Claude Chevalley

Claude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, the theory of algebraic groups. He was a founding member of the Bourbaki group, his father, Abel Chevalley, was a French diplomat who, jointly with his wife Marguerite Chevalley née Sabatier, wrote The Concise Oxford French Dictionary. Chevalley graduated from the École Normale Supérieure in 1929, he spent time at the University of Hamburg, studying under Emil Artin, at the University of Marburg, studying under Helmut Hasse. In Germany, Chevalley discovered Japanese mathematics in the person of Shokichi Iyanaga. Chevalley was awarded a doctorate in 1933 from the University of Paris for a thesis on class field theory; when World War II broke out, Chevalley was at Princeton University. After reporting to the French Embassy, he stayed in the USA, first at Princeton at Columbia University, his American students included Leon Ehrenpreis and Gerhard Hochschild.

During his time in the USA, Chevalley became an American citizen and wrote a substantial part of his lifetime output in English. When Chevalley applied for a chair at the Sorbonne, the difficulties he encountered were the subject of a polemical piece by his friend and fellow Bourbakiste André Weil, titled "Science Française?" and published in the NRF. Chevalley was the "professeur B" of the piece, as confirmed in the endnote to the reprint in Weil's collected works, Oeuvres Scientifiques, tome II. Chevalley did obtain a position in 1957 at the faculty of sciences of the University of Paris, after 1970 at the Université de Paris VII. Chevalley had artistic and political interests, was a minor member of the French non-conformists of the 1930s; the following quote by the co-editor of Chevalley's collected works attests to these interests: "Chevalley was a member of various avant-garde groups, both in politics and in the arts... Mathematics was the most important part of his life, but he did not draw any boundary between his mathematics and the rest of his life."

In his PhD thesis, Chevalley made an important contribution to the technical development of class field theory, removing a use of L-functions and replacing it by an algebraic method. At that time use of group cohomology was implicit, cloaked by the language of central simple algebras. In the introduction to André Weil's Basic Number Theory, Weil attributed the book's adoption of that path to an unpublished manuscript by Chevalley. Around 1950, Chevalley wrote a three-volume treatment of Lie groups. A few years he published the work for which he is best remembered, his investigation into what are now called Chevalley groups. Chevalley groups make up 9 of the 18 families of finite simple groups. Chevalley's accurate discussion of integrality conditions in the Lie algebras of semisimple groups enabled abstracting their theory from the real and complex fields; as a consequence, analogues over finite fields could be defined. This was an essential stage in the evolving classification of finite simple groups.

After Chevalley's work, the distinction between "classical groups" falling into the Dynkin diagram classification, sporadic groups which did not, became sharp enough to be useful. What are called'twisted' groups of the classical families could be fitted into the picture. "Chevalley's theorem" refers to his result on the solubility of equations over a finite field. Another theorem of his concerns the constructible sets in algebraic geometry, i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states. Logicians call this an elimination of quantifiers. In the 1950s, Chevalley led some Paris seminars of major importance: the Séminaire Cartan–Chevalley of the academic year 1955/6, with Henri Cartan, the Séminaire Chevalley of 1956/7 and 1957/8; these dealt with topics on algebraic groups and the foundations of algebraic geometry, as well as pure abstract algebra. The Cartan–Chevalley seminar was the genesis of scheme theory, but its subsequent development in the hands of Alexander Grothendieck was so rapid and inclusive that its historical tracks can appear well covered.

Grothendieck's work subsumed the more specialised contribution of Serre, Goro Shimura, others such as Erich Kähler and Masayoshi Nagata. 1936. L'Arithmetique dans les Algèbres de Matrices. Hermann, Paris. 1940. "La théorie du corps de classes," Annals of Mathematics 41: 394–418. 1946. Theory of Lie groups. Princeton University Press. 1951. "Théorie des groupes de Lie, tome II, Groupes algébriques", Paris. 1951. Introduction to the theory of algebraic functions of one variable, A. M. S. Math. Surveys VI. 1954. The algebraic theory of spinors, Columbia Univ. Press. 1953-1954. Class field theory, Nagoya Univ. 1955. "Théorie des groupes de Lie, tome III, Théorèmes généraux sur les algèbres de Lie", Paris. 1955, "Sur certains groupes simples," Tôhoku Mathematical Journal 7: 14–66. 1955. The construction and study of certain important algebras, Publ. Math. Soc. Japan. 1956. Fundamental concepts of algebra, Acad. Press. 1956-1958. "Classification des groupes de Lie algébriques", Séminaire Chevalley, Secrétariat Math. 11 rue P. Curie, Paris.

