# CLASSES OF ELEMENTS OF SEMISIMPLE ALGEBRAIC GROUPS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 R. STEINBERG 277 CLASSES OF ELEMENTS OF SEMISIMPLE ALGEBRAIC GROUPS ROBERT STEINBERG Given a semisimple algebraic group G, we ask: how are the elements of G partitioned into conjugacy classes and how do these classes fit together to form G? The answers to these questions not only throw light on the algebraic and topological structure of G, but are important for the representation theory and functional analysis of G; conversely, of course, we can expect the representations and functions on G to contribute to the answers to the questions. In what follows, we discuss some aspects of these questions, present some known results, and pose some unsolved problems. To do this, we first recall some basic facts about semisimple algebraic groups. Our reference for all this is [7]. A linear algebraic group G is a.subgroup of some GL n (К) (n>\, К an algebraically closed field) which is the complete set of zeros of some set of polynomials over К in the n 2 matric entries. The Zariski topology, in which the closed sets are the algebraic subsets of G, is used. G is semisimple if it is connected and has no nontrivial connected solvable normal subgroup; it is then the product, possibly with amalgamation of centers, of some simple groups. The simple groups have been classified, in the KilHng- Cartan tradition. Thus there are the classical groups (unimodular, symplectic, orthogonal, spin, etc.) and the five exceptional types. The rank of a semisimple group is the dimension of a maximal torus (a subgroup isomorphic to a product of GL 4 's). We also need the notions of simple connectedness and Lie algebra as they apply to linear algebraic groups, but we omit the definitions. Henceforth К denotes a fixedalgebraically closed field, p its characteristic, G a simply connected semisimple algebraic group over K, and r the rank of Gf The simplest example is G = SL r +i (K)- L denotes the Lie algebra of G. If x is an element of G, we write G x (resp. L x ) for the centralizer of x in G (resp. L). We shorten conjugacy class to class, and dimension to dim. Now we take up the problem of surveying the conjugacy classes 3f G and finding for them suitable representative elements. Recall that an element of G is semisimple if it is diagonalizable, and unipotent if its characteristic values are all 1, and that every element can be decomposed uniquely x =x s x u as a product of commuting semisimple and unipotent elements. An element x will be called regular if dim G x = r, or, equivalently, if dim G x is minimal [12, p. 49]; here x need not be semisimple and may in fact be unipotent. Various chara- :terizations, hence alternate possible definitions, of regular elements

2 278 ПОЛУЧАСОВЫЕ ДОКЛАДЫ HALF-HOUR REPORTS may be found in [12]. The regular (resp. semisimple) elements form a dense open set with complement of codimension 3 (resp. 1) in G [12, 1.3 and 6,8]; thus most elements are regular (resp. semisimple). In [12, 1.4] we have constructed a closed irreducible cross-section, C, isomorphic to affine r-space, for the regular classes of G. Also, we have proved [12, 1.2]: (1) The map x-±- x s sets up a I-I correspondence between the regular and the semisimple classes of G. Thus by choosing the elements of С and their semisimple parts we obtain a system of representatives for the regular classes and for the semisimple classes of G. If G is the group SL n, the representatives of the regular classes chosen turn out to be in one of the Jordan canonical forms. Thus we are led to our first problem. (2) Problem. Determine canonical representatives, simitar to those given by the Jordan normal forms in SL n, for all of the classes of G, not just the regular ones. In proving the results described above, we are naturally led to consider the algebra of regular class functions, those regular functions on G that are constant on the conjugacy classes of G. This is a polynomial algebra generated by the characters Xi>OC2, *> Xr of the fundamental representations of G [12, 6.1]. (If G = SL r+i, the %'s are just the coefficients of the characteristic polynomial with the first and last terms excluded.) First of all these functions give a very useful characterization of regular elements of G: x is regular if and only if the differentials of % u % 2 Xr are linearly independent at x [12, 1.5]. And secondly they are related to our classification problem by the following result [12, 6.17]: (3) The map x -> (% { (x), % 2 (x) % r (x)) on G induces a bijection of the regular classes, and of the semisimple classes, onto the points of affine r-space. (4) Problem.. Assume that x and y in G are such that for every rational representation (p, V) of G the elements p (x) and p (y) are conjugate in GL (V). Prove that x and y are conjugate in G. By (3) this holds if x and y are both regular or both semisimple, and it can presumably de checked when G is a classical group, but we have not done this in all cases. To continue our discussion, we will use the following result. (5) If y is a semisimple element of G, then G y is a connected reductive group (i.e. the product of a semisimple group and a central torus). More generally one can show: The group of fixed points of every semisimple automorphism of G is connected. Such connectedness theorems are important in problems concerning conjugacy classes, as we shall see, and also in problems about cohomology [1, p. 224]. Assume now that x and x' in G have the same semisimple part y. Then x and x' are conjugate in G if and only if x u and x' u are conjugate

