On representations and <Emphasis Type="Italic">K</Emphasis>-theory of the braid groups

 Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group as defined below. The triviality of bundles arising from orthogonal representations of Γ is characterized completely as follows. An orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups. Furthermore, the subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, the subgroup of elements which arises from real orthogonal representations of Γ is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles. Received: 6 February 2002 / Revised version: 19 September 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" Partially supported by the NSF RID="⋆⋆" ID="⋆⋆" Partially supported by grant LEQSF(1999-02)-RD-A-01 from the Louisiana Board of Regents, and by grant MDA904-00-1-0038 from the National Security Agency RID="⋆" ID="⋆" Partially supported by the NSF Mathematics Subject Classification (2000): 20F36, 32S22, 55N15, 55R50     

Linear Representations of the Braid Groups of Some Manifolds

In this paper we prove that the braid group B_n(S²) of 2-sphere, mapping class group M(0,n) of the n-punctured 2-sphere and the braid group B₃(P²) of the projective plane are linear. Partially supported by the Russian Foundation for Basic Research (grant number 02-01-01118).Mathematics Subject Classifications (2000) 20F28, 20F36, 20G35.     

Braid groups in genetic code

For every genetic code with finitely many generators and at most one relation, a braid group is introduced. The construction presented includes the braid group of a plane, braid groups of closed oriented surfaces, Artin— Brieskorn braid groups of series B, and allows us to study all of these groups from a unified standpoint. We clarify how braid groups in genetic code are structured, construct words in the normal form, look at torsion, and compute width of verbal subgroups. It is also stated that the system of defining relations for a braid group in two-dimensional manifolds presented in a paper by Scott is inconsistent. Supported by RFBR grant No. 02-01-01118. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 131–158, March–April, 2006.     

Rational surfaces and regular maps into the 2-dimensional sphere

Let X and Y be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a mapping, , can be approximated by regular mappings in the space of mappings, , equipped with the topology. In this paper, we obtain some results concerning this problem when the target space is the 2-dimensional standard sphere and X has a complexification that is a rational (complex) surface. To get the results we study the subgroup of the second cohomology group of X with integer coefficients that consists of the cohomology classes that are pullbacks, via the inclusion mapping , of the cohomology classes in represented by complex algebraic hypersurfaces. Received December 1, 1998; in final form August 2, 1999     

The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

On the Hochschild Cohomology of Triangular Algebras

Abstract

In order to study the Hochschild cohomology of an n-triangular algebra 𝒯 _n , we construct a spectral sequence, whose terms are parametrized by the length of the trajectories of the quiver associated with 𝒯 _n , and which converges to the Hochschild cohomology of 𝒯 _n . We describe explicitly its components and its differentials which are sums of cup products. In case n = 3 we study some properties of the differential at level 2. We give some examples of use of the spectral sequence and recover formulas for the dimension of the cohomology groups of particular cases of triangular algebras.     

On Carleman formulas for the Dolbeault Cohomology

We discuss the Cauchy problem for the Dolbeault cohomology in a domain of C ⁿ with data on a part of the boundary. In this setting we introduce the concept of a Carleman function which proves useful in the study of uniqueness. Apart from an abstract framework we show explicit Carleman formulas for the Dolbeault cohomology. To the memory of Lamberto Cattabriga     

Linear differential operators with unbounded operator coefficients and semigroups of bounded operators

Associated with a family of evolution operators in a complex Banach space is a linear unbounded operator, which is studied with the aid of a semigroup of difference operators and a difference operator in a sequence space. Some formulas for the spectra of the linear operators in question (in particular, for abstract hyperbolic differential operators) and the spectrum mapping theorem for the semigroup of difference operators are obtained. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 811–820, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00032 and by the International Science Foundation under grant No. NZA000 and grant No. NZA300.     

Cohomology of braid groups with nontrivial coefficients

In the paper, the homology of the braid groups with twisted coefficients and the homology of commutator subgroups of the braid groups are calculated. The main tool is the multiplicative structure on the homology induced by the “addition” of braid groups. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 846–854, June, 1996. This research was partially supported by the International Science Foundation under grant MQO000.     

