Technion, Department of Mathematics

 

Algebra seminar

 

Winter 2010/2011

Spring 2010

Winter 2009/2010

 

Spring semester 2011 - The seminar will meet on Thursdays at 13:30 in Amado 619.

 

 

 

 

 

DEPARTMENT OF MATHEMATICS

                    TECHNION

       ==================================

        Special Number Theory Seminar

                 Two Talks

 

       Note special date, time and room

       ==================================

1.

 

SPEAKER: Yossi Cohen, Technion

 

TITLE:   On the sequence gcd(a^n+1, b^n+1) (In Hebrew).

 

DATE:    Monday, June 13,     2011

 

PLACE:   Room  619    Amado Building, Technion.

 

TIME:    12:30-13:20

 

 

2.

SPEAKER: Jack Sonn, Technion

 

TITLE:   On the sequence gcd(Phi_N(a^n), Phi_N(b^n)); an

         application of the Chebotarev density theorem

 

DATE:    Monday, June 13,     2011

 

PLACE:   Room      Amado Building, Technion.

 

TIME:    13:30-14:20

 

ABSTRACT:

 

 

-          -   -

Speaker: Robert Shwartz, Bar Ilan University.

Title: "Interesting properties of the A-type Affine Coxeter 
groups".

 

-        - - - - -  -

SPEAKER: Darrel Haile, Indiana University and Technion.

 

 

TITLE:  Noninvertible Galois cohomology

 

DATE:    Thursday,   May 12        2011

 

PLACE:   Room   719    Amado Building, Technion.

 

TIME:    13:30

TIME:    13:30

 

ABSTRACT

In this lecture I want to introduce noninvertible Galois cohomology.  
As a motivating problem consider the following situation.  Let K/F be a 
finite separable extension of fields and let A be a central simple 
K-algebra.  We are interested  in F-algebras B that contain A and are 
"tightly connected" to  A.  I will give a formal definition in the 
lecture but an example would be a central simple F-algebra B containing 
A such that A is the full centralizer of K in B.  Such a central simple 
F-algebra B may not exist.  The classification of the F-algebras B that 
do exist, for a fixed A,  is an application of the noninvertible Galois 
cohomology I will discuss.

-        - - - - -- -

TECHNION

 

FACULTY OF MATHEMATICS

 

                        ALGEBRA SEMINAR

 

SPEAKER: Avi Goren, Technion.

 

TITLE:

 

Upper bound for the relative part of stabilizers in conjugacy classes of finite permutation

groups.

 

DATE: Thursday, May. 5, 2011

 

TIME: 13:30

 

PLACE: Amado 719

 

Abstract:

 

Let G be a finite transitive permutation group on a finite set S. Let A be a nonempty subset of S

and denote the pointwise stabilizer of A in G by C(A). Let T be a normal subset of G not

containing 1. Let U be a G-invariant subset o f S.

 

Well demonstrate a connection between the relative part of A in U and the relative part of C(A) in

T.

 

Same connection holds if G is a simple group .

--- ---

 

 

 

SPEAKER: Shlomi Gelaki, Technion

 

TITLE:   Virtually indecomposable tensor categories

 

DATE:    Thursday,  April 28th     2011

 

PLACE:   Room   719    Amado Building, Technion.

 

TIME:    13:30

 

ABSTRACT:

 

J-P. Serre proved the following: 1) For any group G, the spectrum of the Grothendieck ring of its representation category Rep(G) over any field k, is connected. 2) The same is true when G is replaced by any Lie algebra and k has characteristic 0.

 

J-P. Serre asked whether this remains true for any Tannakian category, and in particular for Lie algebras over k with positive characteristic (a case of particular interest to him).

 

Inspired by the this, we say that a tensor category over k is virtually indecomposable if its Grothendieck ring contains no nontrivial central idempotents. In my talk I will describe our following results: tensor categories with the Chevalley property, representation categories of affine (super)group schemes and formal (super)groups, and symmetric tensor categories of exponential growth in characteristic 0, are all virtually indecomposable tensor categories.

 

As special cases of these results, we obtain an alternative proof to Serre's theorem and a positive answer to his questions.

           

 

-----------

SPEAKER: Danny Neftin

 

TITLE:   Admissibility of finite groups over number fields

 

DATE:    Thursday, April 14        2011

 

PLACE:   Room   719    Amado Building, Technion.