3 R. STEINBERG 279 in G y, in fact, by (5), in the semisimple part of G y. The latter group may not be simply connected, but this is immaterial for the study of unipotent elements. Thus the study of arbitrary classes is reduced to the study of unipotent classes, and these are the clases we now consider: At present not much is known about the classification of the unipotent classes of G. Observe, though, that (1) implies that regular unipotent elements exist and that they in fact form a single conjugacy class. (6) Problem. Classify the unipotent classes and determine canonical representatives for them. Up =0, the study of unipotent classes in G is equivalent to that of nilpotent classes in L, and a classification of these latter classes together with representative elements may be found in [3; 4]. The result, however, is in the form of a list, which, though finite, is very long, thus subject to error and inconvenient for applications. The main procedure in the construction of the list, that of imbedding each nilpotent element in an algebra of type s/ 2 and then using the representation theory of this algebra, or the corresponding procedure with G in place of L, is not available if p фо [12, p. 60]. In studying the nilpotent classes of L (still for p =0) one is naturally led to study the class H of G-harmonic polynomials on L: indeed these polynomials are in natural correspondence with the regular functions on the variety of all nilpotent elements of L, by results in [61. To our knowledge, no one has succeeded in effectively relating H to the classification problem at hand. (7) P г о b 1 e m. Do this. (8) P г о b 1 e m. Do the same for the unipotent classes of G (for p arbitrary), having first found an analogue for H. Now it follows from (5) that a unipotent element centralized by a semisimple element not in the center of G is contained in a proper semisimple subgroup of G. Thus an important special case of (6) is: (9) P г о b 1 e m. Same as (6) for the classes of unipotent elements x such that G x is the product of the center of G and a unipotent subgroup. As is easily seen, the class of regular unipotent elements always satisfies this condition. If p = 0, then relatively few other classes do [3, Th. 10.6].. A more modest problem is as follows. (10) Problem. Prove that the number of unipotent classes of G is finite. Here only classes which satisfy (9) need be considered. Observe that a proof of (4) would yield the finiteness together with a bound on the number of classes. From the work of Kostant one obtains a rather lengthy proof of (10) together with the bound 3 r, in case /7=0. Richardson [6] has found a simple, short proof that works not only if p =0, but, more generally, if p does not divide any coefficient of

