arXiv research

Lie-Rinehart algebras over $C^\infty$-rings defined and studied.

A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson K-ring are proved and the Poisson K-rings are calcul…

It is proved that the ring of invariants of the standard smooth completion of a Kac-Moody Lie algebra is functionally generated by two elements: the coefficient of the center and the Killing form.

Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.

We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group.

For every abelian compact Lie group A, we prove that the homotopical A-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the A-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism r…

Defines vector fields and differential forms on local C-infinity-ringed spaces.

Abstract Lie algebroids generalize Lie algebroids to abstract categories.

Let $G$ be a connected semisimple Lie group with its maximal compact subgroup $K$ being simply-connected. We show that the twisted equivariant $KK$-theory $KK^{\bullet}_{G}(G/K, τ_G^G)$ of $G$ has a ring structure induced from the renowned ring structure of the twisted equivariant $K$-theory $K^{\bullet}_{K}(K, τ_K^K)$…

Differential K-theory gets a $λ$-ring structure.

Contact Lie algebras have specific properties related to stabilizers and invariant polynomials.

We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topol…

Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo…

For any n>1 we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group Sp(n,1). We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all n>1.

Triality connects three polynomial bases in Lie algebra studies.

New Euler characteristic and Burnside group defined for definable groupoids.

In this paper, motivated by Chen--Ruan's stringy orbifold theory on almost complex orbifolds, we construct a new cohomology ring $\mathscr H^\ast_{G,cs}(X)$ for an equivariant almost complex pair $(X,G)$, where $X$ is a compact connected almost complex manifold, $G$ is a connected compact Lie group which acts on $X$ an…

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H. Gloeckner and K.-H. Neeb), without any restriction on the dimension or on the cha…

Based on a pair of cohomology operations on so called $δ-2$-formal spaces, we construct the integral cohomology rings of the classifying spaces of the Lie groups $Spin(n)$ and $Spin^{c}(n)$. As applications, we introduce characteristic classes for the reduced topological $K_{Spin}$ theory, determine the ring of integra…

We compute the equivariant $K$-theory $K_G^*(G)$ for a simply connected Lie group $G$ (acting on itself by conjugation). We prove that $K_G^*(G)$ is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group $G$, namely PSU(3),…

Constructs 2-vector bundles and 2K-theory for Lie groupoids and 2-equivariant settings.

We compute the equivariant complex K-theory ring of a cohomogeneity-one action of a compact Lie group at the level of generators and relations and derive a characterization of K-theoretic equivariant formality for these actions. Less explicit expressions survive for a range of equivariant cohomology theories including …

Let M be a compact simply connected hyperkähler (or holomorphically symplectic) manifold, \dim H^2(M)=n. Assume that M is not a product of hyperkaehler manifolds. We prove that the Lie algebra so(n-3,3) acts by automorphisms on the cohomology ring H^*(M). Under this action, the space H^2(M) is isomorphic to the fundame…

Researchers lift knot coloring polynomial to Habiro ring.

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the \emph{algebra of symmetries} $\mathscr{S}(\Box^r)$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of highest weight repres…

Consider a complex analytic manifold $X$ and a coherent Lie subalgebra $\shi$ of the Lie algebra of complex vector fields on $X$. By using a natural $\shd_X$-module $\shm_\shi$ naturally associated to $\shi$ and the ring (in the derived sense) $\rhom[\shd_X](\shm_\shi,\shm_\shi)$, we associate integers which measure th…

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie alge…

Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.

Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.

Study vector fields and flows on singular spaces like submanifolds.

The paper investigates exotic smooth structures on manifolds with group actions.

We show that the equivariant cohomology of any hyperpolar action of a compact and connected Lie group on a symmetric space of compact type is a Cohen-Macaulay ring. This generalizes some results previously obtained by the authors.

The paper finds torsion in Johnson homomorphisms' cokernels for large genus surfaces.

We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map $T_1: H^*(Γ;A) \to H^{*-1}((N\rtimes Γ;A)$ for any crossed module $N\to Γ$ and prove…

This paper is devoted to an elementary new construction of $1$-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring $\mathcal D_o$ together with the vector space $\mathcal S=\mathcal S(\mathcal D_o)$ with basis $\mathcal B_o = \mathcal B(\mathcal D_o)$ formed from some local distributi…

Probabilistic theory counts intersections in Riemannian spaces.

After the torch of Anders Kock [Taylor series calculus for ring objects of line type, Journal of Pure and Applied Algebra, 12 (1978), 271-293], we will establish the Baker-Campbell-Hausdorff formula as well as the Zassenhaus formula in the theory of Lie groups.

New method computes automorphisms of surface groups using skein algebras.

We set up foundations of representation theory over $S$, the sphere spectrum, which is the `initial ring' of stable homotopy theory. In particular, we treat $S$-Lie algebras and their representations, characters, $gl_n(S)$-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, w…

This article discusses the twisted adjoint action $\mathrm{Ad}_{g}^κ:G\rightarrow G$, $x\mapsto gxκ(g^{-1})$ given by a Dynkin diagram automorphism $κ\in\mathrm{Aut}(G)$, where $G$ is compact, connected, simply connected and simple. The first aim is to recover the classification of $κ$-twisted conjugacy classes by elem…

Constructs a new graded variety from algebraic data.

We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how thes…

Let $G$ be a compact connected Lie group acting on a stable complex manifold $M$ with equivariant vector bundle $E$. Besides, suppose $φ$ is an equivariant map from $M$ to the Lie algebra $\mathfrak{g}$. We can define some equivalence relation on the triples $(M, E, φ)$ such that the set of equivalence classes form an …

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces ov…

Constructs positive energy representations from Toda equations Stokes data.

We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M. More generally the Thom spectrum E^{-TM} is a ring spectrum. Similarly a fiberwise …

Pulling back the weight system associated with the spinor representation of the Lie algebra so(7) by the universal Vassiliev-Kontsevich invariant yields a numerical link invariant with values in formal power series. Computing some skein relations satisfied by this invariant, I derive a recursive algorithm for its evalu…

Paper develops equivariant basic cohomology for Lie groupoids.