 

TIME:    13:30

 

ABSTRACT:

 

A finite group G is called K-admissible if there is a Galois G-extension L/K and a division algebra D with center K for which L is a maximal subfield of D (in which case, D is called a G-crossed product division algebra).

 

Over number fields K, we present necessary conditions for K-admissibility, called K-preadmissibility, that are easier to verify. We shall show that in many cases preadmissibility implies admissibility, discuss the case of abelian groups A in detail and use tools from class field theory to determine the special cases in which A is K-preadmissibile but not K-admissible. Finally, we shall use our knowledge on preadmissibility to study an arithmetic equivalence relation induced by admissibility.

--------------------

SPEAKER: Crystal Hoyt, Bar-Ilan University

 

TITLE:   Finite W-algebras and primitive ideals

 

DATE:    Thursday, April 7    2011

 

PLACE:   Room   719    Amado Building, Technion.

 

TIME:    13:30

 

I will begin with a definition of finite W-algebras and then give a partial survey of the literature.  A finite W-algebra U(g,e) is a certain associative algebra which can be constructed from the universal enveloping algebra U(g) of a semisimple Lie algebra g and  nilpotent element e of g.  These algebras first arose from the work of Kostant on Whittaker modules in 1978.  It was shown in 2002 by Premet and Gan-Ginzburg that U(g,e) can be realized as a quantization of theSlodowy slice to the nilpotent Ad G orbit O_e.

 I will discuss the connection of finite W-algebras to primitive ideals of U(g) and Goldie rank polynomials, and in particular, the recent work of Premet, Losev and Brundan.

 

--------------------

SPEAKER:   Doron Puder, Hebrew University

 

TITLE:  On Primitive Free Words: A New Algorithm to Detect them and Measure Preservation.

 

DATE:  Thursday, March 31, 2011

 

PLACE:  Room 719 Amado Building, Technion.

 

TIME:    13:30

 

ABSTRACT: An element $w \in F_k$, the free group on $k$ generators, is called primitive* if it belongs to a basis (i.e. a minimal generating set) of $F_k$. In the first part of the talk I will present briefly a new elementary algorithm to determine whether a given word is primitive.

 

In the second part, the results from the first one will be used to show a tight connection between primitivity and measurepreservation. A free word $w \in F_k$ is called *measure preserving* if for every finite group $G$, the word map $w: G^k \to G$ induces uniform distribution on $G$ (given uniform distribution on $G^k$).

 

It is an easy observation that a primitive word is measure preserving. It is conjectured that the converse is also true. We prove it for $F_2$. In fact, both notions (primitivity, measure preservation) can be naturally extended to sets of free words. The strategy weuse actually shows that for sets of size at least $k-1$ in $F_k$, the two notions indeed coincide.

 

-------

·       Thursday, March 24, 13:30. Room: Amado 719.

Luda Marcus, Technion

 

Word Problem for Inverse Monoids Presented by a Single Relator.

This is joint work with Mark Berman, Uri Onn, and Pirita Paajanen.

Since Magnus it has been well known that one-relator groups have a decidable word problem. However, solvability of the word problem in one-relator monoids is far from being completely studied. Only few examples of inverse monoids with solvable word problem are known. Recently, the solvability of the word problem in inverse monoids with a single sparse relator has been announced by Hermiller, Lindblad and Meakin.

We consider certain one-relator inverse monoids. In our attempt to solve the word problem, we relay on the result of Ivanov, Margolis and Meakin which states that the word problem for the inverse $E$-unitary one-relator monoid is decidable if the membership problem for the prefix monoid is decidable. Thus, we first solve the membership problem for the prefix monoid and then apply the theorem to solve the word problem. Our methods employ van Kampen diagrams and word combinatorics.Thursday, March 17, 13:30. Room: Amado 719.

·       Thursday, March 17, 13:30. Room: Amado 719.

Michael Natapov, Technion

A graph theoretic approach to graded identities

·       Thursday, March 10, 13:30. Room: Amado 719.

Jamshid Derakhshan, Oxford

Uniform cell decomposition and applications to Chevalley groups.

This is joint work with Mark Berman, Uri Onn, and Pirita Paajanen.

We express definable integrals uniformly over non-Archimedean local fields extending results of Pas, Macintyre, and Denef--Loeser. We apply this to integrals over integral p-adic points of Chevalley groups, in particular proving that zeta functions counting conjugacy classes or dimensions of Hecke algebras in congruence quotients depend only on the size of the residue class field, for sufficiently large residue characteristic.

