Second part of proving linearization theorem for sl2(C).
problem Proving linearization theorem for sl2(C).
method Developed Nash-Moser method for functions flat at a point.
result Linearization result for a more general class of Lie algebras.
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48 results for “Lie algebra sl2(C)”
This paper upgrades Khovanov homology to an L-infinity module structure.
problem Exploring Khovanov homology with L-infinity algebra structures.
method Developed an L-infinity algebra structure on sl2(∧) and showed annular Khovanov homology is an L-infinity module over it.
result The annular Khovanov homology of a link L is an L-infinity module over sl2(∧) up to quasi-isomorphism.
We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Che…
Study quantized SL2-character variety of a once-punctured torus, finding three Coulomb branch isomorphisms.
problem Understanding the structure of quantized SL2-character variety of a once-punctured torus.
method Analyzing the quantized algebra and its subalgebras isomorphic to Coulomb branches.
result Three -invariant subalgebras of the quantized algebra are isomorphic to Coulomb branches.
We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming para…
Let p an integer. We define a family of idempotents (and nilpotents) in the Temperley - Lieb algebras at 4p-th roots of unity which generalize the usual Jones-Wenzl idempotents. These new idempotents correspond to finite dimentional simple and projective indecomposable representations of the restricted quantum group Uq…
Paper extends algebraic geometry results to hyperbolic link complements.
problem Understanding algebraic and number-theoretic properties of canonical curves.
method Generalizes Chinburg-Reid-Stover's results to hyperbolic link complements.
result Azumaya algebra does not extend to canonical surfaces.
New bounds on diameters and generators for specific lattices and graphs.
problem Finding bounds on diameters and generators for arithmetic lattices and Ramanujan graphs.
method Analyzing arithmetic lattices from Eichler orders in quaternion algebras, applying techniques to definite quaternion algebras.
result Bounds on diameters and generators for arithmetic lattices and Ramanujan graphs.
Constructs new explicit proper r-harmonic functions on Thurston geometries.
problem Developing proper r-harmonic functions on specific 3D geometries.
method Explicit construction of new functions for Thurston geometries.
result Explicit construction of new proper r-harmonic functions on various Thurston geometries.
The paper defines a new algebra structure on skein modules of 3-manifolds and shows it's isomorphic to a known algebra.
problem Defining and studying algebraic structures on skein modules of 3-manifolds.
method Introducing and analyzing a new algebra structure based on links in -manifolds.
result The algebra is naturally isomorphic to a known algebra when the parameter is .
We describe a method to compute the Culler-Shalen seminorms of a knot, using the (-3,3,4) pretzel knot as an illustrative example. We deduce that the SL2(C)-character variety of this knot consists of three algebraic curves and that it admits no non-trivial cyclic or finite surgeries. We also summarize similar results f…
Unified invariant of knots derived from Verma modules.
problem Constructing a unified invariant of knots from quantum sl2.
method Braid groups' action on tensors of Verma modules.
result Unified invariant interpolates colored Jones and ADO polynomials.
Study non-acyclic SL2-representations of twist knots and their L-functions.
problem Characterize SL2-representations of twist knots and their properties.
method Character variety, Reidemeister torsion, Chebyshev polynomials, and L-functions.
result Non-acyclic SL2-representations lie on the line x=y in character variety, and their orders are related to (-3)-Dehn surgery.
Polynomial error equidistribution for SL2 groups.
problem Equidistribution of orbits in arithmetic quotients.
method Margulis function, incidence geometry, spectral gap.
result Polynomial error rate for equidistribution.
New anomaly cancellation formulas for E8*E8*E8 gauge group.
problem Anomaly cancellation for E8*E8*E8 gauge group.
method Constructed modular forms over SL2(Z) to get new anomaly cancellation formulas.
result New anomaly cancellation formulas for characteristic forms.
Study periodic geodesics on contact 3D manifolds, proving existence and precise properties.
problem Existence and properties of periodic geodesics in contact sub-Riemannian metrics.
method Develops two independent subjects: existence of spiraling geodesics and precise study of geodesics on quotient of SL2(R).
result Proves existence and precise properties of periodic geodesics.
We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the geometric interpretation of their centers. The gauge invariance of the link invariant …
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…
The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.
problem Characterizing and understanding k-para-Kähler Lie algebras.
method Generalization of para-Kähler Lie algebras to k-para-Kähler Lie algebras, introduction of new structures, determination of Lie algebras.
result Determination of all k-symplectic Lie algebras of dimension (k+1) and six-dimensional 2-para-Kähler Lie algebras.
