Graded Poisson structures on the algebra of differential forms

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.     

L'algebre de hopf bitensorielle

For any field k of zero characteristic we give a functor from the category of k-vector spaces into the category of k-Hopf algebras, attaching to any vector space V its bitensorial pointed Hopf algebra A^v. This Hopf algebra is graded, fulfills a universal property, and contains a remarkable subspace P of primitive elements, which as a conjecture may generate the Lie algebra Prim A^v. In case V is finite-dimensional we exhibit a Hopf pairing between A^vand A^v-whose kernel contains the (Hopf) ideal generated by the elements of P of degree ? 2.     

On the Divided Power Algebra and the Symplectic Group in Characteristic 2

Let V be an even dimensional vector space over a field K of characteristic 2 equipped with a nondegenerate alternating bilinear form f. The divided power algebra DV is considered as a complex with differential defined from f. We examine the cohomology modules as representations of the corresponding symplectic group.     

On the combinatorics of the graph-complexSur la combinatoire du complexe de graphes

In their recent preprint [3] Kontsevich and Shoikhet have introduced two graph-complexes: the complex on the even (resp. odd) space in order to study the cohomology of the Lie algebra Ham₀ (resp. Ham₀^odd) of Hamiltonian vector fields vanishing at the origin on the infinite-dimensional even (resp. odd) space. We construct an isomorphism between those two graph-complexes, proving in particular that their cohomologies coincide. This solves a problem posed by Shoikhet. To cite this article: B. Lass, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1–6     

Quadratic descent of hermitian forms

Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined result is also obtained for hermitian (resp. skew hermitian) forms over a quaternion algebra with symplectic (resp. orthogonal) involution.     

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V,?Q) with a non-degenerate quadratic form Q of any signature (p,?q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra C?(V*,?Q) of linear functionals (multiforms) acting on the universal Clifford algebra C?(V,?Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of C?(V,?Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of C?(V,?Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Ab?amowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].     

Root Systems In Finite Symplectic Vector Spaces

We study realizations of root systems in possibly degenerate symplectic vector spaces over finite fields, up to symplectic isomorphisms. The main result of this article is the classification of such realizations for the field 𝔽₂. Thereby, each root system requires a specific degree of degeneracy of the symplectic vector space. Our main motivation for this article is that for each such realization of a root system we can construct a Nichols algebra over a nonabelian group.     

Group actions on algebras and the graded Lie structure of Hochschild cohomology

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.     

Quantum Giambelli formulas for isotropic Grassmannians

Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the authors?? cohomological Giambelli formulas.     

Noncommutative Enumeration in Graded Posets

We define a noncommutative algebra of flag-enumeration functionals on graded posets and show it to be isomorphic to the free associative algebra on countably many generators. Restricted to Eulerian posets, this ring has a particularly appealing presentation with kernel generated by Euler relations. A consequence is that even on Eulerian posets, the algebra is free, with generators corresponding to odd jumps in flags. In this context, the coefficients of the cd-index provide a graded basis.     

Bireflectionality of orthogonal and symplectic groups of characteristic 2

Let V be a finite dimensional vector space over a field K of characteristic 2. Let O(V) be the orthogonal group defined by a nondegenerate quadratic form. Then every element in O(V) is a product of two elements of order 2, unless all nonsingular subspaces of V are at most 2-dimensional. If V is a nonsingular symplectic space, then every element in the symplectic group Sp (V) is a product of two elements of order 2, except if dim V = 2 and |K| = 2.     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

Coherent Transforms and Irreducible Representations Corresponding to Complex Structures on a Cylinder and on a Torus

We study a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution: a cylinder or a torus. Over each of such surfaces we introduce a family of complex structures and Hilbert spaces of antiholomorphic sections in which the irreducible Hermitian representations of the original algebra are realized. The reproducing kernels of these spaces are expressed in terms of the Riemann theta function and its modifications. They generate quantum Kähler structures on the surface and the corresponding quantum reproducing measures. We construct coherent transforms intertwining abstract representations of an algebra with irreducible representations, and these transforms are also expressed via the theta function.     

Exponentiation of infinite dimensional Z-graded lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace £ of the dual space V ^*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.     

The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces

We compute the center and nilpotency of the graded Lie algebra for a large class of formal spaces X. The latter calculation determines the rational homotopical nilpotency of the space of self-equivalences aut₁(X) for these X. Our results apply, in particular, when X is a complex or symplectic flag manifold.     

On Odd Symplectic Schur Functions

Proctor defined combinatorially a family of Laurent Polynomials, called odd symplectic Schur functions, indexed by pairs (λ, c), where λ is partition andcis a column length of λ. A conjecture of Proctor (Invent. Math.92,1988, 307–332) includes the statement that the odd symplectic Schur functions are actually characters ofSp(2n + 1, C). The purpose of the present note is to prove this.     

\mathcal{A}(p) generators for H^*V and Singer's homological transfer

We list explicitly a minimal set of generators for the cohomology of an elementary abelian p-group, V, of rank 1 or 2, as a module over the mod p Steenrod algebra, for an odd prime p. Following Singer, we then construct a transfer map to the vector space spanned by such generators, where V now has arbitrary rank, from the homology of the Steenrod algebra. We show that this map takes images in the subspace of GL(V)-invariants and that it is an isomorphism for V having rank 1 or 2. Received June 11, 1996; in final form June 9, 1997     

The exterior derivative as a Killing vector field

Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the graded sense, and have a graded Laplacian operator that annihilates the whole algebra of differential forms. Partially supported by DGICYT grants #PB91-0324, and SAB94-0311; CONACyT grant #3189-E9307.     

Hopf Algebras on Schurian Quivers

Let H be a Hopf algebra with a finite-dimensional, nontrivial space of skew primitive elements, over an algebraically closed field of characteristic zero. We prove that H contains either the polynomial algebra as a Hopf subalgebra, or a certain Schurian simple-pointed Hopf subalgebra. As a consequence, a complete list of the locally finite, simple-pointed Hopf algebras is obtained. Also, the graded automorphism group of a Hopf algebra on a Schurian Hopf quiver is determined, and the relation between this group and the automorphism groups of the corresponding Hopf quiver, is clarified.     

Canonical forms of skew polynomial rings

We consider special relations in a skew polynomial ring with the following property: every commutation relation between the elements of the ring basis and the elements of the ring of coefficients can be calculated with the help of these special relations. Such relations are called canonical forms of the skew polynomial ring. For example, the Weyl relation is a canonical form for the Weyl algebra. Skew polynomial rings with such canonical forms can be applied, for example, to the representation theory and to mathematical physics. Bibliography: 10 titles.Published in Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 40–57.This revised version was published online in April 2005 with a corrected cover date and article title.