We also show that certain zeta functions associated with Chevalley groups defined over a non-Archimedean local field are given by a rational function satisfying a functional equation, and we obtain the same dependence only on the size of the residue class field.

 

·     Thursday, March 3,  13:30,  Room Amado  719.

 

 Avi Goren, Technion

 

Upper bound for the size of stabilizers in finite simple or transitive permutation groups.

 

1)     Let G be a finite transitive permutation group on a finite set S. Let A be a nonempty subset of S and denote the pointwise stabilizer of A in G by C(A). We'll find an upper bound for |C(A)| which depends on  |A| (and of course depends also on |G| and |S|)

Same bound holds if G is a simple group .

 

2)  Let G be a finite transitive permutation group on a finite set S. Let A be a nonempty subset of S and denote the pointwise stabilizer of A in G by C(A). Let T be a normal subset of G not containing 1. Let U be a G-invariant subset of S. We'll demonstrate a connection between the relative part of A in U and the relative part of C(A) in T.

Same connection holds if G is a simple group .

 

Organized by: David Chillag

 

__________________________________________________________________________________

 

Winter semester 2010/2011

 

 

·       Thursday, Jan 20, 12:30, Room: Amado 719

 

Jung-Miao Kuo, National Chung Hsing University, Taiwan 

A universality property of the Clifford algebras of ternary cubic forms 

         Abstract: In this talk, we introduce an algebra associated to a cubic curve

      *C* defined over a field *F* of characteristic not two or three. This algebra

      is Azumaya of rank 9 and its center is the affine coordinate ring of an

      elliptic curve, namely the Jacobian of the cubic curve *C*. The induced

      function from the group of *F*-rational points on the Jacobian into the

      Brauer group of *F* is a group homomorphism with image precisely the

      relative Brauer group of classes of central simple *F*-algebras split by the

      function field of *C*. Also, this algebra is split if and only if the cubic

      curve *C* has an *F*-rational point. Finally, we present a universality

      property of the family of such algebras. These results generalize Haile's

      work on the Clifford algebra of a binary cubic form.

 

 

·       Thursday, Jan 13, 12:30, Room: Amado 719

     

      RonyBitan, Technion

     

      The discriminant of an algebraic torus.

 

 Abstract:  Let T be an algebraic torus defined over a local field K with discrete valuation ring O  and finite residue field k. As T(K) is locally compact, it admits a multiplicative Haar measure which is unique up to a scalar multiplication.  

Let T(O) be the maximal compact subgroup of T(K). Its volume with respect to such a normalized measure is an arithmetic invariant which plays an important role in the quasi-discriminant formulated by Ono and Shyr for tori defined over global fields.  We use the standard integral model defined by Voskresenskii and its reduction over k in order to describe the structure of T(O) and measure it.

Due to a construction of Kottwitz, the result is expressed in terms of the cocharacter group of T.



·        Thursday, Jan 6, 12:30, Room: Amado 719

     

      Lucas Fresse, Hebrew University

     

      Smooth orbital varieties and orbital varieties with a dense B-orbit

        

          Abstract: To a nilpotent element x in a reductive Lie algebra, one can attach several algebraic varieties which play roles in geometric representation theory: its nilpotent orbit; the intersection of its nilpotent orbit with the nilradical of a Borel subalgebra (the irreducible components of this intersection are called orbital varieties); the fiber over x of the Springer resolution. There is a close relation between the Springer fiber over x and the orbital varieties attached to x. In this talk, we rely on this relation in order to study two properties of orbital varieties: the smoothness, and the property to contain a dense B-orbit. We concentrate on the type A. We provide several results which suggest that the two mentioned properties are related. This is a joint work with Anna Melnikov.

 

·       Thursday, Dec 30, 12:30, Room: Amado 719

 

      Yevgeny Musicantov, Technion

 

      On p^n dimensional semisimple Hopf algebras using reduction of extensions to equivariantizations

 

         Abstract: Let G be a finite group and let D be a fusion category. Consider the following situation, C is a G-extension of D and M is an indecomposable C-module category, such that, its restriction to D remains indecomposable. We prove that the dual C*_M is a G-equivariantization of D*_M. We apply the above theorem to the classification problem of the p^n-dimensional semisimple Hopf algebras. We completely classify the p^3-dimensional Hopf algebras, recovering the result of Masuoka.

