Non-Associative Algebras and Related Topics: NAART II, Coimbra, Portugal, July 18–22, 2022 (Springer Proceedings in Mathematics & Statistics, 427)

Organization
Preface
Contents
Lie Algebras, Superalgebras and Groups
Local Derivations of Classical Simple Lie Algebras
	1 Introduction
	2 Main Result
		2.1 A Classical Simple Lie Algebra
		2.2 Local Derivations of Algebras of Type A
		2.3 The Proof of the Main Result
		2.4 Local Derivations on mathfrakpsl3(mathbbF) Over a Field of Characteristic 3
		2.5 Local Derivations on 3-Dimensional Simple Lie Algebra Over a Field of Characteristic 2
		2.6 Local Derivations on Borel Subalgebras
	References
Examples and Patterns on Quadratic Lie Algebras
	1 Introduction
	2 Patterns and Duality
	3 Local Quadratic Algebras
	4 Final Comments
	References
Reductive Homogeneous Spaces of the Compact Lie Group G2
	1 Introduction
	2 Background on Octonion Algebras and mathfrakg2
		2.1 Octonion Division Algebra
		2.2 Cross Products and 3-Forms
		2.3 The Automorphism Group
		2.4 The Exceptional Lie Algebra mathfrakg2
		2.5 Reductive Subalgebras of mathfrakg2
		2.6 The Principal Three-Dimensional Subalgebra of mathfrakg2
	3 Homogeneous Spaces of G2
		3.1 G2 as a Hypersurface of V7,3
		3.2 The Symmetric Space Ms.5-.5.5-.5.5-.5.5-.5G2/H1.5-.5.5-.5.5-.5.5-.5G2/SO(4)
		3.3 The 6-Dimensional Sphere mathbbS6.5-.5.5-.5.5-.5.5-.5G2/H6.5-.5.5-.5.5-.5.5-.5G2/SU(3)
		3.4 The Unit Fiber Bundle Over the Six Dimensional Sphere V7,2.5-.5.5-.5.5-.5.5-.5G2/H3.5-.5.5-.5.5-.5.5-.5G2/SU(2)l.5-.5.5-.5.5-.5.5-.5mathcalUmathbbS6
		3.5 The Complex Projective Quadric  widetildeGr7,2.5-.5.5-.5.5-.5.5-.5G2/H2.5-.5.5-.5.5-.5.5-.5G2/U(2)l.5-.5.5-.5.5-.5.5-.5Q5
		3.6 The Unknown Quotient Min.5-.5.5-.5.5-.5.5-.5G2/SO(3)
		3.7 The Twistor Space of Complex Structures Mr.5-.5.5-.5.5-.5.5-.5G2/U(2)r
		3.8 The Twistor Space of Quaternionic Structures Pr.5-.5.5-.5.5-.5.5-.5G2/SU(2)r
		3.9 The Irreducible Isotropy
	4 Conclusions
	References
On Certain Algebraic Structures Associated with Lie (Super)Algebras
	1 Introduction
	2 Definitions and Results
	3 Examples of (ε,δ)-JTS
	4 Examples of Lie (Super)Algebras Associated with (ε,δ)-Freudenthal-Kantor Triple Systems
	5 Concluding Remarks
	6 History from a Certain Personal Viewpoint
	References
Schreier\'s Type Formulae and Two Scales for Growth of Lie Algebras and Groups
	1 Analogue of Schreier\'s Formula for Free Lie Algebras
	2 Scale 1 and Growth of Free Solvable (Polynilpotent) Finitely Generated Lie Algebras and Groups
	3 Exponential Analogue of Schreier\'s Formula for Free Lie Algebras
	4 Scale 2 for the Codimension Growth of Lie PI-Algebras
		4.1 The Codimension Growth for Another Classes of Linear Algebras
	5 Explicit Formulae for Complexity Functions and Asymptotic for Lie PI-Algebras
	References
Leibniz Algebras
Universal Central Extensions  of Compatible Leibniz Algebras
	1 Introduction
	2 Preliminaries on Leibniz Algebras
	3 Compatible Leibniz Algebras
	4 Homology with Trivial Coefficients
	5 Universal Central Extensions
	References
On Some Properties of Generalized Lie-Derivations of Leibniz Algebras
	1 Introduction
	2 Preliminaries on Leibniz Algebras
	3 Generalized Lie-Derivations and Lie-Centroids of Low-Dimensional Leibniz Algebras