4 280 ПОЛУЧАСОВЫЕ ДОКЛАДЫ HALF-HOUR REPORTS the highest root of any simple component of G, this root being expressed in terms of the simple ones. In the sequel such a value of p will be called good. Richardson's method depends eventually on the fact that an algebraic set has only a finite number,of irreducible components; thus it yields no bound on the number of unipotent classes, but it does yield the following useful result. (11) If p is good and G has no simple component of type A n, then L x is the Lie algebra of G x for every x in G. (12) P г о b 1 e m. Extend (11) to all p and G, with the conclusion replaced by: L x is the sum of the Lie algebra of G x and the center of L. This is easy if x is semisimple and known if x is regular (cf. [ 12,4.3] ). Finally let us mention an old problem which seems to be closely related to (6). (13) Problem. Determine, within a general framework, the conjugacy classes of the Weyl group W of G, and assuming that W acts, as usual, on a real r-dimensional space V, relate them to the W-harmonic polynomials on V. Next we consider some results and problems about centralizers of elements of G. We have already stated, in (5) and (11) above, two such results. Springer [10] has proved that for any x in G the group G x contains an Abelian subgroup of dimension r. It follows that if x is regular, then the identity component G x of G x is Abelian. (14) P г о b 1 e m. // x is regular, prove that G x is Abelian. (15) Problem. Prove conversely that if G x is Abelian, then x is regular. These results would yield an abstract characterization of the regular elements, but other such characterizations are known [ 12, 3.14]. Both problems may easily be reduced to the unipotent case. If x is a regular unipotent element, then, as already mentioned, G x is the product of the center of G and a unipotent group U x. Springer [11,4.Ill has proved: (16) If p is good, then U x is connected (hence Abelian). Thus (14) holds in this case. If p is bad, (16) is false, in fact x \$ U%. (17) Problem. For p bad determine the structure of U x /U x and whether U x is Abelian. A solution to (16) would complete the proof of (14), and quite likely would also have cohomological applications (cf. [11, 3]). We do not know whether (15) is true even if p =0. (18) Problem. Prove that dim G x r is always even. In other word, the dimension of each conjugacy class is even. This has been proved in the following cases: if p = 0 [5, Prop. 15], if p is good (because of (11) the same type of proof as for p =0 can be given), if x is semisimple (this is easy); and presumably can be checked when G is a classical group.

5 R. STEINBERG 281 All of the above problems are special cases of one big final problem about central izers. (19) P T о b 1 e m. For every element x of G determine the structure >fg x. Next we discuss briefly the individual classes, and their closures. Each class is, of course, an irreducible subvariety of G (because G is connected). A class is closed if and only if it is semisimple [12, 6.13]. Vlore generally, the closure of a class consists of a number of classes )f which exactly one is semisimple [12, 6.11]. Whether the number s always finite or not depends on (10). The most important case >ccurs when we start with the class U Q of regular unipotent elenents; then the closure U is just the variety of all unipotent elements )f G. Although some results about U are known, e.g. U is a complete ntersection specified by the equations % t (x) = % t (l) (l<t<r) (the t's are still the fundamental characters), and the codimension >f U U Q in U is at least 2 [12, 6.11], we feel that U should be tudied thoroughly. (20) Problem. Study U thoroughly. Of course this problem is related to (8). Using the properties of U just mentioned and also (16), Springer las proved the following result. (21) If p is good, then the variety U (of unipotent elements of G) is amorphic, as a G-space, to the variety N of nilpotent elements fl. If p is bad, this is false, because (16) is. If p = 0, Kostant [5] ias obtained results about the structure and cohomology of N, in articular that N and any Cartan subalgebra of L intersect at 0 with multiplicity equal to the order of the Weyl group W. (22) Problem. Find a natural action of W on affine dim G r)-space so that the quotient variety is isomorphic to N. So far we have been considering G over an algebraically closed eld K- From now on we shall assume that G is defined over a perfect eld k which has К as an algebraic closure; thus the polynomials rtiich define G as an algebraic group can be chosen so that their oefficients are in k. The problem is to study the classes of G h, the group f elements of G which are defined over k, i.e. which have their coorinates in k. The main idea is to relate the classes of G h to those of! with the help of the Galois theory. We will discuss mainly the prolem of surveying, and finding canonical representatives for, the lasses of G Ä. Recalling our solution to this problem for the regular md semisimple) classes of G, we naturally try to adapt our crossaction С of the regular classes to the present situation. Consider a reguir class of G which meets G Ä. This class, call it R, is necessarily defined is a variety) over k. How can we ensure that the representative eleîent С П R of R is in G Ä? The obvious way is to construct С so that

