THE ELEVENTH AMITSUR MEMORIAL SYMPOSIUM RamatGan, Monday, 26 June and Tuesday, 27 June, 2006


The annual algebra symposium in memory of Professor Shimshon Amitsur will take place this year at BarIlan University in RamatGan.
The symposium is partially sponsored by The Gelbart Research Institute for Mathematical Sciences.
Please register by sending the following registration form. There are no fees.
Invited Speakers:
Alexei Belov (The Hebrew University and BarIlan)
Yuval Ginosar (Haifa University)
Istvan Heckenberger (University of Leipzig)
Michael Larsen (Indiana University)
Susan Montgomery (University of Southern California)
Michael Natapov (BarIlan University)
Amitai Regev (Weizmann Institute)
Yoav Segev (BenGurion University)
David Saltman (University of Texas at Austin)
Yehuda Shalom (TelAviv University)
Lance Small (University of California, San Diego)
Amnon Yekutieli (BenGurion University)
Location:
BarIlan University campus, Beck Auditorium
(Building 410, ground floor). Here is a map.
9:30  10:00  Reception 
10:00  11:00  David Saltman 
Division algebras over surfaces  
11:15  12:00  Yoav Segev 
On homomorphic images of G(K) for an absolutely simple Kgroup G  
12:15  13:00  Yehuda Shalom 
Elementary linear groups and Kazhdan's property (T)  
13:00  14:30  Lunch break 
14:30  15:15  Amitai Regev 
Some numerical invariants for p.i. algebras  
15:30  16:15  Istvan Heckenberger 
Nichols algebras of diagonal type  
16:15  16:45  Tea break 
16:45  17:30  Michael Larsen 
Abelian varieties over cyclic fields  
18:30  Symposium dinner 
9:30  10:00  Reception 
10:00  11:00  Susan Montgomery 
Representations of semisimple Hopf algebras  
11:15  12:00  Alexei Belov 
Automorphisms of the semigroup of endomorphisms of free associative algebras  
12:15  13:00  Michael Natapov 
On simple algebras and their fine gradings  
13:00  14:30  Lunch break 
14:30  15:15  Yuval Ginosar 
Hereditary crossed product orders  
15:30  16:15  Amnon Yekutieli 
Deformation quantization in algebraic geometry  
16:15  16:45  Tea break 
16:45  17:45  Lance Small 
Amitsur's tricks 
Alexei Belov  Automorphisms of the semigroup of endomorphisms of free associative algebras 
Let A=A(x_1, ...,x_n) be a free associative algebra freely generated over
a field K by a set X= We prove that the group Aut End A is generated by semiinner and mirror automorphisms of End A and the group Aut Ass# is generated by semiinner and mirror automorphisms of the category Ass#. Joint with A. Berzins and R. Lipyanski 
Yuval Ginosar  Hereditary crossed product orders  Let $S$ be the integral closure of a discrete valuation ring $R$ in a finite Galois extension $L/K$ ($K$ the quotient field of $R$) with Galois group $G$. Then any 2cocycle $f:G\times G\rightarrow S^*$ gives rise to a crossed product $S^f*G$, which is an $R$order inside the central simple algebra $L*G$. The question of when such classical crossed products are hereditary was considered in several works since the 60's. However, the wildly ramified case was surprisingly neglected, possibly because of an error in one of the textbooks. In this talk we give a criterion for the heredity of $S^f*G$, showing that it may hold even when $S/R$ is not tamely ramified. Joint work with A. Braun and A. Levy. 
Istvan Heckenberger  Nichols algebras of diagonal type  Nichols algebras of diagonal type play a crucial role in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. In this talk the astounding relation of these algebras to semisimple (super) Lie algebras will be enlightened by the introduction of arithmetic root systems and Weyl groupoids. The latter objects have a very rich structure and admit complete classification under some finiteness hypotheses. In the talk some of the methods and results will be explained. 
Michael Larsen  Abelian varieties over cyclic fields  Let K be a field of characteristic zero such that Gal(\bar K/K) is cyclic. Then every abelian variety over K has infinite rank. I will sketch the proof (which relies on group theory and geometry) and discuss some open problems. 
