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Bott–Samelson Resolution
In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by in the context of compact Lie groups. The algebraic formulation is independently due to and . Definition Let ''G'' be a connected reductive complex algebraic group, ''B'' a Borel subgroup and ''T'' a maximal torus contained in ''B''. Let w \in W = N_G(T)/T. Any such ''w'' can be written as a product of reflections by simple roots. Fix minimal such an expression: :\underline = (s_, s_, \ldots, s_) so that w = s_ s_ \cdots s_. (''ℓ'' is the length of ''w''.) Let P_ \subset G be the subgroup generated by ''B'' and a representative of s_. Let Z_ be the quotient: :Z_ = P_ \times \cdots \times P_/B^\ell with respect to the action of B^\ell by :(b_1, \ldots, b_\ell) \cdot (p_1, \ldots, p_\ell) = (p_1 b_1^, b_1 p_2 b_2^, \ldots, b_ p_\ell b_\ell^). It is a smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schubert Variety
In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using linear algebra, inside a fixed vector subspace ''W''. Here ''W'' may be a vector space over an arbitrary field, though most commonly over the complex numbers. A typical example is the set ''X'' whose points correspond to those 2-dimensional subspaces ''V'' of a 4-dimensional vector space ''W'', such that ''V'' non-trivially intersects a fixed (reference) 2-dimensional subspace ''W''2: :X \ =\ \. Over the real number field, this can be pictured in usual ''xyz''-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of \mathbb(W), we obtain an open subset ''X''° ⊂ ''X''. This is isomorphic to the set of all lines ''L'' (not necessarily through the origin) which m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolution Of Singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic ''p'' it is an open problem in dimensions at least 4. Definitions Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety ''X'', in other words a complete non-singular variety ''X′'' with the same function field. In practice it is more convenient to ask for a different condition as follows: a variety ''X'' has a resolution of singularities if we can find a non-singular variety ''X′'' and a proper birational map from ''X′'' to ''X''. The condition that the map is proper is needed to exclude trivial solutions, such as taking ''X′'' to be the subvariety of non- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Lie Group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces. Compact Lie groups Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include * the circle group T and the torus groups T''n'', * the orthogonal group O(''n''), the special orthogonal group SO(''n'') and its covering spin group Spin(''n''), * the unitary group U(''n'') and the special unitary group SU(''n''), * the compact forms of the exceptional Lie groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductive Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a numbe ... [...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] |
Borel 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] |
Maximal Torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefore isomorphic to the standard torus T''n''). A maximal torus is one which is maximal among such subgroups. That is, ''T'' is a maximal torus if for any torus ''T''′ containing ''T'' we have ''T'' = ''T''′. Every torus is contained in a maximal torus simply by dimensional considerations. A noncompact Lie group need not have any nontrivial tori (e.g. R''n''). The dimension of a maximal torus in ''G'' is called the rank of ''G''. The rank is well-defined since all maximal tori turn out to be conjugate. For semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram. Examples The unitary group U(''n'') has as a maximal torus the subgroup of all diagonal matrices. That is, : T = \left\. ''T'' is clearl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Length Of A Weyl Element
In mathematics, the length of an element ''w'' in a Weyl group ''W'', denoted by ''l''(''w''), is the smallest number ''k'' so that ''w'' is a product of ''k'' reflections by simple roots. (So, the notion depends on the choice of a positive Weyl chamber.) In particular, a simple reflection has length one. The function ''l'' is then an integer-valued function of ''W''; it is a length function In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group. Definition A length function ''L'' : ''G'' → R+ on a group ''G'' is a function satisfy ... of ''W''. It follows immediately from the definition that ''l''(''w''−1) = ''l''(''w'') and that ''l''(''ww'''−1) ≤ ''l''(''w'') + ''l''(''w' ''). References * {{cite book, last1=Kac, first1=Victor G., title=Infinite dimensional Lie algebras, date=1994, publisher=Cambridge University Press, location=Cambridge, isbn=9780521466936, ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Variety
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. Definition First, let ''X'' be an affine scheme of finite type over a field ''k''. Equivalently, ''X'' has a closed immersion into affine space ''An'' over ''k'' for some natural number ''n''. Then ''X'' is the closed subscheme defined by some equations ''g''1 = 0, ..., ''g''''r'' = 0, where each ''gi'' is in the polynomial ring ''k'' 'x''1,..., ''x''''n'' The affine scheme ''X'' is smooth of dimension ''m'' over ''k'' if ''X'' has dimension at least ''m'' in a neighborhood of each point, and the matrix of derivatives (∂''g''''i''/∂''x''''j'') has rank at least ''n''−''m'' everywhere on ''X''. (It follows that ''X'' has dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Singularities
In mathematics, more particularly in the field of algebraic geometry, a scheme X has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map :f \colon Y \rightarrow X from a regular scheme Y such that the higher direct images of f_* applied to \mathcal_Y are trivial. That is, :R^i f_* \mathcal_Y = 0 for i > 0. If there is one such resolution, then it follows that all resolutions share this property, since any two resolutions of singularities can be dominated by a third. For surfaces, rational singularities were defined by . Formulations Alternately, one can say that X has rational singularities if and only if the natural map in the derived category :\mathcal_X \rightarrow R f_* \mathcal_Y is a quasi-isomorphism. Notice that this includes the statement that \mathcal_X \simeq f_* \mathcal_Y and hence the assumption that X is normal. There are related notions in positive and mixed characteristic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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