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In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of vector spaces; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.
In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of .
In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras.
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and semisimple algebraic groups are reductive.
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras.
In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra is a nilpotent Lie algebra if and only if for each , the adjoint map
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if is a finite-dimensional representation of a solvable Lie algebra, then stabilizes a flag ; "stabilizes" means for each and i.
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group.
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another structure.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles (1984). The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.
In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology of G into account. More precisely, Λ(G) is the inverse limit of the group rings Zp(G/H) as H runs through the open normal subgroups of G. Commutative Iwasawa algebras were introduced by Iwasawa (1959) in his study of Zp extensions in Iwasawa theory, and non-commutative Iwasawa algebras of compact p-adic analytic groups were introduced by Lazard (1965).
In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.
In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra . There is a closely related theorem classifying the irreducible representations of a connected compact Lie group . The theorem states that there is a bijection
Igor Dmitrievich Ado was a Soviet mathematician. He was born into the family of a state employee and he lived in Kazan till the end of his life. After leaving school Igor Ado entered the faculty of mathematics and physics at Kazan State University, named after V. I. Lenin, from which he graduated successfully in 1931. He was admitted to the PhD study at the Chair of Mathematics under the supervision of Nikolai Chebotaryov. Igor Ado finished successfully his PhD study by preparing a scientific qualifying work for the degree of a Candidate (PhD) of physical and mathematical sciences. The University board awarded him for this work the degree of Doctor nauk of physic-mathematical sciences. This is an analogue to the European Habilitation and is very unusual to receive for a PhD work.
Basic Number Theory is an influential book by André Weil, an exposition of algebraic number theory and class field theory with particular emphasis on valuation-theoretic methods. Based in part on a course taught at Princeton University in 1961-2, it appeared as Volume 144 in Springer's Grundlehren der mathematischen Wissenschaften series. The approach handles all 'A-fields' or global fields, meaning finite algebraic extensions of the field of rational numbers and of the field of rational functions of one variable with a finite field of constants. The theory is developed in a uniform way, starting with topological fields, properties of Haar measure on locally compact fields, the main theorems of adelic and idelic number theory, and class field theory via the theory of simple algebras over local and global fields. The word `basic’ in the title is closer in meaning to `foundational’ rather than `elementary’, and is perhaps best interpreted as meaning that the material developed is foundational for the development of the theories of automorphic forms, representation theory of algebraic groups, and more advanced topics in algebraic number theory. The style is austere, with a narrow concentration on a logically coherent development of the theory required, and essentially no examples.
References
- Ado, Igor D. (1935), "Note on the representation of finite continuous groups by means of linear substitutions", Izv. Fiz.-Mat. Obsch. (Kazan'), 7: 1–43. (Russian language)
- Ado, Igor D. (1947), "The representation of Lie algebras by matrices", Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk (in Russian), 2 (6): 159–173, ISSN 0042-1316, MR 0027753 translation in Ado, Igor D. (1949), "The representation of Lie algebras by matrices", American Mathematical Society Translations, 1949 (2): 21, ISSN 0065-9290, MR 0030946
- Iwasawa, Kenkichi (1948), "On the representation of Lie algebras", Japanese Journal of Mathematics, 19: 405–426, MR 0032613 CS1 maint: discouraged parameter (link)
- Harish-Chandra (1949), "Faithful representations of Lie algebras", Annals of Mathematics , Second Series, 50: 68–76, doi:10.2307/1969352, ISSN 0003-486X, JSTOR 1969352, MR 0028829 CS1 maint: discouraged parameter (link)
- Hochschild, Gerhard (1966), "An addition to Ado's theorem", Proceedings of the American Mathematical Society , 17: 531–533, doi: 10.1090/s0002-9939-1966-0194482-0 CS1 maint: discouraged parameter (link)
- Nathan Jacobson, Lie Algebras, pp. 202–203