 

 

 

·       Thursday, Dec 23, 12:30, Room: Amado 719

     

      Dmitry Kerner, U. Toronto 

     

      On local determinantal representations of hypersurfaces, aka maximally Cohen Macaulay modules

 

      Abstract:

 

      Let M be a square matrix whose entries are series in several variables. M is considered up to multiplication by locally invertible matrices from both sides.

In classical algebraic geometry such an object is called the local determinantal representation of the corresponding hypersurface {det(M)=0}. In commutative algebra such objects are called maximally Cohen Macaulay modules. In local geometry they correspond to specific elements of the local class group.

When the hypersurface singularity is of ADE type the classification
of such modules is finite. For higher singularities such modules have
moduli and the classification problem is wild.

I will present various points of view on these objects. Time permitting I will formulate some recent results: various necessary and sufficient criteria of decomposability of modules over the hypersurfaces.

 

 

 

·        Thursday, Dec 16, 12:30, Room: Amado 719

     

      Gary Zeitz, U. Oregon

 

      Unipotent elements in algebraic groups and nilpotent elements in

      their Lie algebras

 

 

·        Thursday, Dec 9, 12:30, Room: Amado 719

     

      Zur Itzhakian, Tel Aviv U.

     

      A Glimpse at Supertropical Algebra 

 

      Abstract:

     

      The objective of the talk is to introduce an algebraic structure

      rich enough to support algebraic formulations of properties of

      tropical geometry.  In order to bypass the lack of additive

      inverses in idempotent semirings, we consider a ``cover'' semiring

      structure that has a distinguished ``ghost ideal'' taking the

      place of the zero element in many of the theorems.

 

      This structure permits a systematic development both of polynomial

      algebra and of matrix algebra, yielding direct analogs  to many

      results and notions from commutative algebra. These establish the

      connection between algebra and geometry in the supertropical

      setting, leading also to a natural approach to linear algebra and

      semigroup representations.

 

 

 

·        Thursday, Dec 2, 12:30, Room: Amado 719

 

      Darrell Haile, U. Indiana and Technion

 

      The Clifford algebra of a quartic curve of genus one

 

 

      Abstract:

 

      This is joint work with Ilseop Han. For each irreducible quartic f over a

      field k, we construct a k-algebra A_f associated to the hyperelliptic

      affine curve C:y^2=f(x). We prove that A_f has many interesting

      properties. For example it is an Azumaya algebra of rank 4 over its center

      and its center is the coordinate ring of the affine elliptic curve E

      related to the Jacobian of C. Each simple image of A_f is a quaternion

      algebra. The simple images with center k then come from the rational

      points on E and the resulting function from the group of rational points

      on E to the Brauer group of k is a group homomorphism. We also prove that

      A_f is split if and only if the curve C has a k-rational point.

 

 

·        Thursday, Nov 18, 12:30, Room: Amado 719

 

      Eli Matzri (Technion)

 

       Non cyclic algebras with n-central elements.

 

      Abstract:


Let D be a division algebra ( i.e. every non zero element of D is invertible) of finite dimension over it's center, F.
It is known that [D:F]=n^2 for some natural number n which is called the degree of D.
For every element d in D, F[d] is a field. We call D a cyclic algebra if it has a subfield of dimension n which is Galois over F with group C_n. More generally we call D a crossed product with respect to a group G is D has a subfield Galois over F with group G.
Cyclic algebras are very important in the theory of finite dimensional division algebras due to a wonderful theorem of Merkurjev and Suslin stating that if the center contains enough roots of unity M_k(D) for some natural k is isomorphic to the tensor product of cyclic algebras. For D of degree a prime p, it is a well known theorem of Albert that says that D is cyclic iff it contains a non cental element d whoes p-th power is in the center. This theorem is clear if F contains a p-th root of unity but is true even if it does not.
P.Roquette asked if Alberts criterion is true even for non prime degrees, and gave a counter example in degree four.
 In a recent work with L.H. Rowen and U. Vishne we give examples of algebras D, of degree p^2 and a non central element whose p^2 power is in the center but  D is not cyclic thus showing Alberts criterion for cyclicity  is false for algebras of degree p^2 for every prime p.

 

 

·        Thursday, Nov 4, 12:30, Room: Amado 719

 

      Prof. Antonio Giambruno  (University of Palermo, Italy) 

 

      Polynomial identities and codimension growth of algebras

 

      Abstract:

 

      The determination of the precise polynomial identities of a given PI
 algebra  (and even a generating set) seems to be an extremely hard
problem in general  and so people tried, instead, to estimate the "number" of
identities or what  is better behaved, the number of "non identities" of a certain
degree of a given algebra.  In the lecture I will explain the essence of the codimension

theory and present results for different type of PI algebras
as associative algebras, $G$-graded algebras, Lie algebras, arbitrary nonassociative.

