The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
Classifies semisimple pairs in complex and quaternionic hyperbolic spaces.
problem Classifying semisimple pairs in Lie groups.
method Using configuration spaces and conjugacy classes.
result Local parametrization of representations of semisimple pairs.
Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.
problem Proving the existence and uniqueness of harmonic metrics on Higgs bundles.
method Analyzing Higgs bundles equipped with a non-degenerate symmetric pairing on non-compact Riemann surfaces.
result Proving the existence and uniqueness of compatible harmonic metrics under certain conditions.
The paper generalizes Hodge theory to semisimple local systems and proves a geometric Decomposition theorem.
problem Generalizing Hodge theory to semisimple local systems.
method Establishing a canonical isomorphism and proving a global invariant cycle theorem.
result A new geometric proof of the Decomposition theorem for semisimple local systems.
Study uses Vinberg pairs for Higgs bundles, revealing their role.
problem Understanding the role of Vinberg pairs in Higgs bundle theory.
method Exploring Vinberg pairs defined by cyclic gradings of a Lie algebra in Higgs bundle theory.
result Vinberg pairs play a significant role in Higgs bundle theory.
We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-…
We describe a class (called regular) of invariant generalized complex structures on a real semisimple Lie group G. The problem reduces to the description of admissible pairs (\gk, ω), where \gk is an appropriate regular subalgebra of the complex Lie algebra \gg^{C} associated to G and ωis a closed 2-form on \gk, such t…
For a Lie group G, we seek the right definition of a "moment space" for G. One axiom is clear, involving a closed equivariant three-form. We construct this form for symmetric spaces associated to a symmetric pair (H,G) with an additional structure. Furthermore, we prove a decomposition theorem for these pairs over a co…
Let g be a vector space and [,],[,]′ be a pair of Lie brackets on g. By definition they are compatible if [,]+[,]′ is again a Lie bracket. Such pairs play important role in bihamiltonian and r-matrix formalisms in the theory of integrable systems. We propose an approach to a long standin…
New invariant calculates 4-manifolds using trisection diagrams and combings.
problem Calculating non-semisimple 4-manifold invariants.
method Using trisection diagrams and combings of the trisection surface.
result Invariant calculated for Stein nuclei, generalizing earlier semisimple version.
New proof and formula linking fusion trees to quantum knot invariants.
problem Quantum knot invariants encoding in non-semisimple TQC.
method Connection between fusion trees and Lawrence representations, using graphical calculus.
result Explicit encoding of quantum knot invariants via fusion trees.
New infinite-dimensional representations with bounded multiplicity found for Lie groups.
problem Finding representations with bounded multiplicity for Lie groups.
method Proving existence of infinite-dimensional irreducible representations with bounded multiplicity property.
result Infinite-dimensional irreducible representations with bounded multiplicity found for non-compact semisimple Lie groups.
Extends Kostant's results to symmetric pairs in Clifford algebras.
problem Analyzing k-invariants in Clifford algebras of symmetric pairs. method Proves Cartan theorem, transgression theorem, Harish-Chandra isomorphism, and Clifford algebra conjecture for relative case.
result Establishes a relative transgression theorem and Harish-Chandra isomorphism for Clifford algebras.
New algebraic fundamental groups identified for fake projective planes.
problem Characterizing algebraic fundamental groups of fake projective planes.
method Analysis of complex conjugate pairs and explicit finite étale covers.
result Forty-six distinct isomorphism classes of algebraic fundamental groups.
Let p be a Lie subalgebra of a semisimple Lie algebra g and (G,P) be the corresponding pair of connected Lie groups. A Cartan geometry of type (G,P) associates to a smooth manifold M a principal P-bundle and a Cartan connection, and a parabolic geometry is a Cartan geometry where P is parabolic. We show t…
Study real structures on Higgs pairs over Klein surfaces, proving a correspondence.
problem Real structures on Higgs pairs over Klein surfaces.
method Establish Hitchin-Kobayashi correspondence, homeomorphism between moduli spaces.
result Real G-Higgs bundles appear as fixed points of involutions. Defines rho numbers for metrics with positive scalar curvature.
problem Computing index classes for manifolds with boundary and cuspidal parabolics.
method Uses orbital integrals, delocalized eta invariants, and index theorems.
result Defines higher rho numbers for metrics with positive scalar curvature.