6 282 ПОЛУЧАСОВЫЕ ДОКЛАДЫ HALF-HOUR REPORTS it also is defined over k. For this to be possible, the unique unipotent element of С must be in G A, and then the unique Borel (i.e. maximal connected solvable) subgroup of G containing this element [12, 3.2 pnd 3.3] must de defined over k; i.e. G must contain a Borel subgroup defined over k. This last condition is a natural one which arises in other classification problems, and, although a bit restrictive, holds for a significant class of groups, e.g. for the so-called Chevalley groups 12], of which one is the group SL n. Conversely, if this condition holds, then it turns out that with a few exceptions (certain groups of type A m (but not SL m +i) are forbidden as simple components of G) С can be constructed over k [12, 9]. Thus if we form С (] G k, we obtain a set of canonical representatives for those regular classes of G which meet G k ; but not necessarily for the regular classes of G k since a single class of G may intersect G k in several classes. In other words, elements of G A conjugate in G may not be conjugate in G k. Looking for an idea to remedy this situation, we consider the case G = SLn,. We have the Jordan normal forms for all the classes of SL n (K), no! just the regular ones, and, though this fails in SL n (K), it does hold in GLn(k). Now the action of GL n (k) by conjugation is that oi PGLn (k), i.e. of ö h, if we let ù denote the adjoint group of G, i.e. the quotient of G over its center, which is PSL n in the present case. Thus we are led to the following problem. (23) Problem. Asume that G contains a Borel subgroup definec over k (and perhaps that G has no simple component of type A m ) (a) Prove that every class of G defined over k meets G k, i.e. contains an eh ment defined over k. (b) Prove that two elements of G k which are conjugatt in G are conjugate under G h. As we have seen, (a) holds for regular classes. It holds also foi semisimple classes, without any restriction on the components o: type A m [12, 1.7]. (24) Assume that G contains a Borel subgroup defined over k. Thet every semisimple class of G defined over k meets G h. (And conversely*.] This result has a number of applications to classification problem: [12, p. 51-2], especially if (cohomological) dim fe<l. We recal [8, p. 11-8] that dim fe<l is equivalent to: every finite dimensiona division algebra over k is commutative. This holds in several casei of interest in number theory, in particular if k is finite [8, p ] As a consequence of (24), we haveji2, 1.9]: (25) // dim &< 1 and H is any connected linear group defined over k then the Galois cohomology H 1 (k, H) is trivial. This result and an extension due to Springer [8, p ] lea to the classification of semisimple groups defined over k, if dim &< '. (we get [12, 10.2] that there is always a Borel subgroup defined ove k), and answer most of the questions raised above. Concerning (23a), w< get [12, 10.2]:

7 R. STEINBERG 283 (26) J/ k and H are as in (25), then every class of H defined over k meets H h. Concerning (23b), we get [21, 10.3]: (27) // dim <l, then two semisimple elements of G k which are conjugate in G are conjugate in G h. This result is false for arbitrary elements, e.g. for regular ones. To show a typical kind of argument, we give the proof of (27). Let y and z be semisimple elements of G k conjugate in G; thus ayar 1 = z with a in G. Combining this equation with the one obtained by applying a general element y of the Galois group of К over k, we see that ar x y (a), call it x y, is in G y. As we check at once, x satisfies the cocycle condition х у & = x y y (x 6 ), hence represents an element of Н г (к, G y ), which is trivial, by (25), because G is connected, by (5), Thus x y = by (b" 1 ) for all у and some b in G y. We conclude that у and z are conjugate in G h, by the element ab in fact. The same argument, with (16) in place of (5), shows that regular unipotent elements of ô k which are conjugate in ô (the adjoint group) are conjugate in ô h, if p is good (here dimfe<l is not necessary, but p good is). By combining (26) and (27), we get a complete survey of the semisimple classes of G A, if dim <l; in particular, these classes correspond to the rational points of affine r-space, suitably defined over k (cf. [12, 10.3]). Thus, if A is a finite field of q elements, so that G k is a simply"connected version of one of the simple finite Chevalley groups or their twisted analogues, the number of such classes is q T, a useful fact in the representation theory of these groups [ 11]. A related result, whose proof, unfortunately, does not use the same ideas, states that the number of unipotent elements of G h is (/ d,mg - r [13]. Finally we conclude our paper with another problem. (28) Problem. If G is defined over k (any perfect field), prove that every unipotent class of G is defined over k. This would yield a proof of (10). In fact, assuming p Ф 0, as we may, choosing k as the fieldof p elements, applying (28) with G suitably defined, and then using (26), we would get (10) with the number bounded by \G k \. Dept. of Mathematics, University of California, Los Angeles, USA REFERENCES [1] Borel A., Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tôhoku Math. J., 13 (1961), [2] С h e v a 1 1 e y C, Sur certains groupes simples, Tôhoku Math. J., 7 (1955), [3] Дынкин E. Б., Полупростые подалгебры полупростых алгебр Ли, Матем. сб., 30 (72) (1952),

