A Study On Gr(?)bner-Shirshov Basis Theory Of Some Associative And Nonassociative Algebras

Grobner basis theory was established by Buchberger,Shirshov and Bergman.The Grobner basis theory for commutative algebras was introduced by Buchberger and it pro-vides an effective algorithm to the reduction problem in commutative algebras.Bergman generalized Buchberger's theory on associative algebras.A parallel theory for Lie alge-bras was introduced by Shirshov.Later Bokut proved that the Grobner basis theory of Buchberger and Bergman is a special case of Shirshov's method for Lie algebras.So now this theory is also called Grobner-Shirshov basis theory.The central result of Grobner-Shirshov basis theory is the Composition-Diamond lemma which enables us to get a linear basis of the corresponding quotient algebra,whenever we have a Grobner-Shirshov basis.Now,the Grobner-Shirshov basis theory has been widely used in many branches of math-ematics and other relevant subjects.The notion of Leibniz algebra was introduced by Loday as a non-commutative general-ization of Lie algebra.Later,Loday generalizes the notion of Leibniz algebra to the Leibniz n-algebra(n ? 2),and Leibniz 2-algebra is exactly the Leibniz algebra itself.Because of the important applications in the nano-mechanics,Leibniz algebra quickly became a pop-ular research field in algebra.Later,Dzhumadil'daev and others introduced q-Leibniz algebra and restricted Leibniz algebra,and got very good results on them.Loday also introduced a Koszul dual of Liebniz algebr-a,called Zinbiel algebr-a(the word Zinbiel is also the reverse write of Liebniz)and Loday and other mathematicians got some important properties and interesting examples of the Zinbiel algebra.Commutative algebra,Grassmann algebra and path algebra are associative algebras.The path algebras play a central role in the repreentation theory of finite-dimensional algebras and the theory of Grobner-Shirshov basis for path algebras has been given by Farkas,Feustel and E.L.Green.Grassmann(Exterior)algebra is an important tool in linear algebra and geometry.In particular,it is a natural domain in which to state and prove many theorems in linear and affine geometry.I.M.Gelfand and Dorfman,in connection with Hamiltonian operators in the for-mal calculus of variations,invented a new class of non-associative algebras,which is a right-symmetric(right pre-Lie)algebra whose left multiplication operators commute(left-commutative algebra).At the same time,Novikov,independently,invented the same algebras in connection with linear Poisson brackets of hydrodynamic type.Osborn called this new algebra as Novikov algebra and began to classify simple Novikov algebra as well as its irreducible modules.Now,this algebra is called right Gelfand-Dorfman-Novikov al-gebra(or right GDN algebra for short).S.I.Gelfand pointed out that any associative com-mutative differential algebra is a GDN algebra under the new multiplication xoy = x(Dy).The structure theory of GDN algebras was started by Zelmanov.Bokut,Chen,Zhang established the Grobner-Shirshov basis theory for GDN algebra and using this theory proved that any GDN algebra is embeddable into a differential commutative-associative algebra.They also proved the Composition-Diamond lemma for GDN algebra.The quantum groups were introduced by Drinfeld and Jimbo independently and get-ting vast attentions from algebraists.Now the quantum groups already became a hot research area in mathematics.In the beginning of nineties of the last century,Ringel in-troduced the Ringel-Hall algebra by using the representation theory of finite dimensional algebra and consequently gave a realization of the positive part of quantum groups.Later it is acknowledged that the Ringel-Hall algebra approach is one of the most successful model of the quantum groups.The Grobner-Shirshov basis theory for the quantum en-veloping algebra(that is the quantum group)is established by Bokut and Malcolmson and by using the relations of Yamane,they provided an explicit Grobner-Shirshov basis for the quantum group of type An(q8 ? 1).In this thesis,we mainly study Grobner-Shirshov basis theory for 1/2-Leibniz algebra,the Composition-Diamond lemma for right ideals of free Leibniz algebra,the Composition-Diamond lemma for Zinbiel algebra over a commutative algebra,and establish Composition-Diamond lemma for multiple tensor product of some associative algebras.We also study the Grobner-Shirshov basis of irreducible module Vq(?)of quantum group of type An by using the compositions of associative algebras,relevant results in the representation theory of algebras and the method of double free module.Finally,as an application of Composition-Diamond lemma of GDN algebra,we compute a non-associative Grobner-Shirshov basis of a free GDN algebra.This thesis is divided into four chapters.Their main contents are as followsIn the first chapter we introduce some backgrounds,the aim and the meaning of the study of the Grobner-Shirshov basis theory and Composition-Diamond Lemma.We also sketched the history of development and current research status of the Grobner-Shirshov basis theory and Composition-Diamond Lemma of different classes of algebras,and the main work in this thesis.Furthermore,we give our research contents and the chapter arrangements of the thesis.In the second chapter,we prove a Composition-Diamond lemma for two sided ideals of so called a(right)1/2-Leibniz polynomial algebra 1/2-Lei(X)= Magma(X |(uv)w =(uw)v + u(vw),v>w)in deg-lex ordering of non-associative words.Then,we prove a Composition-Diamond lemma for right ideals of a free(right)Leibniz algebra Lei(X)=Magma(X | u(vw)=(uv)w-(uw)v)in deg-rlex ordering of words(rlex means reverse lexicographic or right lexicographic),by using Aymon-Grivel theorem,that is any Leibniz algebra is embeddable into a dialgebra.Finally,we establish the Composition-Diamond lemma for Zinbiel algebra Zin(X)over a commutative algebra k[Y].In the third chapter,first,we establish the Composition-Diamond lemma for multiple tensor products of commutative algebra k[Y],free associative algebras k(X(i),Grassman algebras Gk(Z(j)and path algebras pathk(P(l)over a field k.Then,we give a Grobner-Shirshov basis of finite dimensional irreducible module Vq(?)of Uq(An),the Drinfeld-Jimbo quantum group of type An,by using the double free module method and the known Grobner-Shirshov basis of Uq(An).Finally,by specializing a suitable version of Uq(An)at q = 1.we get a Grobner-Shirshov basis of U(An),and get a Grobner-Shirshov pair for the finite dimensional irreducible module V(?)of U(An).In the fourth chapter,as an application of Composition-Diamond lemma of non-associative algebra GDN(X),we compute a Grobner-Shirshov basis for GDN(X).More precisely,first,by checking composition of the defining relations,we found a non-associative polynomial set S1 such that Irr(S1)is exactly the set of right GDN tableaux over a well-ordered set X.Then,we prove the set Irr(S1)forms a linear basis of the free right GDN algebras generated by X,and so by Composition-Diamond lemma,we prove that S1 is the non-associative Grobner-Shirshov basis of the free GDN algebra.