We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space . 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…
Study on pre-Lie structures for semisimple Lie algebras over C.
problem Admissibility of pre-Lie structures in semisimple Lie algebras.
method Examined properties of anti-flexible algebras (AFAs), computed Lie-admissibility criteria, and provided examples.
result Explicit counterexample of an AFA admissible by sl(2, C).
We construct the Lie algebra of an n-Lie algebra and we also define the notion of cohomology of an n-Lie algebra.
The aim of this note is to introduce the notion of a -Lie algebra and to prove some elementary properties of -Lie algebras, the category of -Lie algebras, the category of modules on a -Lie algebra and extensions of -Lie algebras. …
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the -category of curved Lie algebras using homotopy theory of algebras over a complete operad.
result Equivalence of -categories between Lie algebroids and certain kinds of curved Lie algebras.
In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of …
A pseudo -type Lie algebra naturally gives rise to a conformal pseudo-subriemannian fundamental graded Lie algebras. In this paper we investigate the prolongations of the associated fundamental graded Lie algebra and the associated conformal pseudo-subriemannian fundamental graded Lie algebra. In particular, we show…
If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g_i's, we say that g is simply assembled from the g_i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the g_i's, one obtains a Lie algebra assembled in two step…
New Lie algebras from knot homology.
problem Defining Lie algebras from knot homology.
method Using group homology, analogous to Goldman Lie algebra.
result Relations among new Lie algebras discussed.
A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…
Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
Proofs centerless unimodular contact Lie algebras.
problem Characterizing centerless unimodular contact Lie algebras.
method Elementary proof and introduction of DS-contact Lie algebras.
result The only centerless unimodular examples are and .
Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…
Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebrasmath-ph
The paper connects Lie bialgebras, Rota-Baxter Lie algebras, and their properties.
problem Exploring connections between Lie bialgebras and Rota-Baxter Lie algebras.
method Introducing quadratic Rota-Baxter Lie algebras, matched pairs, bialgebras, and Manin triples.
result Established a correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras.
Study coKähler structures on Lie algebras using Fino-Vezzoni correspondence.
problem Characterize coKähler structures on Lie algebras.
method Use Fino-Vezzoni correspondence to relate coKähler Lie algebras to Kähler Lie algebras.
result Complete the flat case for odd-dimensional Lie algebras, proving coKähler structures exist.
Study on generalized derivations in polynomial vector fields Lie algebras.
problem Understanding generalized derivations in specific Lie sub-algebras of polynomial vector fields.
method Analysis of Lie sub-algebras containing constant and Euler vector fields, under specified conditions.
result Characterization of generalized derivations in the studied Lie sub-algebras.
Born Lie algebrasmath.DG
Born Lie algebras classified up to 6D, with integrable metrics studied.
problem Classifying and understanding Born Lie algebras.
method Bicross product construction from pseudo-Riemannian Lie algebras.
result Classification of Lie algebras up to 6D with integrable Born structures.
Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.
problem Understanding cohomology of hemistrict Lie 2-algebras.
method Functorial construction and isomorphism proof of cohomology.
result Cohomology of hemistrict Lie 2-algebras is isomorphic to Chevalley-Eilenberg cohomology.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.
problem Understanding the relationship between Lie algebras and symplectic nilmanifolds.
method Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds constructed.
result Covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.
The paper studies the center of the Goldman Lie algebra and its properties.
problem Identifying the center of the Goldman Lie algebra and its properties.
method Analyzing the Goldman Lie algebra as a Z_2-graded Lie algebra and using properties of the even part.
result The center of the even part of the Goldman Lie algebra is generated by specific classes of loops.
The paper investigates gradings of complex simple Lie algebras, focusing on -gradings and their algebraic structures.
problem Investigating the algebraic structure of -gradings of complex simple Lie algebras.
method Completely determining the possible reductive algebras and proving the uniqueness of a specific free nilpotent Lie algebra.
result The only free nilpotent Lie algebra of step 3 that appears as the negative part of a -grading is the usual -grading of the exceptional Lie algebra .
This research introduces Lie brackets on spaces of biderivations in Lie algebras.
problem Understanding higher-order infinitesimal symmetries in Lie algebras.
method Study of right biderivations and Lie brackets on their spaces.
result New Lie algebra framework for biderivations with applications in deformation theory.