Christian Kassel

Cyclic homology, comodules, and mixed complexes

relating cyclic and Hochschild homology, one can see that any formula for HC,(A @A’) will not only involve HC,(A) and HC,(A’), but also the “periodicity” operator S and the Hochschild groups. Fortunately, if the map S brings in some complications, it also endows HC,(A) with a comodule structure over the cyclic homology of the ground ring k. Actually, the comodule structure exists already on the complex level: a convenient complex on which S has the form of a canonical surjection is Connes’s double complex with differentials b and B. Now the idea is to view the cyclic homology of an algebra as a composite functor

Cyclic homology of differential operators, the Virasoro algebra and a q-analogue

We show how methods from cyclic homology give easily an explicit 2-cocycle ϕ on the Lie algebra of differential operators of the circle such that ϕ restricts to the cocycle defining the Virasoro algebra. The same methods yield also aq-analogue of ϕ as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity. We use an algebra ofq-difference operators andq-analogues of Koszul and the Rham complexes to construct these “quantum” cocycles.

L'homologie cyclique des algèbres enveloppantes

SummaryFor any Lie algebra g, we compute the Hochschild and cyclic homology groups of its enveloping algebra in terms of the canonical Lie-Poisson structure on the dual g*. We also discuss the collapsing of Connes spectral sequence for cyclic homology, particularly in the case of semisimple Lie algebras.

The singular locus of a Schubert variety

Abstract The singular locus of a Schubert variety Xμ in the flag variety for GL n ( C ) is the union of Schubert varieties Xν, where ν runs over a set Sg(μ) of permutations in Sn. We describe completely the maximal elements of Sg(μ) under the Bruhat order, thus determining the irreducible components of the singular locus of Xμ.

CHORD DIAGRAM INVARIANTS OF TANGLES AND GRAPHS

The notion of a chord diagram emerged from Vassilievs work Vas90], Vas92] (see also Gusarov Gus91], Gus94] and Bar-Natan BN91], BN95]). Slightly later, Kontsevich Kon93] deened an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagrams modulo the four-term relation. This knot invariant establishes an isomorphism of a projective limit of algebras generated by the Vassiliev equivalence classes of knots onto the algebra of chord diagrams. Kontsevich originally deened his invariant of knots via a multiple integral given by an explicit but complicated analytic expression. This expression, however beautiful, does not reveal the combinatorial nature of the invariant. (A similar situation would occur if the linking number of knots were introduced via the Gauss integral formula without a combinatorial calculation). A combinatorial reformulation of the Kontsevich integral appeared in the works of Bar-Natan BN94], Cartier Car93], Le and Murakami LM93], Piunikhin Piu95] (see also Kas95, Chapter XX]). On the algebraic side, this reformulation uses the notions of braided and innnitesimal symmetric categories as well as the notion of an associator introduced by Drinfeld Dri89] in his study of quasitriangular quasi-Hopf algebras. On the geometric side, one uses categories of tangles, as introduced by Yetter and Turaev (see Tur94]). Note also that a counterpart of the Kontsevich knot invariant in the theory of braids was discovered earlier by Kohno Koh85] who considered an algebraic version of chord diagrams. In this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms are spanned by tangles and chord diagrams, respectively. More precisely, we x a commutative ring R and consider categories T (R) and A(R) whose morphisms are formal linear combinations of framed oriented tangles and chord diagrams with coeecients in R, cf. Section 2. The set of morphisms in T (R) has a canonical ltration given by the powers of an ideal I which we call the augmentation ideal. Functions on morphisms in T (R) vanishing on I m+1 are exactly the Vassiliev invariants of degree m for framed oriented tangles. Completing T (R) at the ideal I, we obtain the pro-unipotent completion b T (R) = lim ?m T (R)=I m+1. Our main result (Corollary 2.5) states that, if R contains the eld Q of rational numbers, then b T (R) is isomorphic to a suitable completion b A(R) of the category …

POLYNOMIAL IDENTITIES AND NONCOMMUTATIVE VERSAL TORSORS

Abstract To any cleft Hopf Galois object, i.e., any algebra H α obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” A H α and U H α . The algebra A H α is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, A H α is a cleft H-Galois extension of a “big” commutative algebra B H α . Any “form” of H α can be obtained from A H α by a specialization of B H α and vice versa. If the algebra H α is simple, then A H α is an Azumaya algebra with center B H α . The algebra U H α is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of H α are satisfied. We construct an embedding of U H α into A H α ; this embedding maps the center Z H α of U H α into B H α when the algebra H α is simple. In this case, under an additional assumption, A H α ≅ B H α ⊗ Z H α U H α , thus turning A H α into a central localization of U H α . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.

Sturmian morphisms, the braid group B 4 , Christoffel words and bases of F 2

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F2) of automorphisms of the rank two free group F2 and show that it can be realized as a monoid in the group B4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F2 lifting any given basis of the free abelian group Z2. We further give an algorithm allowing to decide whether two elements of F2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.

Quantum Principal Bundles up to Homotopy Equivalence

Hopf-Galois extensions are known to be the right generalizations of both Galois field extensions and principal G-bundles in the framework of non-commutative associative algebras. An abundant literature has been devoted to them by Hopf algebra specialists (see [11], [14], [15] and references therein). Recently there has been a surge of interest in Hopf-Galois extensions among mathematicians and theoretical physicists working in non-commutative geometry a la Connes and a la Woronowicz (cf. [2], [3], [6], [7], [8], [9]). In their work Hopf-Galois extensions are considered in the setting of “quantum group gauge theory.”

Twisting algebras using non-commutative torsors: explicit computations

Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. We prove a theorem that affords a presentation by generators and relations for the algebras obtained by such twisting. We give a number of examples, including new constructions of the quantum affine spaces and the quantum tori.

Generic Hopf Galois extensions

In previous joint work with Eli Aljadeff we attached a generic Hopf Galois extension \( A_H^\alpha \) to each twisted algebra \( {}^\alpha H \) obtained from a Hopf algebra H by twisting its product with the help of a cocycle α. The algebra \( A_H^\alpha \) is a flat deformation of \( {}^\alpha H \) over a “big” central subalgebra \( B_H^\alpha \) and can be viewed as the noncommutative analogue of a versal torsor in the sense of Serre. After surveying the results on \( A_H^\alpha \) obtained with Aljadeff, we establish three new results: we present a systematic method to construct elements of the commutative algebra \( B_H^\alpha \), we show that a certain important integrality condition is satisfied by all finite-dimensional Hopf algebras generated by grouplike and skew-primitive elements, and we compute \( B_H^\alpha \) in the case where H is the Hopf algebra of a cyclic group.