 

 

 

·        Thursday, Oct 28, 12:30, Room: Amado 719

     

      Shifra Reif (Weizmann Institute of Science)

 

      Denominator identities for Lie superalgebras

     

      Abstract:

      In 1972 Macdonald generalized the Weyl denominator identity to affine root systems. The simplest example of these identities turned out to be the famous Jacobi triple product identity. In 1994 V. G. Kac and M. Wakimoto stated an analog for some affine Lie superalgebras and showed that it has applications in number theory. We provide identities for the rest of the (non-twisted) affine Lie superalgebras and deduce the Jacobi formula for counting the number of presentations of an integer as a sum of 8 squares (joint with M. Gorelik).

 

 

 

·       Thursday, Oct 21, 12:30, Room: Amado 719

 

      Crystal Hoyt (Bar Ilan University)

     

      Good gradings of basic Lie superalgebras

 

Abstract:  Z-gradings of finite-dimensional simple Lie algebras were classified by V.G. Kac and A.G. Elashvili in 2005.  This problem arose in connection to W-algebras. We will discuss the classification of good Z-gradings for basic Lie superalgebras. A finite-dimensional simple Lie superalgebra g=g_0 + g_1 is called basic if g_0 is a reductive Lie algebra  and there exists a nondegenerate invariant supersymmetric bilinear form on g.   Basic Lie superalgebras were classified by V.G. Kac in 1977.  These are:

sl(m|n) : m not equal to n, psl(n|n), osp(m|2n), D(2,1,a), F(4) and G(3).

 

A Z-grading g= \sum_{j in Z} g(j)  is called good if there exists e in g_0(2) such that the map (ad e):g(j)-->g(j+2) is injective for j less than or equal to -1 and is surjective for j greater than or equal to -1. If e in g_0 belongs to an sl(2)-triple {e,f,h} where [e,f]=h, [h,e]=2e and [h,f]=-2f, then the Z-grading of g given by the eigenspaces of (ad h) is a good Z-grading for e, and is called a Dynkin grading.

All good gradings of the exceptional Lie superalgebras, F(4), G(3), and D(2,1,a), are Dynkin gradings. The good Z-gradings of sl(m|n), psl(n|n) and osp(m|2n) are classified using pyramids, analogously to the Lie algebra setting.

 

 

Organized by: Jack Sonn

 

__________________________________________________________________________________

 

Spring semester 2010

 

 

·        Monday, June 21, 13:30, Room: TBA

 

Lior Bary-Soroker (IEM, Universität Duisburg-Essen)

 

       Irreducible values of polynomials

 

    Abstract: Does there exist a polynomial f(X) such that all polynomials

f(X), f(X)+1, f(X)+2, ..., f(X)+285 are irreducible? Clearly the

answer depends on the field the coefficients are taken from. We will

discuss a generalization of this problem (aka Schinzel's hypothesis H

for polynomial rings),  some recent results, and the connection with

Hilbert's irreducibility theorem.

 

·         Thursday, June 3, 13:30, Room: Amado 919

           

Eyal Subag (Technion)

 

       Inonu-Wigner contractions as direct limits

 

    Abstract: It is often the case that one physical theory with a symmetry group approximates in a certain limit another physical theory which also possesses a symmetry group. In a famous paper of Inonu and Wigner from 1953 a new notion of contraction of Lie algebras and their representations

was used to construct several such examples. While the process of contracting Lie Algebras was completely rigorous and clear, the process

of contraction of representations was very sketchy and involved various kinds of limit formulas.

In my talk I will describe a new definition of contraction of Lie algebra representations in which the representation space of the "limit" representation is a direct limit of a certain family of linear spaces. Using this definition I will explain the various formulas of contraction of Lie algebra representations which were given by Inonu and Wigner. If time permits I shall also describe some new examples.

 

 

·         Monday, May 31, 14:30, Room: Amado 815  (SPECIAL SEMINAR—NOTE CHANGE OF TIME AND PLACE)

 

Ehud Meir (Technion)

 

Wall's complex, projective resolutions and complexity of modules.

