Abdukadir Obul

Gröbner–Shirshov Basis of Quantum Group of Type G 2

By using the generating sequence and relations given by Ringel for his Ringel–Hall algebra in [8], we give a Gröbner–Shirshov basis for quantum group of type G 2.

Gröbner-Shirshov bases for free Gelfand-Dorfman-Novokov algebras and for right ideals of free right Leibniz algebras

ABSTRACT In this paper, we first found a magmatic (i.e., absolutely non-associative) Gröbner-Shirshov basis of a free Gelfand-Dorfman-Novikov algebra GDN(X) such that the corresponding set of irreducible magmatic words is the Dzhumadildaev-Löfwall linear basis of the GDN(X). Then, we prove a Composition-Diamond lemma for right ideals of a free right Leibniz algebra Lei(X).

Gröbner–Shirshov basis method for multiple tensor products of some associative algebras

In this paper, we establish 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 path...

Gröbner–Shirshov pair of irreducible modules over quantized enveloping algebra of type An

In this paper, first, we give a Grobner–Shirshov pair of finite-dimensional irreducible module Vq(λ) over Uq(An), the quantized enveloping algebra of type An, by using the double free module method and the known Grobner–Shirshov basis of Uq(An). Then, 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 (λ) over U(An).

Gröbner-Shirshov Basis of Twisted Generic Composition Algebras of Affine Type

In this paper, by using PBW bases for the twisted generic composition algebras of affine type, we prove that the set of the skew-commutator relations of the iso-classes of indecomposable representations forms a minimal Grobner-Shirshov basis for the twisted generic composition algebras of affine type.

Grobner-Shirshov Basis of Quantum Groups ⁄

In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal Grobner-Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn.