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Query: Who is considered the founder of algebra?
In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" and in context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".[14][15][16][17] A debate now exists whether who (in general sense) is more entitled to be known as "the father of algebra". Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.[18] Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to,[19] and that he gave an exhaustive explanation of solving quadratic equations,[20] supported by geometric proofs, while treating algebra as an independent discipline in its own right.[21] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".[22]
An algebra "A" over a field "K" is in particular a "K"-vector space and so one can consider the associative algebra End("A") of "K"-linear vector space endomorphism of "A". We can associate to the algebra structure on "A" two subalgebras of End("A"), the derivation algebra and the (associative) enveloping algebra.
Thābit ibn Qurrah made important discoveries in algebra, geometry, and astronomy. In astronomy, Thābit is considered one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.
Michio Jimbo considered the algebras with three generators related by the three commutators
When formula_6, these reduce to the commutators that define the special linear Lie algebra formula_4. In contrast, for nonzero formula_8, the algebra defined by these relations is not a Lie algebra but instead an associative algebra that can be regarded as a deformation of the universal enveloping algebra of formula_4.
The super-Poincaré algebra was first proposed in the context of the Haag–Łopuszański–Sohnius theorem, as a means of avoiding the conclusions of the Coleman–Mandula theorem. That is, the Coleman–Mandula theorem is a no-go theorem that states that the Poincaré algebra cannot be extended with additional symmetries that might describe the internal symmetries of the observed physical particle spectrum. However, the Coleman–Mandula theorem assumed that the algebra extension would be by means of a commutator; this assumption, and thus the theorem, can be avoided by considering the anti-commutator, that is, by employing anti-commuting Grassmann numbers. The proposal was to consider a supersymmetry algebra, defined as the semidirect product of a central extension of the super-Poincaré algebra by a compact Lie algebra of internal symmetries.
Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra. An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra "T". Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure "E". The quotient algebra "T"/"E" is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator "m", taking two arguments, and the inverse operator "i", taking one argument, and the identity element "e", a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables "x", "y", "z", etc. the term algebra is the collection of all possible terms involving "m", "i", "e" and the variables; so for example, "m(i(x), m(x,m(y,e)))" would be an element of the term algebra. One of the axioms defining a group is the identity "m(x, i(x)) = e"; another is "m(x,e) = x". The axioms can be represented as trees. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group.
The Israel Halperin Prize is awarded every five years by the Canadian Annual Symposium on Operator Theory and Operator Algebras to a member of the Canadian mathematical community who has recently obtained a doctorate and has made contributions to operator theory or operator algebras. It honors Israel Halperin, the founder of a group of researchers in operator algebras and operator theory at the University of Toronto who strongly influenced the field across Canada. First awarded in 1980.
Exterior algebras of vector bundles are frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the Serre–Swan theorem. More general exterior algebras can be defined for sheaves of modules.
Antonio Monteiro showed that for every monadic Boolean algebra one can construct a trivalent Łukasiewicz algebra (by taking certain equivalence classes) and that any trivalent Łukasiewicz algebra is isomorphic to a Łukasiewicz algebra thus derived from a monadic Boolean algebra. Cignoli summarizes the importance of this result as: "Since it was shown by Halmos that monadic Boolean algebras are the algebraic counterpart of classical first order monadic calculus, Monteiro considered that the representation of three-valued Łukasiewicz algebras into monadic Boolean algebras gives a proof of the consistency of Łukasiewicz three-valued logic relative to classical logic."
In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering "Lie algebras" over the integers.
The Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra, and act on the tensor product of the corresponding Hilbert spaces. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The type of the tensor product of two von Neumann algebras (I, II, or III) is the maximum of their types. The commutation theorem for tensor products states that
Those known as the father, mother, or considered a founder in a Humanities field are those who have made important contributions to that field. In some fields several people are considered the founders, while in others the title of being the "father" is debatable. Some of the people who have humanity are given in REFERENCES.
A real (possibly infinite-dimensional) Lie algebra is also considered a Kac–Moody algebra if its complexification is a Kac–Moody algebra.
