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Nilpotent Orbit
In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. Definition An element ''X'' of a semisimple Lie algebra ''g'' is called nilpotent if its adjoint endomorphism : ''ad X'': ''g'' → ''g'', ''ad X''(''Y'') = 'X'',''Y'' is nilpotent, that is, (''ad X'')''n'' = 0 for large enough ''n''. Equivalently, ''X'' is nilpotent if its characteristic polynomial ''p''''ad X''(''t'') is equal to ''t''dim ''g''. A semisimple Lie group or algebraic group ''G'' acts on its Lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action. A nilpotent orbit is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent. Examples Nilpotent n\times n matrices with complex entries form the main motivating case for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras. Examples *This definition can be applied in particular to square matrices. The matrix :: A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because A^3=0. See nilpotent matrix for more. * In the factor ring \Z/9\Z, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9. * Assume that two elements a and b in a ring R satisfy ab=0. Then the element c=ba is nilpotent as \beginc^2&=(ba)^2\\ &=b(ab)a\\ &=0.\\ \end An example with matrices (for ''a'', ''b''):A = \begin 0 & 1\\ 0 & 1 \end, \;\; B =\begin 0 & 1\\ 0 & 0 \end. Here AB=0 and BA=B. *By definition, any element of a nilsemigroup is nilpotent. Properties No nilpotent el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Closure
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Endomorphism
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g(h) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Group
In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph '' The Classical Groups''. The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern phy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominance Order
In discrete mathematics, dominance order (synonyms: dominance ordering, majorization order, natural ordering) is a partial order on the set of partitions of a positive integer ''n'' that plays an important role in algebraic combinatorics and representation theory, especially in the context of symmetric functions and representation theory of the symmetric group. Definition If ''p'' = (''p''1,''p''2,…) and ''q'' = (''q''1,''q''2,…) are partitions of ''n'', with the parts arranged in the weakly decreasing order, then ''p'' precedes ''q'' in the dominance order if for any ''k'' ≥ 1, the sum of the ''k'' largest parts of ''p'' is less than or equal to the sum of the ''k'' largest parts of ''q'': : p\trianglelefteq q \text p_1+\cdots+p_k \leq q_1+\cdots+q_k \text k\geq 1. In this definition, partitions are extended by appending zero parts at the end as necessary. Properties of the dominance ordering * Among the partitions of ''n'', (1,…,1) is the smalles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Murray Gerstenhaber
Murray Gerstenhaber (born June 5, 1927) is an American mathematician and professor of mathematics at the University of Pennsylvania, best known for his contributions to theoretical physics with his discovery of Gerstenhaber algebra. He is also a lawyer and a lecturer in law at the University of Pennsylvania Law School. Early life and education Gerstenhaber was born in Brooklyn, New York, to Pauline (née Rosenzweig; who was born in Romania; died in 1978) and Joseph Gerstenhaber (who was born in 1892 in Romania; died in 1975). His father was trained as a jeweler, "but being unable to find work in this line he ookemployment in a factory making airplane precision instruments”. As to his mother, in 2015 he noted: "For someone born into a minority family without means, I have been exceedingly lucky. The problems faced by talented but disadvantaged children today are preventing many who could be important contributors to the sciences, arts, and society in general, from achieving thei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Relation
In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram. Definition Let X be a set with a partial order \le. As usual, let < be the relation on such that |
Algebraically Closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation ''x''2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Poset
In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) ''P'' equipped with a rank function ''ρ'' from ''P'' to the set N of all natural numbers. ''ρ'' must satisfy the following two properties: * The rank function is compatible with the ordering, meaning that for all ''x'' and ''y'' in the order, if ''x'' < ''y'' then ''ρ''(''x'') < ''ρ''(''y''), and * The rank is consistent with the covering relation of the ordering, meaning that for all ''x'' and ''y'', if ''y'' covers ''x'' then ''ρ''(''y'') = ''ρ''(''x'') + 1. The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''inco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', or equivalently, such that conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the set of elements \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring ''R'' is a subring of ''R''. This article also deals with centralizers and norma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sl2-triple
In the theory of Lie algebras, an ''sl''2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra ''sl''2. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to their nilpotent orbits. Definition Elements of a Lie algebra ''g'' form an ''sl''2-triple if : ,e= 2e, \quad ,f= -2f, \quad ,f= h. These commutation relations are satisfied by the generators : h = \begin 1 & 0\\ 0 & -1 \end, \quad e = \begin 0 & 1\\ 0 & 0 \end, \quad f = \begin 0 & 0\\ 1 & 0 \end of the Lie algebra ''sl''2 of 2 by 2 matrices with zero trace. It follows that ''sl''2-triples in ''g'' are in a bijective correspondence with the Lie algebra homomorphisms from ''sl''2 into ''g''. The alternative notation for the elements of an ''sl''2-triple is , with ''H'' corresponding to ''h'', ''X'' corresponding to ''e'', and ''Y'' corresponding to ''f''. H is ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