Susan Montgomery  Representations of semisimple Hopf algebras  Let H be a finitedimensional semisimple Hopf algebra. In this talk we will discuss the FrobeniusSchur indicator of an irreducible Hmodule, which extends the classical notion in group representations. This invariant (and its generalizations) is one of the few invariants known for Hopf algebras and has already proved to be quite useful: it has been used in classification theory and in results about the prime divisors of the exponent of H. Thus it is of interest to compute the indicator explicitly. Let H = D(G), the Drinfel'd double of the group algebra of G. In joint work with Kashina and Mason, we showed that if G is the symmetric group S_n, then the indicator of every irreducible D(G)module V is +1; in particular they are all selfdual and ``real''. More recently, with R. Guralnick, we extend this fact to D(G) for any finite real reflection group G. These results generalize the classical results for G itself. 
Michael Natapov  On simple algebras and their fine gradings  We consider a matrix algebra M_n(C), over the field of complex numbers, graded by a group G with a fine Ggrading. We construct an Azumaya algebra which is universal with respect to all Ggraded forms of M_n(C). The central localization of this algebra is a simple algebra of dimension n^2 over its center and graded by the group G. This Universal central simple algebra is a division algebra if and only if G is a member of a short list of special groups. Also we give a finite generating set of the Tideal of Ggraded polynomial identities for M_n(C). This is joint work with Eli Aljadeff and Darrell Haile. 
Amitai Regev  Some numerical invariants for p.i. algebras.  The sequence of codimensions of a p.i. algebra $A$ carries significant information about the identities of $A$. We review some major results about the asymptotics of these sequences. We also discuss some recent results here  when the algebra $A$ satisfies a Capelli identity. This recent work is joint with A. Berele. 
David Saltman  Division algebras over surfaces  A great deal of the theory of division algebras is the theory of these objects in, roughly speaking, one dimensional situations. In Amitsur's 1989 conference I talked about division algebras over some very special surfaces. Here I would like to talk more on this same theme, with more kinds of surfaces. Particular attention will be paid to the problem of whether prime degree division algebras are cyclic. 
Yoav Segev  On homomorphic images of G(K) for an absolutely simple Kgroup G  Let G be an absolutely simple simply connected algebraic group defined over the field K. A fantastic theorem (because it is so general!) due to Jacque Tits says that when G(K) contains unipotent element (i.e. it is Kisotropic) G(K)^+ is projectively simple (i.e. normal subgroups are contained in the center) for any field K of size larger or equal 4. Here G(K)^+ is roughly the subgroup generated by unipotent elements. The problems that remain are: (1) What is the structure of G(K)/G(K)^+ (the KneserTits problem) and (2) What happens with G(K) when G(K) does not have unipotent elements (i.e. G is Kanisotropic). I will survey the current state of these problems, the conjectures related to them and some recent developments 
Yehuda Shalom  Elementary linear groups and Kazhdan's property (T)  A group is said to have Kazhdan's property (T) if every isometric action of it on a Hilbert space has a global fixed point. Assuming no prior familiarity with this notion, we will discuss the proof of the following result: For any finitely generated commutative ring R with 1, the group EL(n,R) generated by the elementary nxn matrices over R, has Kazhdan's property (T) once n > 1 + Krull dim R. We will particularly try to explain how Bass important ring theoretic notion of stable range becomes a key tool in the proof of this result. 
Lance Small  Amitsur's tricks  Two old results of Amitsur on algebras over uncountable fields will be applied to problems about stably noetherian rings and the behavior of primitive ideals in certain affine algebras. The principal results in these areas are due to Bell and Bell and Rogalski. 
Amnon Yekutieli  Deformation quantization in algebraic geometry  We study deformation quantization of the structure sheaf O_X of a smooth algebraic variety X in characteristic 0. The universal deformation formula of Kontsevich gives rise to an L_infinity quasiisomorphism between the pullbacks of the DG Lie algebras T_{poly,X} and D_{poly,X} to the bundle of formal coordinate systems of X. Using simplicial sections we obtain an induced twisted L_infinity quasiisomorphism between the mixed resolutions Mix(T_{poly,X}) and Mix(D_{poly,X}). If certain cohomologies vanish (e.g.\ if X is Daffine) it follows that there is a canonical function from the set of gauge equivalence classes of formal Poisson structures on X to the set of gauge equivalence classes of deformation quantizations of O_X. This is the quantization map. When X is affine the quantization map is in fact bijective. This is an algebrogeometric analogue of Kontsevich's celebrated result. Eprint: math.AG/0310399. 