8 284 ПОЛУЧАСОВЫЕ ДОКЛАДЫ HALF-HOUR REPORTS [4] Kostant В., The principal three-dimensional subgroup and thebetti numbers of a complex simple Lie group, Amer. J. Math., 81 (1959), [5] К о s t a n t В., Lie group representations on polynomial rings, Amer. J. Math., 86 (1963), [6] R i c h a r d s o n R. W., Conjugacy classes in Lie algebras and algebraic groups, to appear. * [7] Séminaire С. Chevalley, Classification des Groupes de Lie Algébriques (two volumes), Paris (1956-8). [8] Serre J.-P., Cohomologie Galoisienne, Lecture Notes, Springer- Verlag, Berlin. [9] S p r i n g e r T. A., Some arithmetical results on semisimple Lie algebras, Publ. Math. I.H.E.S., 30 (1966), [10] Springer T. A., A note Чэп centralizers in semisimple groups, to appear. [11] Steinberg R., Representations of algebraic groups, Nagoya Math. J., 22 (1963), [12] Steinberg R., Regular elements of semisimple algebraic groups, Publ. Math. I.H.E.S., No. 25 (1965), [13] Steinberg R., Endomorpisms of linear algebraic groups, Memoirs Amer. Math. Soc, to appear. О НЕКОТОРЫХ ПРОБЛЕМАХ БЕРНСАЙДОВСКОГО ТИПА Е. С. Г О Л О Д Целью доклада является описание приема, который позволяет строить контрпримеры для некоторых проблем бернсайдовского типа [1] в случае «неограниченного» показателя. Как известно, в случае «ограниченного» показателя большинство из этих проблем имеет положительное решение [2, 3, 4], за исключением собственно проблемы Бернсайда о периодических группах [5]. Далее будет доказана следующая Теорема. Пусть k произвольное поле. Существует А-алгебра А (без единицы) с d > 2 образующими, которая бесконечномерна как векторное пространство над k, в которой всякая подалгебра с числом образующих <d нильпотентна и которая такова, что П А п = (0). n=i Частными случаями этой теоремы являются результаты, полученные тем же методом в [6], а также следующие утверждения: Следствие 1. Существует финитно-аппроксимируемая бесконечная р-группа с d > 2 образующими, в которой всякая подгруппа с числом образующих <С ^.койечна *). х ) Как доказал С. П. Струнков [7], если в группе с числом образующих d> 3 всякая собственная подгруппа конечна, то и сама группа также конечна.

### On Bhargava s representations and Vinberg s invariant theory

On Bhargava s representations and Vinberg s invariant theory Benedict H. Gross Department of Mathematics, Harvard University Cambridge, MA 02138 gross@math.harvard.edu January, 2011 1 Introduction Manjul

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### Lie groups and Lie algebras.

