Witt algebra acts on categorified quantum groups in type A.
problem Action of Witt algebra on categorified quantum groups.
method Construction of action on categorified quantum group and foams.
result Action of Witt algebra on foams recovers previous results.
Defines and characterizes operators on Lie ∞-algebras with respect to actions.
problem Characterizing operators on Lie ∞-algebras with respect to actions.
method Using higher derived brackets construction and Maurer-Cartan elements.
result Determines the Lie ∞-algebra controlling the deformation of operators.
We define the notion of action of an L-infinity algebra g on a graded manifold M, and show that such an action corresponds to a homological vector field on g[1]×M of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particula…
Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.
problem Determining when degree-2 Azumaya algebras with C2-actions extend to entire curves.
method Criteria for extension of algebra and new condition for extension with action, testable by computer algebra systems.
result New conditions for extending degree-2 Azumaya algebras with C2-actions over curves.
Skein algebra action is faithful if quantum parameter isn't a root of 1.
problem Faithfulness of geometric action of skein algebras.
method Examined the Kauffman bracket skein algebra and its action on handlebody modules.
result The action is faithful if and only if the quantum parameter is not a root of 1.
Action of loop groups on Cuntz algebras constructs geometric twists.
problem Defining actions of loop groups on Cuntz algebras.
method Using representations of Cuntz algebras and analytic loop groups to construct actions and bundles.
result Explicit construction of geometric twists in higher K-theory. Develops a spectral sequence for Lie group actions on manifolds.
problem Understanding cohomology of manifolds with Lie group actions.
method Introduces a spectral sequence relating manifold cohomology to Lie algebra cohomology.
result Establishes a new description of de Rham cohomology for manifolds with Lie group actions.
New categorical actions link topological and algebraic structures.
problem Understanding relationships between topological and algebraic structures.
method Categorical actions of type B braid group on homotopy categories.
result Proves Rouquier's conjecture on faithfulness of Type B 2-braid group.
Study boundary actions on CAT(0) spaces, proving topological freeness.
problem Understanding boundary actions on CAT(0) spaces.
method Verification of freeness of Myrberg points on boundaries.
result Large class of boundary actions are topologically free.
We study properly discontinuous and cocompact actions of a discrete subgroup Γ of an algebraic group G on a contractible algebraic manifold X. We suppose that this action comes from an algebraic action of G on X such that a maximal reductive subgroup of G fixes a point. When the real rank of any simple subg…
Convex cores found for group actions on median spaces.
problem Understanding group actions on median spaces without metric or topology.
method Introduced convex cores for actions on finite-rank median algebras.
result Actions on median spaces have nonempty convex cores.
Study boundary actions of CAT(0) spaces and their C∗-algebras.
problem Investigate boundary actions of CAT(0) spaces and their associated C∗-algebras. method Topological dynamics and C∗-algebras, focusing on actions of specific groups and their properties. result Established (strongly) pure infiniteness results for reduced crossed product C∗-algebras of boundary actions. New algebra connects braid group actions to disc curves.
problem Understanding braid group actions on disc curves.
method Constructed a type B zigzag algebra and showed its categorical action on projective modules.
result Type B braid group action on homotopy category of projective modules.
Lifts an sl2 action to annular Khovanov homology's stable refinement.
problem Stable refinement of annular Khovanov homology's sl2 action. method Lifts actions of sl2 generators to maps of spectra, using cancellations in cube of resolutions. result Commutativity of sl2 action with Steenrod algebra action. Foams have Lie algebra symmetries that simplify web state spaces.
problem Understanding symmetries in foam structures.
method Defined an action of a Lie subalgebra on foams compatible with glN-foam evaluation. result Endows glN-web state spaces with sl2-action. Construct spectral triples on C*-algebras with group actions.
problem Building spectral triples on C*-algebras with group actions.
method Systematic construction of spectral triples on A using geometry of AG and G. result Comparison with established examples of spectral triples.
The paper studies Morse theory for Lie algebra actions on Riemannian foliations.
problem Morse theory for Lie algebra actions on Riemannian foliations.
method Equivariant Morse-Bott theory on leaf space.
result Established foliated versions of Morse-Bott lemma and handle presentation theorem.
Odd Khovanov homology gets a new algebraic action from super foams.
problem Understanding the algebraic structure of odd Khovanov homology.
method Introducing a local gl1∣1-action on odd Khovanov homology via super foams. result The action of gl1∣1 on odd Khovanov homology is shown to arise from super foams. The actions of a half Virasoro algebra have appeared in many integrable systems. In this paper we show that there is an action of a (Half) Virasoro algebra on the space of (2+0) harmonic maps into a Lie group. This action is generated by a natural action on the frames. A similar calculation on the space-time (1+1) harm…
This paper shows how post-Lie algebra structures can be induced by simply transitive NIL-affine actions.
problem Understanding which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Introducing post-Lie algebra structures and showing their correspondence with simply transitive actions.
result Simply transitive NIL-affine actions induce complete post-Lie algebra structures in the 2-step nilpotent case.
Let Q denote a smooth manifold acted upon smoothly by a Lie group G. The G-action lifts to an action on the total space T of the cotangent bundle of Q and hence on the standard symplectic Poisson algebra of smooth functions on T. The Poisson algebra of G-invariant functions on T yields a Poisson structure on the space …
Study of algebraic links in lens spaces, proving they are fibered and finding examples.
problem Understanding algebraic links in lens spaces.
method Investigation of polynomials invariant under a Zp-action on the 3-sphere, studying links and their properties. result All algebraic links in lens spaces are fibered, with results on Seifert genus.