These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields. Hence such things as group theory and ring theory took their places in pure mathematics. The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building up on the work of Ernst Kummer, 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".
Most authors consider as an algebraic surface only algebraic varieties of dimension two, but some also consider as surfaces algebraic sets all of whose irreducible components have the dimension two.
C-algebras (pronounced "C-star") are subjects of research in functional analysis, a branch of mathematics. A C*-algebra is a complex algebra "A" of continuous linear operators on a complex Hilbert space with two additional properties:C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of C*-algebras which are now known as von Neumann algebras.
More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve. The simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and even algebraic curves that are defined independently to any embedding in an affine or projective space. This leads to the most general definition of an algebraic curve:
Any monadic Boolean algebra can be considered to be an interior algebra where the interior operator is the universal quantifier and the closure operator is the existential quantifier. The monadic Boolean algebras are then precisely the variety of interior algebras satisfying the identity "x" = "x". In other words, they are precisely the interior algebras in which every open element is closed or equivalently, in which every closed element is open. Moreover, such interior algebras are precisely the semisimple interior algebras. They are also the interior algebras corresponding to the modal logic S5, and so have also been called S5 algebras.
In mathematics, a projectionless C*-algebra is a C*-algebra with no nontrivial projections. For a unital C*-algebra, the projections 0 and 1 are trivial. While for a non-unital C*-algebra, only 0 is considered trivial. The problem of whether simple infinite-dimensional C*-algebras with this property exist was posed in 1958 by Irving Kaplansky, and the first example of one was published in 1981 by Bruce Blackadar. For commutative C*-algebras, being projectionless is equivalent to its spectrum being connected. Due to this, being projectionless can be considered as a noncommutative analogue of a connected space.
Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues.[28] George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).[29]
Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.
The main figure responsible for the birth of commutative algebra as a mature subject was Wolfgang Krull, who introduced the fundamental notions of localization and completion of a ring, as well as that of regular local rings. He established the concept of the Krull dimension of a ring, first for Noetherian rings before moving on to expand his theory to cover general valuation rings and Krull rings. To this day, Krull's principal ideal theorem is widely considered the single most important foundational theorem in commutative algebra. These results paved the way for the introduction of commutative algebra into algebraic geometry, an idea which would revolutionize the latter subject.
According to Guerraggio & Nastasi (page 9, 2005) Luigi Cremona is "considered the founder of the Italian school of algebraic geometry". Later they explain that in Turin the collaboration of Enrico D'Ovidio and Corrado Segre "would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity". A one-time student of Segre, H.F. Baker wrote (1926, page 269), [Corrado Segre] "may probably be said to be the father of that wonderful Italian school which has achieved so much in the birational theory of algebraical loci." On this topic, Brigaglia & Ciliberto (2004) say "Segre had headed and maintained the school of geometry that Luigi Cremona had established in 1860." Reference to the Mathematics Genealogy Project shows that, in terms of "Italian doctorates", the real productivity of the school began with Guido Castelnuovo and Federigo Enriques. In the USA Oscar Zariski inspired many Ph.D.s.
The algebra M("n", C) of "n" × "n" matrices over C becomes a C*-algebra if we consider matrices as operators on the Euclidean space, C, and use the operator norm ||·|| on matrices. The involution is given by the conjugate transpose. More generally, one can consider finite direct sums of matrix algebras. In fact, all C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The self-adjoint requirement means finite-dimensional C*-algebras are semisimple, from which fact one can deduce the following theorem of Artin–Wedderburn type:
More significantly, the Banach–Stone theorem suggests the philosophy that one can replace a "space" (a geometric notion) by an "algebra", with no loss. Reversing this, it suggests that one can consider algebraic objects, even if they do not come from a geometric object, as a kind of "algebra of scalars". In this vein, any "commutative" C*-algebra is the algebra of scalars on a Hausdorff space. Thus one may consider "non"commutative C*-algebras (and their Spec) as non-commutative spaces. This is the basis of the field of noncommutative geometry.