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 40, Number 2, Pages 253 257 S 0273-0979(03)00979-0 Article electronically published on February 12, 2003 Essays in the history of Lie groups

### GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

### ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### Analytic cohomology groups in top degrees of Zariski open sets in P n

Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction

### Spherical representations and the Satake isomorphism

Spherical representations and the Satake isomorphism Last updated: December 10, 2013. Topics: otivation for the study of spherical representations; Satake isomorphism stated for the general case of a connected

### COHOMOLOGY OF GROUPS

Actes, Congrès intern. Math., 1970. Tome 2, p. 47 à 51. COHOMOLOGY OF GROUPS by Daniel QUILLEN * This is a report of research done at the Institute for Advanced Study the past year. It includes some general

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### Quantum Mechanics and Representation Theory

Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Commutative Algebra seminar talk

Commutative Algebra seminar talk Reeve Garrett May 22, 2015 1 Ultrafilters Definition 1.1 Given an infinite set W, a collection U of subsets of W which does not contain the empty set and is closed under

### ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

### Quadratic Equations in Finite Fields of Characteristic 2

Quadratic Equations in Finite Fields of Characteristic 2 Klaus Pommerening May 2000 english version February 2012 Quadratic equations over fields of characteristic 2 are solved by the well known quadratic

### Galois representations with open image

Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group

### FINITE NONSOLVABLE GROUPS WITH MANY DISTINCT CHARACTER DEGREES

FINITE NONSOLVABLE GROUPS WITH MANY DISTINCT CHARACTER DEGREES HUNG P. TONG-VIET Abstract. Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. Let cd(g) be

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS

GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS GUSTAVO A. FERNÁNDEZ-ALCOBER AND ALEXANDER MORETÓ Abstract. We study the finite groups G for which the set cd(g) of irreducible complex

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### BASIC FACTS ABOUT WEAKLY SYMMETRIC SPACES

BASIC FACTS ABOUT WEAKLY SYMMETRIC SPACES GIZEM KARAALI 1. Symmetric Spaces Symmetric spaces in the sense of E. Cartan supply major examples in Riemannian geometry. The study of their structure is connected

### EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

### A NOTE ON TRIVIAL FIBRATIONS

A NOTE ON TRIVIAL FIBRATIONS Petar Pavešić Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija petar.pavesic@fmf.uni-lj.si Abstract We study the conditions

### Matrix generators for exceptional groups of Lie type

J. Symbolic Computation (2000) 11, 1 000 Matrix generators for exceptional groups of Lie type R. B. HOWLETT, L. J. RYLANDS AND D. E. TAYLOR School of Mathematics and Statistics, University of Sydney, Australia

### Notes on Localisation of g-modules

Notes on Localisation of g-modules Aron Heleodoro January 12, 2014 Let g be a semisimple Lie algebra over a field k. One is normally interested in representation of g, i.e. g-modules. If g were commutative,

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree

### LOW-DEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO

LOW-DEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO PETER MÜLLER AND MICHAEL E. ZIEVE Abstract. Planar functions over finite fields give rise to finite projective planes and other combinatorial objects.

### Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces

Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

### Allen Back. Oct. 29, 2009

Allen Back Oct. 29, 2009 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Notation:(anachronistic) Let the coefficient

### 4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

### ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS

ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS DOV TAMARI 1. Introduction and general idea. In this note a natural problem arising in connection with so-called Birkhoff-Witt algebras of Lie

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### POSTECH SUMMER SCHOOL 2013 LECTURE 4 INTRODUCTION TO THE TRACE FORMULA

POSTECH SUMMER SCHOOL 2013 LECTURE 4 INTRODUCTION TO THE TRACE FORMULA 1. Kernel and the trace formula Beginning from this lecture, we will discuss the approach to Langlands functoriality conjecture based

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### OPERS. The geometric Langlands correspondence conjectures a correspondence

OPERS JONATHAN BARLEV The geometric Langlands correspondence conjectures a correspondence Qcoloc-sysL Gx)) = D-modBun G )) on the level of derived categories. As remarked previously in theseminar,toeach

