mathBlog » 2006 » March

March 2006


Here are some projects and teaching materials that I have worked on while I was a graduate student at the University of Arizona.
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Here are some instructions on using a mathML enabled thunderbird in Fedora (tested on Fedora core Beta 5).
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Generalized Matric Massey Products for Graded Modules.

Arvid Siqveland

The theory of generalized matric Massey products has been applied for sometime to A -modules M , A a k -algebra. The main application is to computethe local formal moduli H ^ M , isomorphic to the local ring of the moduliof A -modules. This theory is also generalized to 𝒪 X -modules , X a k - scheme. In these notes we consider the definition ofgeneralized Massey products and the relation algebra in any obstructionsituation (a differential graded k -algebra with certain properties), andprove that this theory applies to the case of graded R -modules, R a graded k -algebra, k algebraically closed. When the relation algebra isalgebraizable, that is the relations are polynomials rather than power series,this gives a combinatorial way to compute open (\’{e}tale) subsets of themoduli of graded R -modules. This also gives a sufficient condition for thecorresponding point in the moduli of 𝒪 Proj (R) -modules to besingular. The computations are straight forward, algorithmic, and an example onthe postulation Hilbert scheme is given.

PBW-deformation theory and regular central extensions.

Thomas Cassidy (Bucknell University), Brad Shelton (University of Oregon)

A deformation U , of a graded K -algebra A is said to be of PBW type if grU is A . It has been shown for Koszul and N -Koszul algebras that thedeformation is PBW if and only if the relations of U satisfy a Jacobi typecondition. In particular, for these algebras the determination of the PBWproperty is a {\it finite} and explicitly determined linear algebra problem. Weextend these results to an arbitrary graded K -algebra, using the notion ofcentral extensions of algebras and a homological constant attached to A whichwe call the {\it complexity} of A .

Hall algebras and the Quantum Frobenius.

Kevin McGerty

Lusztig has constructed a Frobenius morphism for quantum groups at an -th root of unity, which gives an integral lift of the Frobenius map onuniversal enveloping algebras in positive characteristic. Using the Hallalgebra we give a simple construction of this map for the positive part of thequantum group attached to an arbitrary Cartan datum.

Homotopical Algebraic Geometry II: geometric stacks and applications.

Bertrand Toen, Gabriele Vezzosi

This is the second part of a series of papers devoted to develop HomotopicalAlgebraic Geometry. We start by defining and studying generalizations ofstandard notions of linear and commutative algebra in an abstract monoidalmodel category, such as derivations, etale and smooth maps, flat and projectivemodules, etc. We then use the theory of stacks over model categories introducedin \cite{hagI} in order to define a general notion of geometric stack over abase symmetric monoidal model category C, and prove that this notion satisfiesthe expected properties. The rest of the paper consists in specializing C toseveral different contexts. First of all, when C=k-Mod is the category ofmodules over a ring k, with the trivial model structure, we show that ournotion gives back the algebraic n-stacks of C. Simpson. Then we set C=sk-Mod,the model category of simplicial k-modules, and obtain this way a notion ofgeometric derived stacks which are the main geometric objects of DerivedAlgebraic Geometry. We give several examples of derived version of classicalmoduli stacks, as for example the derived stack of local systems on a space, ofalgebra structures over an operad, of flat bundles on a projective complexmanifold, etc. Finally, we present the cases where C=(k) is the model categoryof unbounded complexes of modules over a char 0 ring k, and C=Sp^{\Sigma} themodel category of symmetric spectra. In these two contexts, called respectivelyComplicial and Brave New Algebraic Geometry, we give some examples of geometricstacks such as the stack of associative dg-algebras, the stack ofdg-categories, and a geometric stack constructed using topological modularforms.

Noncommutative complete intersections and matrix integrals.

Pavel Etingof, Victor Ginzburg

We introduce a class of noncommutatative algebras called representationcomplete intersections (RCI). A graded associative algebra A is said to be RCIprovided there exist arbitrarily large positive integers n such that the schemeRep_n(A), of n-dimensional representations of A, is a complete intersection. Wediscuss examples of RCI algebras, including those arising from quivers.

There is another interesting class of associative algebras callednoncommutative complete intersections (NCCI). We prove that any graded RCIalgebra is NCCI. We also obtain explicit formulas for the Hilbert series ofeach nonvanishing cyclic and Hochschild homology group of an RCI algebra. Theproof involves a noncommutative cyclic Koszul complex, K(A), and a matrixintegral similar to the one arising in quiver gauge theory.

Representations Parameterized by a Pair of Characters.

David E. Radford, Hans-Jürgen Schneider

Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy$(u{\otimes}a)(u’{\otimes}a’) = uu’{\otimes}aa’$ if $a = 1$ or $u’ = 1$. For apair of characters $\rho \in \Alg(U, k)$ and $\chi \in \Alg(A, k)$ we define aleft $H$-module $L(\rho, \chi)$. Under reasonable hypotheses the correspondence$(\rho, \chi) \mapsto L(\rho, \chi)$ determines a bijection between characterpairs and the isomorphism classes of objects in a certain category${}_H\underline{\mathcal M}$ of left $H$-modules. In many cases thefinite-dimensional objects of ${}_H\underline{\mathcal M}$ are thefinite-dimensional irreducible left $H$-modules. In math.QA/0603269 we applythe results of this paper and show that the finite-dimensional irreduciblerepresentations of a wide class of pointed Hopf algebras are parameterized bypairs of characters.

