#### Date of Submission

2-28-2002

#### Date of Award

2-28-2003

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Mukherjee, Gautam (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

The main objective of this thesis is to develop an algebraic deformation theury for associative dialgebras, which are binary quadratic algebras discovered by J.-L. Loday in (16). (17), and subisequently, to derive a G-algebra siructure ou the dialgebra colhomology with cocfticients in itself.Deformation theory dates back at Ieast to Riemann's 1837 memoir on alelian fianetions in which he studied IHanifolds of complex dimension one and calculated the mumber of parameters (called moduli) upon which a deformation depends. The modern theory of deformations of structures on manifolds was developed extensively ly Frolicher-Kodaira-Nijenhnis-Nirenberg-Spencer (13], [14], [15). [25|, [26).The study of deformations of algebraic struetures wias initiated by M. Gersten- haber (5). (6). [7). [8). 191. The basic theurems and features of a deformmation thcory are all due to him. The following is a brief deseription of the deformation theory of asSociative algebra, as introducexl by M. Gerstenhaber.Let A be an associative algebra over a fiekl K, with underlying vector space V. Let K|| be the formal power series ting with coelficients in K and Q = K({1)) be the quotient powrT Series field of K| Let V] br the power series module over V and 19 = V[t|| n ter Q. Theu Ve is a vector space over Q. Any bilinear function f :Vx -l extends to a bilinear function Vq xl Vg uver Q. A bilinear function f: V x Vg Vo which is such an extension is said tu be defined over K. Suppose a multiplication h: Vo Â®4 Vá»¥ le is given by a lormal power series of the form fi(a, b) = Fo(a, b) + F(a, b)t + F2(a, b) +. where cach F, is definexd over K and Fo(a, b) = ab, the original multiplication of A. Assume that f, is associative, that is,

#### Control Number

ISILib-TH136

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Majumdar, Anita Dr., "Deformation Theory of Dialgebras." (2003). *Doctoral Theses*. 99.

https://digitalcommons.isical.ac.in/doctoral-theses/99

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842875