### Nonzero degree tangential maps between dual symmetric spaces

ISSN 1472-2739 (on-line) 1472-2747 (printed) 709 Algebraic & Geometric Topology Volume 1 (2001) 709 718 Published: 30 November 2001 ATG Nonzero degree tangential maps between dual symmetric spaces Boris

### Basics of Polynomial Theory

3 Basics of Polynomial Theory 3.1 Polynomial Equations In geodesy and geoinformatics, most observations are related to unknowns parameters through equations of algebraic (polynomial) type. In cases where

### Revision of ring theory

CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

### A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field.

A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field Jean-Pierre Serre Let k be a field. Let Cr(k) be the Cremona group of rank 2 over k,

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### ARITHMETIC SUBGROUPS OF ALGEBRAIC GROUPS

ARITHMETIC SUBGROUPS OF ALGEBRAIC GROUPS BY ARMAND BOREL AND HARISH-CHANDRA Communicated by Deane Montgomery, July 22, 1961. A complex algebraic group G is in this note a subgroup of GL(n, C), the elements

### ON THE RATIONAL REAL JACOBIAN CONJECTURE

UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: FASCICULUS LI, 2013 ON THE RATIONAL REAL JACOBIAN CONJECTURE by L. Andrew Campbell Abstract. Jacobian conjectures (that nonsingular implies a global inverse)

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### ON FINITE GROUPS WITH THE SAME PRIME GRAPH AS THE PROJECTIVE GENERAL LINEAR GROUP PGL(2, 81)

Transactions on Algebra and its Applications 2 (2016), 43-49. ISSN: 6756-2423 ON FINITE GROUPS WITH THE SAME PRIME GRAPH AS THE PROJECTIVE GENERAL LINEAR GROUP PGL(2, 81) ALI MAHMOUDIFAR Abstract. Let

### Appendix A. Appendix. A.1 Algebra. Fields and Rings

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

### Algebraic Geometry. Andreas Gathmann. Notes for a class. taught at the University of Kaiserslautern 2002/2003

Algebraic Geometry Andreas Gathmann Notes for a class taught at the University of Kaiserslautern 2002/2003 CONTENTS 0. Introduction 1 0.1. What is algebraic geometry? 1 0.2. Exercises 6 1. Affine varieties

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### NOTES ON CATEGORIES AND FUNCTORS

NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category

### On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

### Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions

Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International

### OSTROWSKI FOR NUMBER FIELDS

OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the p-adic absolute values for

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Structure of the Root Spaces for Simple Lie Algebras

Structure of the Root Spaces for Simple Lie Algebras I. Introduction A Cartan subalgebra, H, of a Lie algebra, G, is a subalgebra, H G, such that a. H is nilpotent, i.e., there is some n such that (H)

### arxiv:quant-ph/0603009v3 8 Sep 2006

Deciding universality of quantum gates arxiv:quant-ph/0603009v3 8 Sep 2006 Gábor Ivanyos February 1, 2008 Abstract We say that collection of n-qudit gates is universal if there exists N 0 n such that for

### Computer Algebra for Computer Engineers

p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### Fiber bundles and non-abelian cohomology

Fiber bundles and non-abelian cohomology Nicolas Addington April 22, 2007 Abstract The transition maps of a fiber bundle are often said to satisfy the cocycle condition. If we take this terminology seriously

### INTRODUCTION TO MANIFOLDS IV. Appendix: algebraic language in Geometry

INTRODUCTION TO MANIFOLDS IV Appendix: algebraic language in Geometry 1. Algebras. Definition. A (commutative associatetive) algebra (over reals) is a linear space A over R, endowed with two operations,

### THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP

THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP by I. M. Isaacs Mathematics Department University of Wisconsin 480 Lincoln Dr. Madison, WI 53706 USA E-Mail: isaacs@math.wisc.edu Maria

### ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES.

ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES. O. N. KARPENKOV Introduction. A series of properties for ordinary continued fractions possesses multidimensional