This paper develops 4-manifold invariants using Hopf algebras.
problem Creating 4-manifold invariants from Hopf algebras.
method Using Hopf triplets and trisection diagrams, the authors construct 4-manifold invariants.
result Every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds.
Classifies special Lie algebras with semisimple types.
problem Classifying Lie algebras of semisimple type.
method Introduced conformal pseudo-subriemannian fundamental graded Lie algebras and provided their classification.
result Classification of conformal pseudo-subriemannian fundamental graded Lie algebras of semisimple type and their prolongations.
Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space…
Topological model for coloured Alexander invariants from quantum group representations.
problem Quantum invariants from Uq(sl(2)) at roots of unity. method Topological model using graded intersection pairings in a covering space.
result Coloured Alexander invariants can be obtained as homology class pairings.
New skein categories for non-semisimple settings, extending existing theory.
problem Extending skein theory to non-semisimple settings.
method Introducing skein categories based on tensor ideals in linear ribbon categories.
result Skein categories coincide with factorization homology in non-semisimple settings.
ETQFTs created from non-semisimple modular categories.
problem Constructing ETQFTs from non-semisimple modular categories.
method Explicitly identify linear categories and functors in the image of ETQFTs constructed from modular categories.
result The circle category of ETQFTs is equivalent to the full subcategory of projective objects of the underlying modular category, which need not be semisimple.
New invariants from symplectic fermions categorify link and manifold structures.
problem Defining and computing link and manifold invariants from non-semisimple categories.
method Using non-semisimple finite ribbon categories and modified traces, computing invariants for symplectic fermions.
result Invariant values for symplectic fermions categorify homology groups of lens spaces and rational homology spheres.
Let G --> G' be an embedding of semisimple complex Lie groups, let B and B' be a pair of nested Borel subgroups, and let f:G/B --> G'/B' be the associated equivariant embedding of flag manifolds. We study the pullbacks of cohomologies of invertible sheaves on G'/B' along the embedding f. Let O' be a G'-equivariant inve…
Constructs moduli spaces for monopoles with arbitrary symmetry breaking.
problem Finding moduli spaces for monopoles with varying symmetry.
method Defined configuration space with asymptotic conditions, performed quotient construction, used b-calculus and scattering calculus.
result Constructs hyper-Kähler moduli spaces for monopoles with arbitrary symmetry breaking.
Let Γ be a one-ended, torsion-free hyperbolic group and let G be a semisimple Lie group with finite center. Using the canonical JSJ splitting due to Sela, we define amalgam Anosov representations of Γ into G and prove that they form a domain of discontinuity for the action of Out(Γ). In the appendix,…
Semisimple Lie groups act transitively on pseudo-Riemannian manifolds, making them flat.
problem Understanding actions of semisimple Lie groups on pseudo-Riemannian manifolds.
method Analyzing the pseudo-Riemannian Lichnerowicz conjecture in homogeneous settings.
result Compact pseudo-Riemannian manifolds on which a semisimple group acts conformally, essentially and transitively, are conformally flat.
Study non-semisimple TQFT for Burau representation density and unitarity.
problem Density and unitarity of the Burau representation from a non-semisimple TQFT perspective.
method TQFT construction of Squier's Hermitian form on the Burau representation.
result Density of the image of braid group in unitary representations.
New pseudo-Hermitian models from non-semisimple TQFTs.
problem Constructing exactly solvable pseudo-Hermitian spin Hamiltonians.
method Identifying ground states on surfaces using non-semisimple TQFTs.
result Ground states depend only on spatial topology and can be assigned by non-semisimple TQFTs.
