abstract algebra & its applications (1)
TRANSCRIPT
WELCOME
MEGA - 2015(Mathematical Excellence Gears Advancement-2015)
SRI SARADA NIKETAN COLLEGE FOR WOMENAmaravathipudur, Karaikudi -630301 .
DEPARTMENT OF MATHEMATICS
State Level Workshop
‘Abstract Algebra and its Applications’
28th August , 2015.
Presentation on‘Abstract Algebra and its Applications’
Presented by
Dr.S.SelvaRani, Principal Sri Sarada Niketan College For Women
Amaravathipudur
Venue : Nivedita Hall Sri Sarada Niketan College for Women,
Date : 28th August , 2015
Abstract Algebra & its Applications.
Abstract Algebra is the study of algebraic structures.
The term abstract algebra was coined in the early 20th century to distinguish this area of study from the the parts of algebra. Solving of systems of linear equations, which led to linear algebra Linear algebra is the branch of mathematics concerning vector spaces and linear
mappings between such spaces.
•Solving of systems of linear equations, which led to linear algebra •Attempts to find formulae for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry •Arithmetical investigations of quadratic and higher degree forms that directly produced the notions of a ring and ideal.
Algebraic structures include
groups, rings fields modules, vector spaces, lattices and
algebra over a field
Algebraic structures
Leonhard Euler -- algebraic operations on numbers--generalization of Fermat's little theorem Friedric Gauss - cyclic &general abelian groups
In 1870, Leopold Kronecker- abelian group-particularly, permutation groups.
Heinrich M. Weber gave a similar definition that involved the cancellation property.
Lagrange resolvants by Lagrange. The remarkable Mathematicians
are ..Kronecker,Vandermonde,Galois,Augustin Cauchy ,
Cayley-1854-….Group may consists of Matrices.
Early Group Theory
The end of the 19th and the beginning of the 20th century saw a tremendous shift in the methodology of mathematics.
Abstract algebra emerged around the start of the 20th century, under the name modern algebra.
Its study was part of the drive for more intellectual rigor in mathematics.
Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems.
MODERN ALGEBRA
Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra.
These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in Bartel van der Waerden's Moderne algebra.
The two-volume monograph published in 1930–1931 that forever changed for the mathematical world the meaning of the word…
“ algebra “ from the’ theory of equations’ to the ‘ theory of algebraic structures’.
Examples of algebraic structures with a single binary operation are:
Magmas
Quasigroups
Monoids
Semigroups
Groups
More complicated examples include: Rings Fields Modules Vector spaces Algebras over fields Associative algebras Lie algebras Lattices Boolean algebras
Binary operations are the keystone of algebraic structures studied in abstract algebra:
A binary operation is an operation that applies to two quantities or expressions and .
A binary operation on a nonempty set is a map such that 1. is defined for every pair of elements in , and 2. uniquely associates each pair of elements in to some element of .
Binary operations
On the set M(2,2) of 2 × 2 matrices with
real entries, f (A, B) = A + B is a binary
operation since the sum of two such
matrices is another
2 × 2 matrix.
In abstract algebra, a magma (or groupoid) is a basic kind of algebraic structure.
Specifically, a magma consists of a set, M, equipped with a single binary operation,
M × M → M. The binary operation must be closed by definition
but no other properties are imposed.
magma
Group-like structures
Totalityα Associativity Identity Divisibility Commutativity
Semicategory Unneeded Required Unneeded Unneeded Unneeded
Category Unneeded Required Required Unneeded Unneeded
Groupoid Unneeded Required Required Required Unneeded
Magma Required Unneeded Unneeded Unneeded Unneeded
Quasigroup Required Unneeded Unneeded Required Unneeded
Loop Required Unneeded Required Required Unneeded
Semigroup Required Required Unneeded Unneeded Unneeded
Monoid Required Required Required Unneeded Unneeded
Group Required Required Required Required Unneeded
Abelian Group Required Required Required Required Required
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.
A representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication structures.
The most prominent of these (and historically the first) is the representation theory of groups.
Representation theory
Let V be a vector space over a field F. The set of all invertible n × n matrices is a group under
matrix multiplication The representation theory of groups analyses a group by
describing ("representing") its elements in terms of invertible matrices.
This generalizes to any field F and any vector space V over F, with linear maps replacing matrices and composition replacing matrix multiplication:
There is a group GL(V,F) of automorphisms of V an associative algebra EndF(V) of all endomorphisms of V, and
a corresponding Lie algebra gl(V,F).
Definitionn of Representation
Representation theory studies symmetry in Linear spaces.
• It has many applications, ranging from number theory to
geometry, probability theory, quantum mechanics and quantum
field theory.
•Representation theory was born in 1896 in the work of the
German mathematician F. G. Frobenius.
•And major contributors are : Dedekind, Burnside and
A.H.Clifford.
Applications & Contributors
Because of its generality, abstract algebra is used in many fields of mathematics and science.
For instance, algebraic topology uses algebraic objects to study topologies.
The recently (As of 2006) proved Poincaré conjecture asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not.
Algebraic number theory studies various number rings that generalize the set of integers.
Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem.
Applications
In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations.
In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system
The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system;
For instance, the number of force carriers in a theory is equal to dimension of the Lie algebra
And these bosons interact with the force they mediate if the Lie algebra is nonabelian.[2
Applications
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