 

 

Abstract: Let G be a finite group, let k be a field of prime
characteristic, and let M be a finitely generated kG module. The
complexity of M is a numerical invariant which measures how far M is
from being projective (for example- M has complexity 0 if and only if
M is projective).
Like many other homological properties, the complexity of M can be
detected over elementary abelian subgroups of G (This is the content
of Alperin- Evens Theorem).

In this talk we will consider a generalization of the notion of
complexity for infinite groups and arbitrary rings. By considering
finite quotients of a group G, we will be able to prove a
generalization of Alperin- Evens Theorem. We will prove that under
certain assumptions, the complexity of a module M over G can be
detected over some set of finite index subgroups of G. For the proof,
we use in a fundamental way a theorem of Serre and a complex that was
originally constructed by Wall. We will also show how we can use our
results in order to construct resolutions for the integral special
linear groups.

 

 

·         Thursday, May 27, 13:30, Room: Amado 919

 

Ori Parzan (Hebrew U)

 

      Isospectrality and representations of finite groups

 

Abstract:  A simple phenomenon in finite groups stands behind several famous constructions: arithmetically equivalent number fields (Gassman 1925),   isospectral manifolds (Sunada 1985), drums (Gordon et. al 1992) and graphs (Brooks 1997), and Lie groups with the same Witten zeta function (Larsen 2004). The phenomenon is this: $H$ and $H'$ being conjugate subgroups of $G$ implies that ${\rm Ind}_{H}^{G}1\cong{\rm Ind}_{H'}^{G}1$, but for some $G,H,H'$ the converse is false. We explain how to construct isospectral objects exploiting a more general situation: ${\rm Ind}_{R}^{G}R\cong{\rm Ind}_{H'}^{G}R'$ with $R$, $R'$ being any representations of $H$ and $H'$. The talk is intended for general audience and assumes only basic notions of the representation theory of finite groups (e.g. orthogonality of irreducible characters).
This is joint work with Ram Band.

 

 

 

·         Thursday, April 29, 13:30, Room: Amado 919

 

Ehud Meir (Technion)

 

       On the Hopf Schur group of a field     

 

Abstract:

 

     Let k be a field. We ask what k-central simple algebras can we get as
quotients
of Hopf algebras. We call the corresponding subgroup of
Br(k) the Hopf Schur group of k.
This generalizes the question of what central simple k-algebras are
quotients of group algebras (the so called Schur algebras) and the
Schur subgroup of Br(k). Since group algebras are Hopf algebras, every
Schur algebra is a Hopf Schur algebra.
The Schur subgroup might be a very small subgroup of Br(k), and we ask
what other Hopf Schur algebras can we find.

In this talk I will try to explain why actually every k-central simple
algebra is a Hopf Schur algebra up to Brauer equivalence. We shall do
so by giving a representation theoretic definition of Hopf algebras,
and by considering forms of Hopf algebras.
All the relevant notions will be explained. 

 

(Partially based on a joint work with Eli Aljadeff, Juan Cuadra and
Shlomo Gelaki)

 

 

·         Thursday, April 22, 13:30, Room: Amado 919

 

Leonid Makar-Limanov (Wayne State U.)

 

       A new approach to the two-dimensional Jacobian conjecture

 

     Abstract:

 

     In my talk I'll introduce and discuss some properties of

     a three-dimensional polytope which can be attached to a pair of

     polynomials with the constant Jacobian. This approach gives new

     restrictions on a potential counterexample to the Jacobian Conjecture.

 

 

·         Thursday, April 8, 13:30, Room: Amado 919

 

Yuval Ginosar (Haifa U)

 

       On two notions due to von Neumann

 

·         Thursday, Mar 18, 13:30, Room: Amado 919

 

Adi Wolf (Technion)

 

Automorphisms and Forms of Path Algebras

 

Abstract:

Given a Galois extension K/F and a finite dimensional F-algebra A, we consider the question
of classifying K/F-forms of A. When A=Mat_{n}(F) or A=F^{n} these are known to be the central
simple algebras of degree n and the étale algebras of dimension n, respectively.
We show how some algebras whose nilpotent radical is not trivial, namely hereditary path
algebras with acyclicity restriction on the quiver, are classified by certain H^{1}-cohomology
sets and provide a combinatorial description of their K/F-forms in some cases.

This talk is based on a research thesis done under the supervision of Prof. Eli Aljadeff.