Biproducts and Two-Cocycle Twists of Hopf Algebras.

David E. Radford, Hans-Jürgen Schneider

Let $H$ be a Hopf algebra with bijective antipode over a field $k$ andsuppose that $R{#}H$ is a bi-product. Then $R$ is a bialgebra in theYetter–Drinfel’d category ${}_H^H{\mathcal YD}$. We describe the bialgebras$(R{#}H)^{op}$ and $(R{#}H)^o$ explicitly as bi-products $R^{\UOP}{#}H^{op}$and $R^{\UO}{#}H^o$ respectively where $R^{\UOP}$ is a bialgebra in${}^{H^{op}}_{H^{op}}{\mathcal YD}$ and $R^{\UO}$ is a bialgebra in${}^{H^o}_{H^o}{\mathcal YD}$. We use our results to describe two-cocycle twistbialgebra structures on the tensor product of bi-products.

A construction of a quotient tensor category.

Phung Ho Hai

Let f :G→A be a surjective homomorphism of transitive groupoid schemesand let L denote the kernel of f . The exact sequence of groupoid schemes 1 →L→G→A→1 induces a sequence of functors between the categoriesof finite representations of these groupoid schemes Rep f (A)→Rep f(G)→Rep f(L) . We show that the category Rep f (L) is a quotient category of Rep f (G) by Rep f (A) in an appropriate sense. We also generalize this setting tothe framework where the tensor categories are not necessarily Tannakacategories (i.e. not of the form Rep f (G) for some groupoid scheme G ), where we show under certain assumption the uniqueness of the quotienttensor category.

Noncommutative complete intersections and matrix integrals.

Pavel Etingof, Victor Ginzburg

We introduce a class of noncommutatative algebras called representationcomplete intersections (RCI). A graded associative algebra A is said to be RCIprovided there exist arbitrarily large positive integers n such that the schemeRep_n(A), of n-dimensional representations of A, is a complete intersection. Wediscuss examples of RCI algebras, including those arising from quivers.

There is another interesting class of associative algebras callednoncommutative complete intersections (NCCI). We prove that any graded RCIalgebra is NCCI. We also obtain explicit formulas for the Hilbert series ofeach nonvanishing cyclic and Hochschild homology group of an RCI algebra. Theproof involves a noncommutative cyclic Koszul complex, K(A), and a matrixintegral similar to the one arising in quiver gauge theory.

Normal forms for orthogonal similarity classes of skew-symmetric matrices.

Dragomir Z Djokovic, Konstanze Rietsch, Kaiming Zhao

Let F be an algebraically closed field of characteristic different from 2. Weshow that every nonsingular skew-symmetric n by n matrix X over F isorthogonally similar to a bidiagonal skew-symmetric matrix. In the singularcase one has to allow some 4-diagonal blocks as well. If further thecharacteristic is 0, we construct the normal form for O_n(F)-similarity classesof skew-symmetric matrices. In this case the known normal forms (as presentedin the well known book by Gantmacher) are quite different.

Finally we study some related varieties of matrices. We prove that thevariety of normalized nilpotent n by n bidiagonal matrices for n=2s+1 isirreducible of dimension s. As a consequence the skew-symmetric nilpotentbidiagonal n by n matrices are shown to form a variety of pure dimension s.

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Skew Derivations and Deformations of a Family of Group Crossed Products.

Sarah J. Witherspoon

We obtain deformations of a crossed product of a polynomial algebra with agroup, under some conditions, from universal deformation formulas. We show thatthe resulting deformations are nontrivial by a comparison with Hochschildcohomology. The universal deformation formulas arise from actions of Hopfalgebras generated by automorphisms and skew derivations, and are universal inthe sense that they apply to deform all algebras with such Hopf algebraactions.

Hochschild Cohomology and Graded Hecke Algebras.

Anne V. Shepler, Sarah Witherspoon

We develop and collect techniques for determining Hochschild cohomology ofskew group algebras S(V)#G and apply our results to graded Hecke algebras. Wediscuss the explicit computation of certain types of invariants undercentralizer subgroups, focusing on the infinite family of complex reflectiongroups G(r,p,n) to illustrate our ideas. Resulting formulas for Hochschildtwo-cocycles give information about deformations of S(V)#G and, in particular,about graded Hecke algebras. We expand the definition of a graded Hecke algebrato allow a nonfaithful action of G on V, and we show that there existnontrivial graded Hecke algebras for G(r,1,n), in constrast to the case of thenatural reflection representation. We prove that one of these graded Heckealgebras is equivalent to an algebra that has appeared before in a differentform.

Derived equivalences by quantization.

D. Kaledin

We assume given a smooth symplectic (in the algebraic sense) resolution X of an affine algebraic variety Y , and we prove that, possibly after replacing Y with an etale neighborhood of a point, the derived category of coherentsheaves on X is equivalent to the dervied category of finitely generated leftmodules over a non-commutative algebra R , a non-commutative resolution of Y in a sense close to that of M. Van den Bergh. We also prove some applications,such as: two resolutions are derived-equivalent; every resolution X admits a”resolution of the diagonal”; the cohomology groups of the fibers of the map X →Y are spanned by fundamental classes of algebraic cycles.

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