scholarly journals Algebraic loop structures on algebra comultiplications

2019 ◽  
Vol 17 (1) ◽  
pp. 742-757 ◽  
Author(s):  
Dae-Woong Lee
Keyword(s):  
Lie Algebra ◽  
Rational Cohomology ◽  
Algebraic Loop

Abstract In this paper, we study the algebraic loop structures on the set of Lie algebra comultiplications. More specifically, we investigate the fundamental concepts of algebraic loop structures and the set of Lie algebra comultiplications which have inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications up to all the possible quadratic and cubic Lie algebra comultiplications. We also apply those notions to the rational cohomology of Hopf spaces.

scholarly journals Algebraic Inverses on Lie Algebra Comultiplications

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 565 ◽  
Author(s):  
Dae-Woong Lee
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Binary Operation ◽  
Graded Lie Algebra ◽  
Algebraic Loop ◽  
Left And Right

In this note, we investigate algebraic loop structures and inverses of elements of a set of all homomorphisms of Lie algebras with a binary operation derived from a Lie algebra comultiplication. As a symmetry phenomenon, we show that if l ( 1 ) c and r ( 1 ) c are the left and right inverses of the identity 1 : L → L on a free graded Lie algebra L , respectively, based on the Lie algebra comultiplication ψ c : L → L ⊔ L , then we have l ( 1 ) = l ( 1 ) c and r ( 1 ) = r ( 1 ) c , where c : L → L ⊔ L is a commutator.

scholarly journals Colored partitions and a Generalization of the Braid Arrangement

10.37236/1289 ◽  
1996 ◽  
Vol 4 (1) ◽  
Author(s):  
Volkmar Welker
Keyword(s):  
Lie Algebra ◽  
Combinatorial Analysis ◽  
Topological Properties ◽  
Generic Fiber ◽  
Arrangement Of Hyperplanes ◽  
Intersection Lattice ◽  
Colored Partitions ◽  
Rational Cohomology ◽  
Definition Of ◽  
Algebra Homology

We study the topology and combinatorics of an arrangement of hyperplanes in ${\bf C}^n$ that generalizes the classical braid arrangement. The arrangement plays in important role in the work of Schechtman and Varchenko on Lie algebra homology, where it appears in a generic fiber of a projection of the braid arrangement. The study of the intersection lattice of the arrangement leads to the definition of lattices of colored partitions. A detailed combinatorial analysis then provides algebro-geometric and topological properties of the complement of the arrangement. Using results on the character of $S_n$ on the cohomology of these arrangements we are able to deduce the rational cohomology of certain spaces of polynomials in the complement of the standard discriminant that have no root in the first $s$ integers.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

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