 

 

·         Thursday, Mar 11, 13:30, Room: Amado 919

 

Shlomo Gelaki (Technion)

 

Semisimplicity in symmetric tensor categories

 

Abstract:

Let n be a positive integer, and let k be a field of characteristic 0
or p > n. Let G be any group, let C = Rep(G) be the category of finite
dimensional representations of G over k, and let V be an object of C.
 Serre proved that if the nth exterior power of V (respectively, the
nth symmetric power of V) is semisimple and the dimension of V is not
equal in k to 2,3,...,n (respectively, -n,...,-2), then V is semisimple.
 Moreover, Serre explained that the same results hold for Lie algebras,
and conjectured that they should hold more generally in any symmetric
tensor category C over k.
 Recently we proved a generalization of Serre's conjecture. Namely, we
replaced Rep(G) by any symmetric tensor category C over k, as suggested
by Serre, and moreover replaced the nth exterior power and symmetric power
by any Schur functor S_{\lambda} (here \lambda is a partition of n).
In my talk I will state our results and explain its proofs.

 

Organized by: Jack Sonn

 

_________________________________________________________________________________________ 

Winter semester 2009/2010

 

 

 

·         Thursday, Feb. 18, 11:30, Room: Amado 814

 

Emanuele Pacifici (Univ. of Milan)

 

On Zeroes of Characters of Finite Groups

 

 

·         Thursday, Jan 28, 10:30, Room: Amado 814

 

Hila Maayan (Technion)

 

On Zeroes of Characters of Finite Groups

(The talk will be given in Hebrew)

 

 

·         Thursday, Jan 21, 10:30,  Room: Amado 814

 

Ido Efrat (BGU)

 

The descending central sequence of an absolute Galois group

 

Abstract
Given a field $F$, the descending $p$-central sequence of its absolute Galois group
gives a tower of Galois $p$-extensions $F^{(i)}$ of $F$.
We show that the Galois group of the extension $F^{(3)}$ of $F$ already encodes
much of the arithmetical structure of $F$, as well as its mod-$p$ cohomology.
This is used to construct new examples of profinite groups which are not realizable
as absolute Galois groups.

 

 

·         Thursday, Jan 14, 10:30,  Room: Amado 814

 

Rony Bitan (Bar Ilan)

 

The discriminant of an algebraic torus

 

Abstract: 

Let K be a global field.
The relation between arithmetic invariants of K such as the discriminant,
regulator, class number and the residue of the Dedekind Zeta-function is
well-known.  Let T be an algebraic K-torus.
Following Ono and Shyr we use analogues of these invariants of T to
express this relation for T, by gathering the information in all local
fields.  Locally, for any prime p, we compute the discriminant of T using
an integral scheme of T, namely the Neron model of T.
This local information is expressed in terms of the group of cocharacters of
T over K_p.

 

 

·         Thursday, Dec 31, 10:30, Room: Amado 814

 

David Chillag (Technion)

 

Rationality of finite groups and primes dividing their orders

 

Abstract: 

A rational element of a group is an element for which all the generators of the cyclic

subgroup it generates, are conjugate. A group is rational if all its elements are rational.

We discuss rational groups and variations of such, in particular the prime divisors of their orders.
 

 

·       Thursday, Dec 24, 10:30, Room: Amado 814

 

Pavel Shumyatsky (Univ. of Brazilia)

 

 On elements of prime-power index in finite groups
 

 

·       Thursday, Dec 10, 10:30, Room: Amado 814

 

Alexei Belov-Kanel (Bar Ilan U.)

 

The Specht problem in positive characteristic

 

Abstract: 

The Specht problem for an algebra A asks whether the ideal of polynomial

identities of A is finitely based (to be defined in the lecture).

In the lecture I will discuss the Specht problem where A is an associative

algebra over a field of positive characteristic. The main tools used in

the solution are taken from representation theory of associative algebras.

In particular one needs to consider the interaction between the semisimple

part and the radical of the algebra. Similar problems may be posed for

other structures as groups and Lie algebras.

 

 

·       Monday, Nov 30, 10:30, Room: Amado 719

 

Enric Ventura (Universitat Politecnica de Catalunya)

 

Whitehead Minimization in Polynomial Time

 

Abstract: 

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first

fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier (classical) algorithms had an exponential

dependency in the rank of the free group. It follows that the primitivity problem (to decide whether a word is an element of some basis of the free group) and the free factor problem can also be solved in polynomial time.

