Polynomial identities in algebras

Description

This volume contains the proceedings of the INDAM workshop on “Polynomial Identities in Algebras” held in Roma from September 16 to September 20, 2019. The purpose of the workshop was to present the current state of the art in the theory of PI-algebras. Preface Contents About the Editors Some Thoughts on the Current State of the Theory of Identical Relations in Lie Algebras 1 Introduction 2 Finite Basis Problem 3 Engel Lie Algebras 3.1 Global Nilpotence 3.2 Restricted Burnside Problem: Local Nilpotence of Engel Lie algebras 4 Identities of Simple Lie Algebras 4.1 The Isomorphism Problem 4.2 Identities of Cartan Type Lie Algebras 4.3 Capelli Identities 5 Codimension Growth 5.1 Exponential Growth 5.2 Overexponential Growth 5.3 Colength 6 Graded Lie Algebras and Identities 7 Graded Identities of Lie Algebras and Generalizations 7.1 Codimension Growth 7.2 Isomorphism of H-Algebras 8 Special Lie Algebras References Minimal Degree of Identities of Matrix Algebras with Additional Structures 1 Introduction 2 Involution Case 3 Graded Case 4 Graded Involution Case References On the Asymptotics of Capelli Polynomials 1 Introduction 2 Ordinary Case 3 Z2-Graded Case 4 Involution Case 4.1 -Capelli Polynomials and the -Algebra UT(A1, …, An) 4.2 Asymptotics for -Capelli Polynomials References Regev's Conjecture for Algebras with Hopf Actions 1 Introduction 2 Regev's Conjectures for Algebras Without Actions 2.1 Background on Magnums 2.2 Codimension Sequences 3 H-Cocharacters 4 Rationality of Poincaré Series 5 Young Derived Sequences References -Weak Identities and Central Polynomials for Matrices 1 Introduction 2 Preliminaries 2.1 Identities, Central Polynomials and Examples 2.2 Spechtian Polynomials 3 Weak Identities 3.1 Weak and Strong Variables 3.2 Modules of Weak Identities 4 Central Polynomials for Matrices 4.1 -Weak Central Polynomials 5 The Connection to the Representation Theory 5.1 Identities and the Group Algebra 5.2 Identities and Representations 6 Weak Identities and the Case n = 2 6.1 Polynomials of Degree m = 2 6.2 Weak Identities of Degree m = 3 6.3 Weak Identities of Degree m = 4 7 The Weak PI-Degree of M3(F) 7.1 Fields of Characteristic Not 3 7.2 The Case char F=3 8 Weak Identities for M3(F) in Degree 6 8.1 Weak Identities of M3(F) 8.2 Halpin's Identity and Its Projections 8.3 Identities from the Okubo Algebra 9 Matrices of Size n≥4 9.1 Shadows of Identities 9.2 Weak Identities Degree 2n References Computing Multiplicities in the Sign Trace Cocharacters of M2,1(F) 1 Introduction 2 Reduced Notation 3 The Sign Trace Cocharacter of M2,1(F) 3.1 Computation of the Coefficient for {n-2,2} 4 The Final Result References b-Generalized Skew Derivations on Multilinear Polynomials in Prime Rings 1 Introduction 2 Some Results on Differential Identities with Automorphisms 3 Commuting Generalized Derivations and Commuting Generalized Skew Derivations 4 Some Remarks on Matrix Algebras 5 Commuting Inner b-Generalized Skew Derivations 6 Commuting b-Generalized Derivations on Multilinear Polynomials 7 The Main Result 7.1 Let d and δ be C-Linearly Independent Modulo SDint 7.2 Let d and δ be C-Linearly Dependent Modulo SDint 8 Some Open Problems References Relatively Free Algebras of Finite Rank 1 Introduction 2 Models for Relatively Free Algebras 3 General Remarks 4 Polynomial Identities for UT2(G(k)) 4.1 The Case UT2(G(2m)), for 1≤m≤2 5 Polynomial Identities for Fk(UT2(G)) 6 Conclusion References Graded Algebras, Algebraic Functions, Planar Trees, and Elliptic Integrals 1 Introduction 2 Growth of Algebras and Hilbert Series 3 Algebras with Prescribed Hilbert Series 4 PI-Algebras 5 Algebraic and Transcendental Power Series 6 Planar Rooted Trees and Algebraic Series 7 Noncommutative Invariant Theory References Central Polynomials of Algebras and Their Growth 1 Introduction 2 A General Setting 3 Examples of Central Polynomials 4 Algebras Without Proper Central Polynomials 5 The Proper Central Exponent 6 The Central Exponent 7 Non Associative Algebras References Trace Identities on Diagonal Matrix Algebras 1 Introduction 