scholarly journals Gröbner–Shirshov Bases for Replicated Algebras

2017 ◽  
Vol 24 (04) ◽  
pp. 563-576 ◽  
Author(s):  
P.S. Kolesnikov
Keyword(s):  
Lie Algebras ◽  
Word Problem ◽  
Membership Problem ◽  
Leibniz Algebras ◽  
Universal Approach ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).

scholarly journals The Complexity of the Ideal Membership Problem for Constrained Problems Over the Boolean Domain

2021 ◽  
Vol 17 (4) ◽  
pp. 1-29
Author(s):  
Monaldo Mastrolilli
Keyword(s):  
Sufficient Conditions ◽  
Constraint Satisfaction Problems ◽  
Algorithmic Problem ◽  
Membership Problem ◽  
Efficient Computation ◽  
Constrained Problems ◽  
Constraint Language ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

Given an ideal I and a polynomial f the Ideal Membership Problem (IMP) is to test if f ϵ I . This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the IMP for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the borderline of tractability. By using Gröbner bases techniques, we extend Schaefer’s dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems (CSPs) over the Boolean domain to be either in P or NP-hard. Moreover, our result implies necessary and sufficient conditions for the efficient computation of Theta Body Semi-Definite Programming (SDP) relaxations, identifying therefore the borderline of tractability for constraint language problems. This article is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares (SoS) proofs [O’Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the IMP tractability for combinatorial ideals implies bounded coefficients in SoS proofs.

scholarly journals Local zero estimates and effective division in rings of algebraic power series

2018 ◽  
Vol 2018 (737) ◽  
pp. 111-160 ◽  
Author(s):  
Guillaume Rond
Keyword(s):  
Power Series ◽  
Necessary Condition ◽  
Effective Solution ◽  
Gap Theorem ◽  
Division Theorem ◽  
Ideal Membership ◽  
Formal Power ◽  
Algebraic Power Series ◽  
Ideal Membership Problem ◽  
The Ideal

AbstractWe give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact, we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally, we apply these results to prove a gap theorem for power series which are remainders of the Grauert–Hironaka–Galligo Division Theorem.

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

Some new results on Gröbner–Shirshov bases for Lie algebras and around

2018 ◽  
Vol 28 (08) ◽  
pp. 1403-1423
Author(s):  
L. A. Bokut ◽  
Yuqun Chen ◽  
Abdukadir Obul
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Leibniz Algebras ◽  
Novikov Algebras

We review Gröbner–Shirshov bases for Lie algebras and survey some new results on Gröbner–Shirshov bases for [Formula: see text]-Lie algebras, Gelfand–Dorfman–Novikov algebras, Leibniz algebras, etc. Some applications are given, in particular, some characterizations of extensions of groups, associative algebras and Lie algebras are given.

scholarly journals Leibniz Algebras Constructed by Representations of General Diamond Lie Algebras

2017 ◽  
Vol 42 (3) ◽  
pp. 1281-1293 ◽  
Author(s):  
L. M. Camacho ◽  
I. A. Karimjanov ◽  
M. Ladra ◽  
B. A. Omirov
Keyword(s):  
Lie Algebras ◽  
Leibniz Algebras
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