Constructs maps on skein modules using non-semisimple quantum invariants.
problem Constructing maps on skein modules with specific characters.
method Uses UqHsl2 non-semisimple invariants of 3-manifolds. result Maps with any possible abelian non-central character as classical shadow.
Proves Witten-Reshetikhin-Turaev 3-TQFT as a boundary condition of Crane-Yetter 4-TQFT.
problem Proving a boundary condition for Crane-Yetter 4-TQFT.
method Extending ideas of Crane-Yetter and Jordan, proving Crane-Yetter 4-TQFT and its non-semisimple version are once-extended TQFTs, defining a boundary condition.
result Reconstructs Witten-Reshetikhin-Turaev 3-TQFT and its non-semisimple versions using Crane-Yetter 4-TQFT.
Classifies semisimple weakly symmetric pseudo-Riemannian manifolds.
problem Classifying pseudo-Riemannian manifolds with specific properties.
method Developed from compact Lie group cases, analyzed isotropy representation and metric signature.
result Obtained classification of semisimple weakly symmetric manifolds of specific signatures.
We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus g with n marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum of n copies of Ki…
Study para-Sasakian φ-symmetric spaces using Boothby-Wang fibration.
problem Characterize para-Sasakian φ-symmetric spaces.
method Use Boothby-Wang fibration to construct and provide examples.
result Explicit construction and example of para-Sasakian φ-symmetric spaces.
New 3D TQFTs derived from non-semisimple categories.
problem Constructing topological invariants from non-semisimple categories.
method Using modified traces and Lyubashenko's invariants, with additional assumptions for factorizability.
result Produces new 2+1-TQFTs and monoidal extensions of representations.
New non-semisimple Ising anyons enable robust universal quantum computation.
problem Limitation of semisimple theories in universal topological quantum computation.
method Developed non-semisimple Ising anyon model with new anyon types indexed by α. result Robust universality of braiding persists over an open interval of α. Recent work extends Turaev's modular categories to non-semisimple settings.
problem Building TQFTs beyond semisimplicity.
method Generalized modular categories.
result Success in extending Turaev's construction to non-semisimple settings.
The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.
problem Classifying Q-Fano compactifications of semisimple groups with Kähler-Einstein metrics. method Proving finiteness through classification of compactifications.
result There are only finitely many Q-Fano compactifications of semisimple groups with Kähler-Einstein metrics. We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.
Paper finds a cohomological obstruction to compact Clifford-Klein forms existence.
problem Existence of compact Clifford-Klein forms on homogeneous spaces.
method Relating Lie algebra cohomology to de Rham cohomology, deriving upper-bound estimates.
result Examples of homogeneous spaces without compact Clifford-Klein forms.
Classifies compact Clifford-Klein forms for specific Lie algebras.
problem Classifying compact Clifford-Klein forms for given Lie algebra structures.
method Using Onishchik's results on semisimple Lie algebras, the paper classifies forms for triples (g,h,l).
result New examples of reductive homogeneous spaces with non-standard compact Clifford-Klein forms.
We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations. Finally, we calculate the Schlesinger initial conditions for solutions describing…
Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …
The paper classifies orbits of semisimple elements in real semisimple Lie algebras.
problem Classifying orbits of semisimple elements in real semisimple Lie algebras.
method Case by case analysis of complex numbers and Galois cohomology for real numbers.
result Characterization of orbits with real representatives.
We classify in this paper infinitesimal quasitrivial deformations of semisimple bihamiltonian structures of hydrodynamic type.
Introduces admissible skein modules for non-semisimple categories.
problem No specific problem stated; generalization of Kauffman skein algebra.
method Introduces admissible skein modules associated to ideals in pivotal categories.
result These modules generalize Kauffman skein algebra and relate to quantum invariants.
Semisimplicity proven for conformal blocks representations.
problem Semisimplicity of conformal blocks representations.
method Theory of extensions in non-Abelian Hodge theory and Ocneanu rigidity.
result Semisimplicity of braid group and mapping class group representations.