A multiplicatively closed, horizontal foliation on a Lie groupoid may be viewed as a "pseudoaction" on the base manifold M. A pseudoaction generates a pseudogroup of transformations of M in the same way an ordinary Lie group action generates a transformation group. Infinitesimalizing a pseudoaction, one obtains the…
New representations of Lie algebras via monoidal category actions.
problem Constructing representations of Lie algebras using monoidal categories.
method Using crossed homomorphisms and monoidal categories to generate representations.
result Established new bifunctor for weak and admissible representations of Lie-Rinehart algebras.
This paper studies actions of solvable Lie groups on nilpotent Lie groups.
problem Characterizing which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Using Lie algebra properties and semisimple splitting, the paper provides methods to check for such actions.
result A full description of possibilities for actions up to dimension 4.
We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to …
We study isometric actions on Riemannian symmetric spaces of noncompact type which are induced by reductive algebraic subgroups of the isometry group. We show that for such an action there exists a corresponding isometric action on a dual compact symmetric space, which reflects many properties of the original action. F…
We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov …
Reduces observables on multisymplectic manifolds using Lie algebra actions.
problem Reduction of observables on multisymplectic manifolds with Lie algebra actions.
method Development of a reduction scheme for L∞-algebra of observables. result Reproduces symplectic observable reduction in specific cases.
The paper introduces pseudo-quotients for algebraic actions and applies them to character varieties.
problem Characterizing algebraic actions and their quotients.
method Introducing pseudo-quotients as a weak version of quotients for algebraic actions, focusing on purely topological properties.
result Pseudo-quotients are unique up to virtual class in characteristic zero and can be used to compute character varieties.
For a finite dimensional Lie algebra $\g$ of vector fields on a manifold M we show that M can be completed to a G-space in a unversal way, which however is neither Hausdorff nor T1 in general. Here G is a connected Lie group with Lie-algebra $\g$. For a transitive $\g$-action the completion is of the form $G…
This research extends Lie algebra actions to singular foliations.
problem Understanding symmetries in singular foliations without additional assumptions.
method Equivalence of categories between Lie-Rinehart algebras and Lie ∞-algebroids. result Universal Lie ∞-algebroids for singular foliations. We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtai…
We construct an action of the braid group B_N on the twisted quantized enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In the classical limit q -> 1 we recover the action of B_N on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action prese…
Isomorphism found between filtered calculus and crossed products.
problem Tackles isomorphism in filtered calculus and crossed products.
method Uses natural R-action and structure result for C*-algebra of graded nilpotent Lie groups.
result Found isomorphism between kernel of tangent groupoid and crossed product.
The paper adapts differential signatures to algebraic curves under group actions.
problem Equivalence problem for complex plane algebraic curves under group actions.
method Adapting differential signature construction to algebraic curves, using classifying invariants.
result Explicit sets of rational classifying invariants and formulas for signature curve degree.
We consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of Zp action on manifolds with isolated fixed points when p is a prime.
Proves a Thom isomorphism for foliated differential forms.
problem Transverse Lie algebra actions on foliated manifolds and vector bundles.
method Thom isomorphism theorem for differential forms in foliated settings.
result Established a new Thom isomorphism theorem.
We show that many algebraic actions of higher-rank abelian groups on zero-dimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov--Rokhlin formula for half-space entropies.
Witt algebra acts on Khovanov-Rozansky homology of links.
problem Understanding algebraic structures in knot theory.
method Construction of Witt algebra action on Khovanov-Rozansky homology.
result Induced maps between twists of the homology via link cobordisms.
The paper connects polygon spaces with quotient spaces using spin actions and normed division algebras.
problem Understanding correspondences between polygon spaces and quotient spaces.
method Introducing Hopf maps and spin actions on normed division algebras to construct correspondences.
result Extension of polygon space correspondences to higher dimensions and normed division algebras.
Researchers establish a connection between knot homology and Lie algebra actions.
problem Understanding the HOMFLY-PT homology of (n,n+1) torus knots. method Constructing an explicit isomorphism and computing tautological class actions.
result The tautological class action extends to Hamiltonian vector fields and differentials in spectral sequences.
This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the…
New braid group actions on n-adic integers linked to real algebraic links.
problem Understanding braid group actions on n-adic integers. method Constructing an infinite tower of covering spaces over configuration spaces and associating braids to infinite sequences of braids.
result An infinite family of braids close to real algebraic links.
Study group actions on hyperbolic spaces to find algebraic and geometric properties.
problem Understanding algebraic properties of group actions on hyperbolic spaces.
method Developed an algorithm in the group algebra to show free ideals generated by few elements.
result Derived lower bounds on Morse complexity of closed hyperbolic manifolds.
It is known that the hyperbolic plane admits a countable infinity of compactifications into a closed disk such that the isometric action of SL(2;R) acts analytically on the compactified space. We prove that among those compactifications, only the two most classical ones (namely the closures of Poincaré's disk and Klein…
We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov…
The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.
problem Extending invariant theory to non-compact and non-reductive actions.
method Examined two specific settings: discrete subgroups of Lorentz group acting on Rn,1 and cocompact actions on smooth manifolds. result Classification of invariant-theoretic regimes into four categories, identifying boundaries of Hilbert--Weyl and Schwarz theorems.