Cartier, Springer-Verlag, 2005. 1958. Fondements de la géométrie algébrique, Secrétariat Math. 11 rue P. Curie, Paris. Idèle Valuative criterion of properness Chevalley group Chevalley scheme Chevalley–Iwahori–Nagata theorem Beck–Chevalley condition Non-Conformist Mo

Complex number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.

According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.

The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.

For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia

Algebra

Algebra is one of the broad parts of mathematics, together with number theory and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols, it includes everything from elementary equation solving to the study of abstractions such as groups and fields. The more basic parts of algebra are called elementary algebra. Elementary algebra is considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in x + 2 = 5 the letter x is unknown, but the law of inverses can be used to discover its value: x = 3. In E = mc2, the letters E and m are variables, the letter c is a constant, the speed of light in a vacuum.

Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", the word is used, for example, in the phrases linear algebra and algebraic topology. A mathematician who does research in algebra is called an algebraist; the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wa'l-muḳābala by the Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the fifteenth century, from either Spanish, Italian, or Medieval Latin, it referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century; the word "algebra" has several related meanings as a single word or with qualifiers. As a single word without an article, "algebra" names a broad part of mathematics.

As a single word with an article or in plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the author. The structure has an addition, a scalar multiplication; when some authors use the term "algebra", they make a subset of the following additional assumptions: associative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations. With a qualifier, there is the same distinction: Without an article, it means a part of algebra, such as linear algebra, elementary algebra, or abstract algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, an associative algebra, or a vertex operator algebra. Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers. Algebra began with letters standing for numbers.

This allowed proofs of properties. For example, in the quadratic equation a x 2 + b x + c = 0, a, b, c can be any numbers whatsoever, the quadratic formula can be used to and find the values of the unknown quantity x which satisfy the equation; that is to say. And in current teaching, the study of algebra starts with the solving of equations such as the quadratic equation above. More general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions led extending algebra to non-numerical objects, such as permutations, vectors and polynomials; the structural properties of these non-numerical objects were abstracted into algebraic structures such as groups and fields. Before the 16th century, mathematics was divided into only two subfields and geometry. Though some methods, developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century.

From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, all of which used algebra. Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification where none of the first level areas is called algebra. Today algebra in

Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras the classification and representation theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Root systems are important for their own sake, as in spectral graph theory; as a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right. These vectors span the whole space. If you consider the line perpendicular to any root, say β the reflection of R2 in that line sends any other root, say α, to another root. Moreover, the root to which it is sent equals α + nβ.

These six vectors satisfy the following definition, therefore they form a root system. Let E be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by. A root system Φ in E is a finite set of non-zero vectors that satisfy the following conditions: An equivalent way of writing conditions 3 and 4 is as follows: Some authors only include conditions 1–3 in the definition of a root system. In this context, a root system that satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2. In this article, all root systems are assumed to be crystallographic. In view of property 3, the integrality condition is equivalent to stating that β and its reflection σα differ by an integer multiple of α. Note that the operator ⟨ ⋅, ⋅ ⟩: Φ × Φ → Z defined by property 4 is not an inner product, it is not symmetric and is linear only in the first argument. The rank of a root system Φ is the dimension of E. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space.

A root system which does not arise from such a combination, such as the systems A2, B2, G2 pictured to the right, is said to be irreducible. Two root systems and are called isomorphic if there is an invertible linear transformation E1 → E2 which sends Φ1 to Φ2 such that for each pair of roots, the number ⟨ x, y ⟩ is preserved; the root lattice of a root system Φ is the Z-submodule of E generated by Φ. It is a lattice in E; the group of isometries of E generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ. As it acts faithfully on the finite set Φ, the Weyl group is always finite. In the A 2 case, the "hyperplanes" are the lines perpendicular to the roots, indicated by dashed lines in the figure; the Weyl group is the symmetry group of an equilateral triangle. In this case, the Weyl group is not the full symmetry group of the root system. There is only one root system of rank 1, consisting of two nonzero vectors; this root system is called A 1. In rank 2 there are four possibilities, corresponding to σ α = β + n α, where n = 0, 1, 2, 3.

Note that a root system is not determined by the lattice that it generates: A 1 × A 1 and B 2 both generate a square lattice while A 2 and G 2 generate a hexagonal lattice, only two of the five possible types of lattices in two dimensions. Whenever Φ is a root system in E, S is a subspace of E spanned by Ψ = Φ ∩ S Ψ is a root system in S. Thus, the exhaustive list of four root systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank. In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees. If g is a complex semisimple Lie algebra and h is a Cartan subalgebra, we can construct a root system as follows. We say that α ∈ h ∗ is a root of g relative to h if α ≠ 0 {\displaystyle \al