	4 Generalized Lie-Derivations of a Direct Sum
	References
Biderivations of Low-Dimensional Leibniz Algebras
	1 Introduction
	2 Leibniz Algebras
	3 Derivations, Anti-Derivations and Biderivations
	4 Biderivations of Low-Dimensional Leibniz Algebras
		4.1 Biderivations of Two-Dimensional Leibniz Algebras
		4.2 Biderivations of Three-Dimensional Leibniz Algebras
	References
Poisson Structure on the Invariants  of Pairs of Matrices
	1 Introduction
	2 Calogero-Moser Spaces
	3 Poisson Structure on the Invariants and Its Applications
		3.1 Kontsevich Leibniz Algebra and Its Liezation
		3.2 Description of the Ring of the Invariants
		3.3 Description of the Coordinate Ring of the Third Calogero-Moser Space
		3.4 Recursive Relation on the Coefficients of the Characteristic Polynomial
	References
Associative and Jordan Algebras and Related Structures
Automorphisms, Derivations and Gradings of the Split Quartic Cayley Algebra
	1 Introduction
	2 The Split Quartic Cayley Algebra
	3 Automorphisms and Derivations
	4 Gradings
	References
On a Theorem of Brauer-Cartan-Hua Type in Superalgebras
	1 Preliminaries
	2 The Main Result
	References
Growth Functions of Jordan Algebras
	1 Introduction
	2 Jordan Pairs and Tits-Kantor-Koecher Lie Algebras
	3  Special Jordan Algebras
	4 Exceptional Jordan Algebras
	References
The Image of Polynomials in One Variable on the Algebra of 3 times3 Upper Triangular Matrices
	1 Introduction
	2 Preliminaries
	3 The Image of Polynomials in One Variable on Upper Triangular Matrices
	4 From Upper Triangular to Full Matrices
		4.1 Case (I)
		4.2 Case (II)
		4.3 Case (III)
		4.4 Case (IV)
	5 Final Remarks
	6 Funding
	References
Other Non-associative Structures
Simultaneous Orthogonalization of Inner Products Over Arbitrary Fields
	1 Introduction
	2 Preliminaries and Basic Results
		2.1 Evolution Test Ideal
	3 The Nondegenerate Case
	4 The Degenerate Case
	References
Invariant Theory of Free Bicommutative Algebras
	1 Introduction
	2 Preliminaries
	3 The Finite Generation Problem
	4 Analogue of the Molien Formula
	5 The Chevalley-Shephard-Todd Theorem
	6 Symmetric Polynomials in Free Bicommutative Algebras
	References
An Approach to the Classification of Finite Semifields by Quantum Computing
	1 Introduction
	2 Preliminaries
	3 Quantum Preliminaries and Grover\'s Quantum Search
	4 Quantum Computational Search of Finite Semifields
		4.1 General Procedure
		4.2 Semifields of Order 8
		4.3 Description of Semifields of Order 16
		4.4 Estimation of Costs for the General Case, in Terms  of Quantum Gates
	5 Conclusions and Future Work
	References
On Ideals and Derived and Central Descending Series of n-ary Hom-Algebras
	1 Introduction
	2 Definitions and Properties of n-Hom-Lie Algebras
	3 Generalized Ideals
	4 Ideal Properties of Derived Series and Central Descending Series
	5 Some Examples in the Case of n-Hom-Lie Algebras
	6 Study of Derived Series and Central Descending Series of a Class of 3-dimensional Hom-Lie Algebras
	References
Okubo Algebras with Isotropic Norm
	1 Composition Algebras
	2 Classification of Okubo Algebras
		2.1 Characteristic not 3
		2.2 Characteristic 3
	3 Okubo Algebras Over Finite Fields are Split
	4 Special Gradings on Okubo Algebras with Isotropic Norm
	References
Author Index