Abstract algebra & its applications

Abstract algebra & its applications

1. WELCOME

2. 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 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.

3. 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 .

4. Algebraic structures include   groups ,   rings   fields   modules ,   vector spaces ,  latt ices and  algebra over a field Algebraic structures

5. Binary operations are the keystone of algebraic structures studied in  abstract algebra : They are essential in the definitions of  groups ,  monoids ,  semigroups ,  rings , and more. A binary operation on a  set   S  is a  map  which sends elements of the   Cartesian product S  to  S   Binary operations

6. 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.

7. 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

8. 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

9. 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

10. 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 mo nograph  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’.

11. Examples of algebraic structures with a single  binary operation  are: Magmas Quasigroups Monoids Semigroups Groups

12. More complicated examples include: Rings Fields Modules Vector spaces Algebras over fields Associative algebras Lie algebras Lattices Boolean algebras

13. 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

14. 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 ; F or 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

15. Group-like structures Totality 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 ^ α   Closure , which is used in many sources, is an equivalent axiom to totality, though defined differently

16. Group-like structures Totality α Associativity Identity Divisibility Commutativity Unneeded Required Unneeded Unneeded Unneeded Unneeded Required Required Unneeded Unneeded Unneeded Required Required Required Unneeded Required Unneeded Unneeded Unneeded Unneeded Required Unneeded Unneeded Required Unneeded Required Unneeded Required Required Unneeded Required Required Unneeded Unneeded Unneeded Required Required Required Unneeded Unneeded Required Required Required Required Unneeded Required Required Required Required Required

17. 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 The most prominent of these (and historically the first) is the  representation theory of groups . Representation theory

18. 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 End F ( V ) of all endomorphisms of  V , and a corresponding Lie algebra  gl ( V , F ). Definition

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