Cohomologies of the Poisson superalgebra

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on ²ⁿ are investigated under suitable continuity restrictions on the cochains. The first and second cohomology spaces in the trivial representation and the zeroth and first cohomology spaces in the adjoint representation of the Poisson superalgebra are found for the case of a constant nondegenerate Poisson superbracket or arbitrary n > 0. The third cohomology space in the trivial representation and the second cohomology space in the adjoint representation of this superalgebra are found for arbitrary n > 1.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 163–194, May, 2005.     

On the group of conjugating automorphisms of a free group

The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996.     

Geometry for palindromic automorphism groups of free groups

We examine the palindromic automorphism group , of a free group F _n , a group first defined by Collins in [5] which is related to hyperelliptic involutions of mapping class groups, congruence subgroups of , and symmetric automorphism groups of free groups. Cohomological properties of the group are explored by looking at a contractible space on which acts properly with finite quotient. Our results answer some conjectures of Collins and provide a few striking results about the cohomology of , such as that its rational cohomology is zero at the vcd. Received: January 17, 2000.     

On fundamental groups related to the Hirzebruch surface <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript>

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in ℂ² or in ℂℙ². In this article, we show that these groups, for the Hirzebruch surface F _1,(a,b), are almost-solvable. That is, they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces. This work was supported by the Emmy Noether Institute Fellowship (by the Minerva Foundation of Germany) and Israel Science Foundation (Grant No. 8008/02-3)     

Cohomologie des variétés de Deligne-Lusztig

We study the cohomology of Deligne-Lusztig varieties with aim the construction of actions of Hecke algebras on such cohomologies, as predicted by the conjectures of Broué, Malle and Michel ultimately aimed at providing an explicit version of the abelian defect conjecture. We develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them. We are able to compute the cohomology of varieties associated to (possibly twisted) rank 2 groups and powers of the longest element w₀ (some indeterminacies remain for G₂). We use this to construct Hecke algebra actions on the cohomology of varieties associated to w₀ or its square, for groups of arbitrary rank. In the subsequent work [F. Digne, J. Michel, Endomorphisms of Deligne-Lusztig varieties, Nagoya J. Math. 183 (2006)], we construct actions associated to more general regular elements and we study their traces on cohomology.     

Graph cohomology and Kontsevich cycles

We use the duality between compactly supported cohomology of the associative graph complex and the cohomology of the mapping class group to show that the duals of the Kontsevich cycles [W_λ] correspond to polynomials in the Miller-Morita-Mumford classes. We also compute the coefficients of the first two terms of this polynomial. This extends the results of (Combinatorial Miller-Morita-Mumford classes and Witten cycles, math.GT/0207042, 2002), giving a more detailed answer to a question of Kontsevich (Commun. Math. Phys. 147(1) (1992) 1) and verifying more of the conjectured formulas of Arbarello and Cornalba (J. Algebraic Geom. 5 (1996) 705).     

On tensor representations of knot algebras

Summary The fact that a Yang-Baxter operator defines tensor representations of the Artin braid group has been used to construct knot invariants. The main purpose of this note is to extend the tensor representations of the Artin braid group to representations of the braid groupZ B _k associated to the Coxeter graphB _k. This extension is based on some fundamental identities for the standardR-matrices of quantum Lie theory, here called four braid relations. As an application, tensor representations of knot algebras of typeB (Hecke, Temperley-Lieb, Birman-Wenzl-Murakami) are derived.     

On the inverse braid and reflection monoids of type <Emphasis Type="Italic">B</Emphasis>

There are well-known relations between braid and symmetric groups as well as Artin-Brieskorn braid groups and Coxeter groups: the latter are the factor-groups of the Artin-Brieskorn braid groups. The inverse braid monoid is related to the inverse symmetric monoid in the same way. We show that similar relations exist between the inverse braid monoid of type B and the inverse reflection monoid of type B. This gives a presentation of the latter monoid.