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### SOLVING POLYNOMIAL EQUATIONS BY RADICALS

SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang De-Qi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,

### some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

### BP-cohomology of mapping spaces from the classifying space of a torus to some p-torsion free space

BP-cohomology of mapping spaces from the classifying space of a torus to some p-torsion free space 1. Introduction Let p be a fixed prime number, V a group isomorphic to (Z/p) d for some integer d and

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### ABSTRACT ALGEBRA. Romyar Sharifi

ABSTRACT ALGEBRA Romyar Sharifi Contents Introduction 7 Part 1. A First Course 11 Chapter 1. Set theory 13 1.1. Sets and functions 13 1.2. Relations 15 1.3. Binary operations 19 Chapter 2. Group theory

### Factoring Cubic Polynomials

Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

### The Markov-Zariski topology of an infinite group

Mimar Sinan Güzel Sanatlar Üniversitesi Istanbul January 23, 2014 joint work with Daniele Toller and Dmitri Shakhmatov 1. Markov s problem 1 and 2 2. The three topologies on an infinite group 3. Problem

### G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

### SEMINAR NOTES: HITCHIN FIBERS AND AFFINE SPRINGER FIBERS, OCTOBER 29 AND NOVEMBER 5, 2009

SEMINAR NOTES: HITCHIN FIBERS AND AFFINE SPRINGER FIBERS, OCTOBER 29 AND NOVEMBER 5, 2009 ROMAN BEZRUKAVNVIKOV 1. Affine Springer fibers The goal of the talk is to introduce affine Springer fibers and

### COMMUTATIVITY DEGREE, ITS GENERALIZATIONS, AND CLASSIFICATION OF FINITE GROUPS

COMMUTATIVITY DEGREE, ITS GENERALIZATIONS, AND CLASSIFICATION OF FINITE GROUPS ABSTRACT RAJAT KANTI NATH DEPARTMENT OF MATHEMATICS NORTH-EASTERN HILL UNIVERSITY SHILLONG 793022, INDIA COMMUTATIVITY DEGREE,

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### On the exceptional series, and its descendants

On the exceptional series, and its descendants Pierre Deligne a, Benedict H. Gross b Résumé. Les articles [1], [2], [3] exhibent des ressemblances entre les propriétés des représentations adjointes des

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Nilpotent Lie and Leibniz Algebras

This article was downloaded by: [North Carolina State University] On: 03 March 2014, At: 08:05 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered

### Finite groups of uniform logarithmic diameter

arxiv:math/0506007v1 [math.gr] 1 Jun 005 Finite groups of uniform logarithmic diameter Miklós Abért and László Babai May 0, 005 Abstract We give an example of an infinite family of finite groups G n such

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### Two classes of ternary codes and their weight distributions

Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight

### MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials

Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial. The first half of the note is derived from

### QUOTIENTS IN ALGEBRAIC AND SYMPLECTIC GEOMETRY

QUOTIENTS IN ALGEBRAIC AND SYMPLECTIC GEOMETRY VICTORIA HOSKINS 1. Introduction In this course we study methods for constructing quotients of group actions in algebraic and symplectic geometry and the

### ON FOCK-GONCHAROV COORDINATES OF THE ONCE-PUNCTURED TORUS GROUP

ON FOCK-GONCHAROV COORDINATES OF THE ONCE-PUNCTURED TORUS GROUP YUICHI KABAYA Introduction In their seminal paper [3], Fock and Goncharov defined positive representations of the fundamental group of a

### Introduction to finite fields

Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

### Proof. The map. G n i. where d is the degree of D.

7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal

### (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order 4; (1, 0) : order 2; (1, 1) : order 4; (1, 2) : order 2; (1, 3) : order 4.

11.01 List the elements of Z 2 Z 4. Find the order of each of the elements is this group cyclic? Solution: The elements of Z 2 Z 4 are: (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order

### Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

### COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B

### Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products