Gröbner-Shirshov Bases of Irreducible Modules of the Quantum Group of Type G2

First, the authors give a Grbner-Shirshov basis of the finite-dimensional irreducible module Vq(λ) of the Drinfeld-Jimbo quantum group U_q(G_2) by using the double free module method and the known Grbner-Shirshov basis of U_q(G_2). Then, by specializing a suitable version of U_q(G_2) at q = 1, they get a Grbner-Shirshov basis of the universal enveloping algebra U(G_2) of the simple Lie algebra of type G_2 and the finite-dimensional irreducible U(G_2)-module V(λ).     

Skew-commutator relations and Gröbner-Shirshov basis of quantum group of type F 4

We give a Grobner-Shirshov basis of quantum group of type F4 by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel- Hall algebras of type F4 by using an ＇inductive＇ method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew- commutator relations; instead, we compute the ＇easier＇ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ＇deduce＇ others from these ＇easier＇ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal CrSbner-Shirshov basis of the positive part of the quantum group of type F4. Dually, we get a Grobner-Shirshov basis of the negative part of the quantum group of type F4. And finally, we give a Gr6bner-Shirshov basis for the whole quantum group of type F4.     

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 Ringel , C. M. ( 1996 ). PBW-bases of quantum groups . J. Reine Angew. Math. 470 : 51 – 88 .[Web of Science ®] , [Google Scholar]], we give a Gröbner–Shirshov basis for quantum group of type G ₂.     

Hilbert Series of Group Representations and Gröbner Bases for Generic Modules

Each matrix representation :G GL_n() of a finite Group G over a field induces an action of G on the module ⁿ over the polynomial algebra The graded -submodule M() of _n generated by the orbit of is studied. A decomposition of M() into generic modules is given. Relations between the numerical invariants of and those of M(), the latter being efficiently computable by Gröbner bases methods, are examined. It is shown that if is multiplicity-free, then the dimensions of the irreducible constituents of can be read off from the Hilbert series of M(Pi;). It is proved that determinantal relations form Gröbner bases for the syzygies on generic matrices with respect to any lexicographic order. Gröbner bases for generic modules are also constructed, and their Hilbert series are derived. Consequently, the Hilbert series of M(Pi;) is obtained for an arbitrary representation.     

Grbner-Shirshov Basis for Degenerate Ringel-Hall Algebra of Type C_3

The Gr?bner-Shirshov basis of the degenerate Ringel-Hall Algebras of type C_3 is obtained by studying the generic extension monoid algebra.     

Gröbner-Shirshov bases for Rota-Baxter algebras

We establish the composition-diamond lemma for associative nonunitary Rota-Baxter algebras of weight λ. To give an application, we construct a linear basis for a free commutative and nonunitary Rota-Baxter algebra, show that every countably generated Rota-Baxter algebra of weight 0 can be embedded into a two-generated Rota—Baxter algebra, and prove the 1-PBW theorems for dendriform dialgebras and trialgebras.     

Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Gröbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.     

Optimising Gröbner Bases on Bivium

Bivium is a reduced version of the stream cipher Trivium. In this paper we investigate how fast a key recovery attack on Bivium using Gröbner bases is. First we explain the attack scenario and the cryptographic background. Then we identify the factors that have impact on the computation time and show how to optimise them. As a side effect these experiments benchmark several Gröbner basis implementations. The optimised version of the Gröbner attack has an expected running time of 2^39.12 s, beating the attack time of our previous SAT solver attack by a factor of more than 330. Furthermore this approach is faster than an attack based on BDDs, an exhaustive key search, a generic time-memory trade-off attack and a guess-and-determine strategy.     

Gröbner Bases for Ideals of σ-PBW Extensions

In this article, we introduce the σ-PWB extensions and construct the theory of Gröbner bases for the left ideals of them. We prove the Hilbert's basis theorem and the division algorithm for this more general class of Poincaré–Birkhoff–Witt extensions. For the particular case of bijective and quasi-commutative σ-PWB extensions, we implement the Buchberger's algorithm for computing Gröbner bases of left ideals.     

Grbner-Shirshov Basis for Degenerate Ringel-Hall Algebras of Type F_4

In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gr¨obner-Shirshov basis for degenerate Ringel-Hall algebras of type F_4.     

Gröbner–Shirshov Bases for Schreier Extensions of Groups

In this article, by using the Gröbner–Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.     

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

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 bases for extended modular, extended Hecke, and Picard groups

In this paper, Gröbner-Shirshov bases (noncommutative) for extended modular, extended Hecke and Picard groups are considered. A new algorithm for obtaining normal forms of elements and hence solving the word problem in these groups is proposed.     

On Generators and Relations of Simple Groups of Lie Type

1. Introduction Let L be a simple Lie algebra over C, rank L>1, K a field, R.Steinberg proved a theorem (Cf.〔1〕 and 〔2〕 12.1) which leads to the definition ofUniversal Chevalley groups through generators and relations such that a simple Chevalley group is isomrphic to the quotient group of the corresponding Universal Chevalley group modulo its centre. Steiberg′s theorem playsan importent role in dealing with Chevalley groups.     

On the Connection Between Ritt Characteristic Sets and Buchberger–Gröbner Bases

For any polynomial ideal \(\mathcal {I}\), let the minimal triangular set contained in the reduced Buchberger–Gröbner basis of \(\mathcal {I}\) with respect to the purely lexicographical term order be called the W-characteristic set of \(\mathcal {I}\). In this paper, we establish a strong connection between Ritt’s characteristic sets and Buchberger’s Gröbner bases of polynomial ideals by showing that the W-characteristic set \(\mathbb {C}\) of \(\mathcal {I}\) is a Ritt characteristic set of \(\mathcal {I}\) whenever \(\mathbb {C}\) is an ascending set, and a Ritt characteristic set of \(\mathcal {I}\) can always be computed from \(\mathbb {C}\) with simple pseudo-division when \(\mathbb {C}\) is regular. We also prove that under certain variable ordering, either the W-characteristic set of \(\mathcal {I}\) is normal, or irregularity occurs for the jth, but not the \((j+1)\)th, elimination ideal of \(\mathcal {I}\) for some j. In the latter case, we provide explicit pseudo-divisibility relations, which lead to nontrivial factorizations of certain polynomials in the Buchberger–Gröbner basis and thus reveal the structure of such polynomials. The pseudo-divisibility relations may be used to devise an algorithm to decompose arbitrary polynomial sets into normal triangular sets based on Buchberger–Gröbner bases computation.     

Gröbner Bases for the Polynomial Ring with Infinite Variables and Their Applications

We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-to-one correspondences among various sets of partitions by using the division algorithm.     

Gröbner–Shirshov Basis for HNN Extensions of Groups and for the Alternating Group

In this article, we generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Gröbner–Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained.