THE ELEVENTH AMITSUR MEMORIAL SYMPOSIUM
Ramat-Gan, Monday, 26 June and Tuesday, 27 June, 2006

The annual algebra symposium in memory of Professor Shimshon Amitsur will take place this year at Bar-Ilan University in Ramat-Gan. 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 Bar-Ilan)
Yuval Ginosar   (Haifa University)
Istvan Heckenberger   (University of Leipzig)
Michael Larsen   (Indiana University)
Susan Montgomery   (University of Southern California)
Michael Natapov   (Bar-Ilan University)
Amitai Regev   (Weizmann Institute)
Yoav Segev   (Ben-Gurion University)
David Saltman   (University of Texas at Austin)
Yehuda Shalom  (Tel-Aviv University)
Lance Small   (University of California, San Diego)
Amnon Yekutieli   (Ben-Gurion University)

Location:
Bar-Ilan University campus, Beck Auditorium
(Building 410, ground floor). Here is a map.

PROGRAM

Monday, 26 June 2006

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 K-group 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

Tuesday, 27 June 2006

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

ABSTRACTS

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= , End A be the semigroup of endomorphisms of A, and Aut End A be the group of automorphisms of the semigroup End A. Let Ass is the variety of associative algebras over K, Ass# is the category of finitely generated free algebras from Ass and Aut Ass # is the group of automorphisms of the category Ass# . It has been shown by B. Plotkin that categorical and geometrical equivalences of algebras are determined by the structures of the groups Aut End A and Aut Ass #.
We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A and the group Aut Ass# is generated by semi-inner 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 2-cocycle $f:G\times G\rightarrow S^*$ 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 finite-dimensional semi-simple Hopf algebra. In this talk we will discuss the Frobenius-Schur indicator of an irreducible H-module, 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 self-dual 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 G-grading. We construct an Azumaya algebra which is universal with respect to all G-graded 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 T-ideal of G-graded 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 K-group 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 K-isotropic) 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 Kneser-Tits problem) and (2) What happens with G(K) when G(K) does not have unipotent elements (i.e. G is K-anisotropic). 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 quasi-isomorphism 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 quasi-isomorphism between the mixed resolutions Mix(T_{poly,X}) and Mix(D_{poly,X}). If certain cohomologies vanish (e.g.\ if X is D-affine) 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 algebro-geometric analogue of Kontsevich's celebrated result. Eprint: math.AG/0310399.

For more details please contact the organizers at rowen@math.biu.ac.il or vishne@math.biu.ac.il .

Last updated: 11/6/2006.

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= , End A be the semigroup of endomorphisms of A, and Aut End A be the group of automorphisms of the semigroup End A. Let Ass is the variety of associative algebras over K, Ass# is the category of finitely generated free algebras from Ass and Aut Ass # is the group of automorphisms of the category Ass# . It has been shown by B. Plotkin that categorical and geometrical equivalences of algebras are determined by the structures of the groups Aut End A and Aut Ass #. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A and the group Aut Ass# is generated by semi-inner 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 2-cocycle $f:G\times G\rightarrow S^$ gives rise to a crossed product $S^fG$, which is an $R-$order inside the central simple algebra $LG$. 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 $S^fG$, 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 finite-dimensional semi-simple Hopf algebra. In this talk we will discuss the Frobenius-Schur indicator of an irreducible H-module, 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 self-dual 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 G-grading. We construct an Azumaya algebra which is universal with respect to all G-graded 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 T-ideal of G-graded 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 K-group 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 K-isotropic) 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 Kneser-Tits problem) and (2) What happens with G(K) when G(K) does not have unipotent elements (i.e. G is K-anisotropic). 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 quasi-isomorphism 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 quasi-isomorphism between the mixed resolutions Mix(T_{poly,X}) and Mix(D_{poly,X}). If certain cohomologies vanish (e.g.\ if X is D-affine) 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 algebro-geometric analogue of Kontsevich's celebrated result. Eprint: math.AG/0310399.