 

 

 

·       Thursday, Nov 26, 10:30, Amado 814

 

Eli Matzri (Technion)

 

A birational description of the Brauer Severi variety of a CSA in terms of
the reduced norm.

 

Abstract: 

Let A be a central simple algebra over a field F. To every such A one can associate an algebraic variety called the Brauer Severi variety of A and denoted BSV(A), which is the variety of minimal left ideals of A. Two central simple algebras A and B  are isomorphic iff they have isomorphic BSVs.
It was conjectured by Amitsur that A and B  of the same dimension have birational BSV iff they generate the same subgroup of Br(F).
Amitsur proved that if BSV(A) and BSV(B) are birational then <[A]>=<[B]>. The other direction is still open in general but is known for some specific cases. We give a birational description of BSV(A) in terms of the
reduced norm, and use it to reprove Amitsur's conjecture in the case of symbol algebras.
Then we will discuss a possible birational description of generalized
BSVs. 

 

 

 

·       Thursday, Nov 19, 10:30, Amado 814

 

Michael Natapov (Technion)

 

Graded identities of matrix algebras with a fine group grading

 

Abstract: 

Let A be a ring of n-by-n matrices over an algebraically closed field. Any group grading on A is known to be induced from an elementary and a fine gradings.  While there is a wide literature on elementary gradings and the corresponding graded polynomial identities, the fine gradings and their polynomial identities are less studied. In particular, a minimal generating set for the ideal of G-graded polynomial identities of A with a fine G-grading is known in a case of G abelian only. In a joint work with Darrell Haile we consider a certain family of non-abelian groups G, and study the G-graded polynomial identities on A with a fine G-grading in a search for the minimal generating sets of the ideal of identities.

 

 

 

·       Thursday,  Nov 12, 10:30, Amado 919

 

Michael Schein (Bar Ilan)

 

On irreducible supersingular mod p representations of GL_2(F)

 

Abstract:

Let F be a finite extension of Q_p.  The mod p local Langlands correspondence should be a natural bijection between n-dimensional mod p representations r of the absolute Galois group of F and a certain class of irreducible mod p representations L(r) of GL_n(F).  Irreducible Galois representations correspond to supersingular representations of GL_n(F).

The mod p representation theory of GL_n(F) is poorly understood, and, apart from some special cases, few irreducible supersingular representations have been constructed.  One can use generalizations of Serre's conjecture to specify what the socle of the restriction of L(r) to a maximal compact subgroup should be.  We will show that supersingular representations of GL_2(F) with such socles are generically irreducible and discuss work in progress to construct families of such representations.  The relevant notions will be defined.

 

 

 

·       Thursday, Nov 5, 10:30, Amado 919

 

Daniele d'Angeli (Technion)

 

Self-similar groups and finite Gelfand pairs

 

Abstract:

The class of self similar groups has been largely studied in the last decades providing interesting examples of groups with special and exotic properties. These groups are groups of automorphisms of rooted q-ary trees. The theory of finite Gelfand pairs appears in this context when we consider the action of a (level transitive) self-similar group on the levels of the tree and the stabilizer of a fixed vertex. In this talk I will discuss the example of the so called Basilica group, showing that the action on each level gives rise to a symmetric Gelfand pair.

 

 

 

·       Thursday, Oct 29, 10:30, Amado 919

 

John Meakin (University of Nebraska)

 

Submonoids of Groups

 

Abstract:

Semigroups that embed in groups have received considerable attention in the classical and modern literature in both semigroup theory and group theory. For example, one obtains useful information about large classes of groups such as Adian groups, braid groups and Artin groups by studying their positive submonoids. I will discuss some of these results as well as some recent work and unsolved problems linking algorithmic problems for inverse monoids to corresponding problems for submonoids of groups.

 

 

 

·       Thursday, Oct 22, 10:30,  Amado 919

 

Danny Neftin (Technion)

 

On semiabelian groups and the minimal ramification problem

 

Abstract:

Let $p$ be a prime number and $G$ a $p$-group of rank $r$, i.e. $G$ is generated by $r$ elements and not less.  It is conjectured that $G$ can be realized over $Q$ with exactly $r$ ramified primes. Kisilevsky and Sonn showed the conjecture holds for a certain family of semiabelian groups and asked whether this family is the family of all semiabelian groups. This question was answered positively. We shall discuss the proof and how it can be used to simply the proof of Kisilevsky and Sonn and extend it. (Joint work with Hershy Kisilevsky and Jack Sonn)

 

 

Organized by: Jack Sonn