2 Preliminaries 3 Matrix Algebras with Trace 4 Some Results on Dn 5 Trace Identities on D2 6 Trace Identities on D3 References Codimension Growth for Weak Polynomial Identities, and Non-integrality of the PI Exponent 1 Introduction 2 Preliminaries 2.1 Generalities 2.2 Polynomial Identities 2.3 Multilinear Polynomials. Modules over the Symmetric Group 2.4 The Exponent 2.5 The Action of the General Linear Group 2.6 Special Pairs and Pairs of Associative Type 3 Polynomial Growth of the Codimensions 3.1 Slow Growth of the Codimensions 3.2 Characterizing Varieties of Pairs of Polynomial Growth 3.3 Almost Polynomial Growth 3.3.1 The Pair (UT2,UT2(-)) 3.3.2 The Pair (E,E(-)) 3.3.3 The Pair (M2,sl2) 3.3.4 More Examples 4 Graded Pairs and Amitsur's Conjecture 4.1 Weak Graded Polynomial Identities 4.2 Finitely Generated Superpairs 4.3 Non-integral Exponent: An Example References On Codimensions of Algebras with Involution 1 Introduction 2 *-Codimensions and *-Fundamental Algebras 3 Low Exponential Growth References Context-Free Languages and Associative Algebras with Algebraic Hilbert Series 1 Introduction 2 Preliminaries 2.1 Associative Algebras and Their Hilbert Series 2.2 The Homology and Hilbert Series of Monomial Algebras 2.3 Automaton Algebras and Homologically Unambiguous Algebras 3 Finitely Presented Algebras Associated to Context-Free Languages 3.1 The General Construction 3.2 Graded Algebras Examples References On Almost Nilpotent Varieties of Linear Algebras 1 Introduction 2 Almost Nilpotent Varieties Generated by One or Two Dimensional Algebra 3 Almost Nilpotent Varieties and Skew Symmetric Polynomials 4 A Commutative Metabelian Algebra with Skew Symmetric Polynomials (a Jordan Algebra) 5 Anti-Commutative Metabelian Algebra with Skew Symmetric Polynomials 6 A Characterization of Almost Nilpotent Varieties in Different Classes of Algebras 7 An Infinite Series of Almost Nilpotent Metabelian Varieties with Polynomial Growth 8 Almost Nilpotent Varieties with Exponentional Growth References (δ,)-Differential Identities of UTm(F) 1 Introduction 2 The Problem 3 The Coupled Actions of δ and on UTm(F) 4 The Separate Actions of δ and on UTm(F) 5 Behind the Scenes References Identities in Group Rings, Enveloping Algebras and Poisson Algebras 1 Introduction 2 Identical Relations of Group Rings 3 Identical Relations of Enveloping Algebras 4 Poisson Algebras and Their Identities 4.1 Poisson Algebras 4.2 Examples of Poisson Algebras 4.3 Poisson Identities 5 Multilinear Identities of Symmetric Poisson Algebras 6 Lie Identities of Symmetric Poisson Algebras 6.1 Lie Nilpotence of Truncated Symmetric Algebras s(L) 6.2 Solvability of Truncated Symmetric Algebras s(L) 6.3 Nilpotency and Solvability of Symmetric Algebras S(L) 6.4 Delta-Sets and Multilinear Poisson Identical Relations 6.5 Products of Commutators in Poisson Algebras 6.6 Solvability of Symmetric Algebras s(L) and S(L) in Case Char K = 2 References Notes on the History of Identities on Group (and Loop) Algebras 1 Introduction 2 Existence of Polynomial Identities 3 Group Theoretical Properties of Unit Groups 4 Group Indentities 5 Loop Algebras References Cayley Hamilton Algebras 1 Introduction 2 The Cayley–Hamilton Identity 3 The First and Second Fundamental Theorem 3.1 The Free Trace Algebra 3.3 A Characteristic Free Approach 3.12 Symbolic Approach 3.12.1 The Approach of Zieplies and Vaccarino References Growth of Differential Identities 1 Introduction 2 L-Algebras and Differential Identities 3 On Algebras with Derivations of Polynomial Growth 4 The Algebra of 2 x 2 Upper Triangular Matrices and Its Differential Identities 5 On Differential Identities of the Grassmann Algebra References Derived Lengths of Symmetric Poisson Algebras 1 Introduction 2 Definitions and Notation 3 The Lie Structure of Enveloping Algebras 4 Symmetric Poisson Algebras Satisfying a Poisson Identity 5 Lie Nilpotence and Solvability of S(L) and s(L) 6 Derived Lengths of s(L) References Group and Polynomial Identities in Group Rings 1 Introduction 2 -Group Identities for U(FG) 3 Group Identities for Unitary Units of FG References