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Prehomogeneous Vector Space
In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space ''V'' together with a subgroup ''G'' of the general linear group GL(''V'') such that ''G'' has an open dense orbit in ''V''. Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory. The irreducible PVS were classified by Sato and Tatsuo Kimura in 1977, up to a transformation known as "castling". They are subdivided into two types, according to whether the semisimple part of ''G'' acts prehomogeneously or not. If it doesn't then there is a homogeneous polynomial on ''V'' which is invariant under the semisimple part of ''G''. Setting In the setting of Sato, ''G'' is an algebraic group and ''V'' is a rational representation of ''G'' which has a (nonempty) open orbit in the Zariski topology. However, PVS can also be studied from the point of view of Lie theory: for instance, in Knapp (200 ... [...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] |
Homogeneous Polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x^3 + 3 x^2 y + z^7 is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. A form of degree 2 is a quadratic fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan algebra is also denoted ''x'' ∘ ''y'', particularly to avoid confusion with the product of a related associative algebra. The axioms imply that a Jordan algebra is power-associative, meaning that x^n = x \cdots x is independent of how we parenthesize this expression. They also imply that x^m (x^n y) = x^n(x^m y) for all positive integers ''m'' and ''n''. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element x, the operations of multiplying by powers x^n all commute. Jordan algebras were first introduced by to formalize the notion of an algebra of observables in quantum mechanics. They were originally called "r-number systems", but were renamed "Jordan algebras" by , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Symmetric Space
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' Dynkin diagrams will be explicitly so named. Classification of semisimple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Factor
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semisimple subalgebra. One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra. When viewed as a factor-algebra of ''g'', this semisimple Lie algebra is also called the Levi factor of ''g''. To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple. Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form :\exp(\mathrm(z)) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernest Vinberg
Ernest Borisovich Vinberg (russian: Эрне́ст Бори́сович Ви́нберг; 26 July 1937 – 12 May 2020) was a Soviet and Russian mathematician, who worked on Lie groups and algebraic groups, discrete subgroups of Lie groups, invariant theory, and representation theory. He introduced Vinberg's algorithm and the Koecher–Vinberg theorem. He was a recipient of the 1997 Humboldt Prize. He was on the executive committee of the Moscow Mathematical Society. In 1983, he was an Invited Speaker with a talk on ''Discrete reflection groups in Lobachevsky spaces'' at the International Congress of Mathematicians in Warsaw. In 2010, he was elected an International Honorary Member of the American Academy of Arts and Sciences. Ernest Vinberg died from pneumonia caused by COVID-19 on 12 May 2020. Selected publications * * * editor and co-author: (contains ''Construction of the exceptional simple Lie algebras'') * with A. L. Onishchik:2012 pbk edition* with V. V. Gorbatsevi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilradical Of A Lie Algebra
In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical \mathfrak(\mathfrak g) of a finite-dimensional Lie algebra \mathfrak is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the radical \mathfrak(\mathfrak) of the Lie algebra \mathfrak. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra \mathfrak^. However, the corresponding short exact sequence : 0 \to \mathfrak(\mathfrak g)\to \mathfrak g\to \mathfrak^\to 0 does not split in general (i.e., there isn't always a ''subalgebra'' complementary to \mathfrak(\mathfrak g) in \mathfrak). This is in contrast to the Levi decomposition: the short exact sequence : 0 \to \mathfrak(\mathfrak g)\to \mathfrak g\to \mathfrak^\to 0 does split (essentially because the quotient \mathfrak^ is semisimple). See also * Levi decomposition * Nilradical of a ring In algebra, the nilradical of a commutativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup. For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups. Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive) algebraic groups, in Jacques Tits' theory of groups with a (B,N) pair. Here the group ''B'' is a Borel subgroup and ''N'' is the normalizer of a maximal torus contained in ''B''. The notion was introduced by Armand Borel, who played a leading role in the development of the theory of algebraic groups. Parabolic subgroups Subgroups between a Borel subgroup ''B'' and the ambient group ''G'' are called parabolic subgroups. Parabolic subgroups ''P'' are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Wolcott Richardson
Roger Wolcott Richardson (30 May 1930 – 15 June 1993) was a mathematician noted for his work in representation theory and geometry. He was born in Baton Rouge, Louisiana Baton Rouge ( ; ) is a city in and the capital of the U.S. state of Louisiana. Located the eastern bank of the Mississippi River, it is the parish seat of East Baton Rouge Parish, Louisiana's most populous parish—the equivalent of counties ..., and educated at Louisiana State University, Harvard University and University of Michigan, Ann Arbor where he obtained a Ph.D. in 1958 under the supervision of Hans Samelson. After a postdoc appointment at Princeton University, he accepted a faculty position at the University of Washington in Seattle. He emigrated to the United Kingdom in 1970, taking up a chair at Durham University. In 1978 he moved to the Australian National University in Canberra, where he stayed as faculty until his death. Richardson's best known result states that if ''P'' is a parabol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flag Variety
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space ''V'' over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
