توصيفگر ها :
كلاف برداري , كلاف خطي , كلاف برداري بديهي , كلاف برداري غير بديهي , كلاف برداري جهت پذير , نظريه نمايش , جبر لي
چكيده فارسي :
در اﯾﻦ رﺳﺎﻟﻪ ﺳﺆاﻻت ﻃﺒﯿﻌﯽ زﯾﺮ ﻣﻄرح ﻣﯽ ﺷﻮد
1. آيا ارتباطي ميان نمايش‌هاي روي يك جبر لي و فضاي برداري حاصل از برشهاي يك كلاف برداري وجود دارد؟
2. چه ارتباطي ميان تجزيه كلاف‌هاي برداري و نمايش‌هاي روي يك پايه از كلاف برداري وجود دارد؟
3. رابطه ميان همريختي از يك اسكيم هموار به فضاي تصويري $n$ بعدي و نمايش روي جبرهاي لي چيست؟
4. روي چه كلاف‌هاي برداري سوالات فوق مجاز است؟
5. رابطه ميان جهت پذيري كلاف‌هاي برداري و نمايش‌هاي روي يك جبر لي به چه صورت است؟
هدف اصلي اين رساله مطالعه خواص و ويژگي‌هاي برشهاي يك كلاف برداري و به خصوص بررسي سؤالات بالا است. در اين راستا به سؤال $(1)$ و $(2)$ در حالت كلي پاسخ مي‌دهيم و به بقيه سؤالات فوق در حالت خاص جواب مي‌دهيم.
چكيده انگليسي :
The study of vector bundles, Hilbert schemes and representation theory is a fundamental problem in
algebraic geometry. For example, a mathematical version of the S-duality conjecture formulated by
C. Vafa and E. Witten [13] showed a connection between stable vector bundles and representations
of certain infinite dimensional Lie algebras. In the special case, this concludes a relation between
representations of the infinite dimensional Heisenberg algebras and the Hilbert schemes of points on
algebraic surface. Another example between vector bundles and representation theory is the relation
among representation of compact Lie groups, principal bundles over elliptic Calabi-Yau manifolds and
the physics F-theory [6].
A line bundle expresses the concept of a line that varies from point to point of a space. Studying
of line bundles in Algebraic Geometry is a fundamental problem such that they play important role
in Kodaira Embedding Theorem, Hartshorne-Serre Theorem, Poincare’s formula, etc. Also, they can
be use for the study of Picard group and Fano variety. The decomposition of a vector bundle to direct
sum of line bundles is one of the major problems in the theory of vector bundles. A. Grothendieck [10]
showed that for a rational curve every vector bundle is decomposed as a direct sum of line bundles.
For this, let X be the Rimann sphere, G a complex Lie group, and H a cartan subgroup of G. He
proved that there is a map j between the set of classes of G-holomorphic fiber bundles over X and
a quotient space of the set of classes of H-holomorphic fiber bundles over X. Then he proved that j
is bijection when G is reductive. So he concluded a vector bundle over X can be split into a direct
sum of line bundles. In 1974, W. Barth and A. Van de Ven [3] obtained splitting an algebraic vector
bundle of rank 2 into line bundles is equivalent to the complete intersection problem in codimension
2. In 1982, Hazwinkel and Martin [8] gave a short elementary proof of Grothendieck’s theorem on
algebraic vector bundles by using straightforward manipulations of matrices. In 2003, J. A. Leslie
and Q. Yue [11] studied the decomposition of real vector bundles. They gave a general decomposition
result which relates a given vector bundle to some cohomology classes with local coefficients in the
homotopy group of a Grassmann manifold. E. Ballico [1], in 2005, obtained a decomposition criterion
for vector bundles over X where X is a reduced and connected projective variety. Then he proved in [2]
that every holomorphic vector bundle on a non reduced one dimensional complex space under special
criterion is a direct sum of line bundles. I. Biswas and D. S. Nagaraj [4] classified real algebraic vector
bundles over the scheme defined by a nondegerate anisotropic conic. In fact, let X be a nondegerate
anisotropic real conic. They proved that any real vector bundle over X is isomorphic to
(m⊕i=1T ⊗a iX) ⊕ ( VR ⊗ (n⊕j=1T⊗b j X) ) .
Let X be a nonsingular toric variety, and G a complex linear algebraic group. I. Biswas, A. Dey and
M. Poddar [5] classified equivariant principal G-bundles over X. From this they investigated a vector
bundle E over X can be decomposed as a direct sum of line bundles for special G if it satisfied in
special conditions.
The aim of this paper is to make a link between representations over a Lie algebra and line bundles
over a scheme. However, more research is needed before being able to associate between vector bun-
dles and representations over modules. First, we show that, there is a two-sided relationship between
irreducible representations over a solvable lie algebra and nontrivial line bundles. Finally, we prove
that there is a corresponding between decomposition of a vector bundles to line bundles and direct
sum of representations over a solvable Lie algebra.
