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Unipotent Algebraic Group
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipotent matrix if and only if its characteristic polynomial ''P''(''t'') is a power of ''t'' − 1. Thus all the eigenvalues of a unipotent matrix are 1. The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity. In the theory of algebraic groups, a group element is unipotent if it acts unipotently in a certain natural group representation. A unipotent affine algebraic group is then a group with all elements unipotent. Definition Definition with matrices Consider the group \mathbb_n of upper-triangular matrices with 1's along the diagonal, so they are the group of matrices :\mathbb_n = \left\. Then, a unipotent group can be defined as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general. Definition Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism ''F'' is defined by :F(r) = r^p for all ''r'' in ''R''. It respects the multiplication of ''R'': :F(rs) = (rs)^p = r^ps^p = F(r)F(s), and is 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients :\frac, if . Ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Central Series
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algrebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras. A general group possesses a lower central series and upper central series (also called the descending central series and ascending central series, respectively), but these are central series in the strict sense (terminating in the trivial subgroup) if and only if the group is nilpotent. A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable. Definition A central series is a sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Representation
In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity. Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation. Conversely, a semisimple representation of a group ''G'' over a field k is a semisimple module over the group ring ''k'' 'G'' Equivalent characterizations Let ''V'' be a representation of a group ''G''; or more generally, let ''V'' be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety. Some texts do not require a prime ideal, and call ''irreducible'' an algebraic variety defined by a prime ideal. This article refers to zero-loci of not necessarily prime ideals as affine algebraic sets. In some contexts, it is useful to distinguish the field in which the coefficients are considered, from the algebraically closed field (containing ) over which the zero-locus is considered (that is, the points of the affine variety are in ). In this case, the variety is said ''defined over'' , and the points of the variety that belong to are said '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Matrices
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while an example of a 3×3 diagonal matrix is \left begin 6 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end\right/math>. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values. Definition As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix with ''n'' columns and ''n'' rows is diagonal if \forall i,j \in \, i \ne j \implies d_ = 0. However, the main diagonal entries are unrestricted. The term ''diagonal matrix'' may s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Group
In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. Definition The definition uses the idea of a central series for a group. The followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse (logic)
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposition ''All S are P'', the converse is ''All P are S''. Either way, the truth of the converse is generally independent from that of the original statement.Robert Audi, ed. (1999), ''The Cambridge Dictionary of Philosophy'', 2nd ed., Cambridge University Press: "converse". Implicational converse Let ''S'' be a statement of the form ''P implies Q'' (''P'' → ''Q''). Then the converse of ''S'' is the statement ''Q implies P'' (''Q'' → ''P''). In general, the truth of ''S'' says nothing about the truth of its converse, unless the antecedent ''P'' and the consequent ''Q'' are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
