification of Knot Invariants”
A rank-6 Z/2Z x Z/2Z-quadratic fusion category with fusion multiplicities below 20 must have one of nine Grothendieck rings, and one new…
A recursive coalgebra proof turns the splitting criterion into an explicit coordinate change for vector fields and homotopy algebras.
Proof routes real symmetries through complex ones; anti-unitary time-reversal symmetries now fit the same framework.
· “Classification of symmetric fusion categories over mathbb{R}”
One trace-form identity pins down Jacobson, nilradical, and solvable radicals under Hopf actions.
· “Associative/Lie radical (co)invariance under tracial (co)actions”
Unital composition algebras in Ver_4^+ are either explicit 2- or 4-dimensional algebras or ordinary algebras over the field.
A monoidal equivalence with four integer parameters unifies two module categories over BiHom-Hopf algebras.
· “BiHom-four-angle Hopf modules and BiHom-Yetter-Drinfel'd modules”
For genus 3 and up, a trivial Torelli action forces the category to be pointed; genus 2 is a separate, richer case.
· “Modular fusion categories with trivial Torelli group actions”
A geometric invariant is now set for every known example; the cube-root case yields 90 points, then 36, then none.
· “Point modules of Nichols algebras over non-abelian groups”
Eleven equivalent characterizations unify coring duality; Theorem B separates it from Ringel self-duality homologically.
· “Comparing self-dual corings, Frobenius corings and Ringel self-duality”
Hyperbolic torus bundles with equal GL_N skein dimensions at every rank still differ in their SL3 empty-skein summand.
Categories related by zesting have Reshetikhin-Turaev link invariants that are polynomial-time equivalent.
· “Zesting and the relative complexity of Reshetikhin-Turaev invariants”
Twisted and untwisted sectors match TY(R, χ_-, +1), the first direct-integral fusion computation for a conformal net.
Zhu's algebra presentation turns the classification into a two-parameter family plus a finite list at admissible levels.
Maps Taylor coefficients of Feynman-integral function classes to q-analogues, pinned by recurrences and q-shifts.
· “The q-extension of iterated integrals and nested sums in quantum field theory”
Their full cohomology reduces to invariants times an exterior algebra on Frobenius-twisted simple summands.
· “Chevalley-Eilenberg cohomology of linearly reductive Lie algebras in the Verlinde category”
The localized image is exactly residue-cancelling elements; conical branches carry a sphere trace.
One twisted-centrality equation tells whether a cowreath is Frobenius, and classification over E(n) and Clifford algebras follows.
· “On Separable and Frobenius Cowreaths of type (A otimes H^(op), H,psi)”
Canonical graph integrals link mixed Tate motives over Z to the moduli space of abelian varieties.
· “The motivic Lie algebra embeds into the cohomology of the general linear group”
For a lattice vertex algebra over a field of characteristic p, the radical is generated by Schur modes whose index p does not divide.
· “Lattice vertex algebras over a field of positive characteristic: degenerate cases”
Its oriented subgroup hits all almost-classical links; colorings give unitary representations
Explicit braces of order 16 and p^4 break the Three Subobjects Lemma, blocking local algebraic cartesian closedness
Connected Hopf algebras built from Ore steps stay built that way under subalgebras, quotients, and coideal subalgebras.
Projective subsets make classical definitions transfer verbatim and restore lost dualities
An adjunction of enveloping algebras survives first-order post-product twists and becomes an equivalence
· “An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence”
A monad is gabi exactly when the internal hom lifts to its algebra category; Hopf monads are the invertible-mate case.
Why care: every quasimodular form is uniquely built from series labeled by partitions of width three.
A mirror-plus-strands calculus gives a combinatorial handle on modules of quiver-with-involution Yangians.
Every graph parameter of exponentially bounded edge-connection rank (with f(∅)=1) is a mixed partition function, with color dimensions…
This equality makes canonical graph integrals single-valued multiple zeta values, and every such value a Feynman period.
· “Graph integrals, Feynman periods, and single-valued multiple zeta values”
Split Short Five and pullbacks of split epimorphisms now cover quasitriangular Hopf algebra slices.
For finite folded types, the two quantum bases are linked by a triangular matrix with ones on the diagonal.
· “Construction of PBW and canonical bases in the folding extension algebras for symmetrizable types”
For compact Calabi-Yau manifolds, unimodular holomorphic Poisson structures get a canonical noncommutative deformation.
Two proofs show local annulus replacements leave the induced surface map unchanged—and thus can't reveal exotic surfaces in the 4-ball.
Completes the Arakawa–Moreau conjecture by proving all remaining cases: D_l at level -1, D4/E6/E7/E8 for n>0.
· “Completing the Arakawa--Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras”