Identities of Algebras and their Representations

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Translations of Mathematical Monographs 138

S Title

Identities of Algebras and Their Representations

© Copyright 1994 by the American Mathematical Society

     ISBN 0-8218-4608-6

     QA252.3.R3913 1994 512' .55-dc20

     LCCN 94020766

CONTENTS

PREFACE

CHAPTER I  PRELINUNARY RESULTS

     §1. Associative-Lie pairs, identities of pairs, and varieties of pairs. Connections with varieties of representations of Lie algebras

          1.1. The category of associative-Lie pairs. Identities of pairs

          1.2. The category of representations of Lie algebras in associative algebras

          1.3. The category of representations of Lie algebras in linear spaces

          1.4. Connections with identities of Lie algebras and of associative algebras

     §2. Complexity of varieties of Lie algebras and their representations

          2.1. Complexity functions for varieties of associative-Lie pairs

          2.2. Varieties of exponential type

          2.3. Existence of varieties of large growth

     §3. Central closure for semiprime algebras

          3.1. Construction of a central closure

          3.2. Simplest properties of the central closure

          3.3. Centrally closed prime algebras, sufficient closedness conditions

     §4. Capelli identities and the rank theorem

     §5. Isomorphism of centrally prime associative-Lie pairs with the same identities over an algebraically closed field

     Comments

CHAPTER II  CHARACTERS AND a-FUNCTIONS ON 2-WORDS AND VARIETIES OF REPRESENTATIONS OF LIE ALGEBRAS DISTINGUISHED BY THEM

     §6. An important example

          6.1. The solvability problem for 3rd Engel Lie algebras over fields of characteristic 5

          6.2. Existence of nonsolvable (p - 2)th Engel Lie algebras over fields of characteristic p > 5.

          6.3. Identities of the two-dimensional irreducible representation of the simple three dimensional Lie algebra

     §7. Characters on 2-words and a-functions

     §8. The variety of pairs Ba defined by the character a

     §9. The construction of a-functions for any representation of a finite-dimensional Lie algebra possessing a nondegenerate invariant symmetric bilinear form

     §10. The correspondence between the ideals of weak identities and ideals of the commutative algebra defined by a multiplicative character on 2-words

     §11. A general approach and the setting of the problem of studying varieties of pairs by the method of 2-words

     Comments

CHAPTER III  a-FUNCTIONS RELATED TO THE KILLING FORM AND TO IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS. CENTRAL POLYNOMIALS OF IRREDUCIBLE REPRESENTATIONS OF REDUCTIVE LIE ALGEBRAS

     § 12. Statement of the main results of the chapter

     §13. Some remarks on the enveloping algebras of semisimple Lie ' algebras

     §14. Existence of central polynomials in simple enveloping algebras

     §15. Varieties of algebras with three supports var(K, g, U)

     § 16. An auxiliary algebra with three supports (Yo, Z1, F, ) and the extension of an a-function a to the space of generalized 2-elements

     §17. For finite-dimensional U, the identities of the pair (U, g) are determined by its a-function

     §18. Proof of Theorem 12.1

     § 19. For an arbitrary simple algebra U, the identities of the pair (U, g) are determined by its cr-function

     §20. Some consequences of Theorems 12.1 and 16.1

     §21. A construction of a polynomial mapping that recovers the algebra of commutative polynomials from the Lie algebra of all of its derivations

     Comments

CHAPTER IV  a-FUNCTIONS RELATED TO FULL MATRIX ALGEBRAS. TRACE IDENTITIES AND CENTRAL POLYNOMIALS OF FULL MATRIX ALGEBRAS M AND MATRIX SUPERALGEBRAS M,,,k

     §22. Main results of the chapter

          22.1. Main notation.

     §23. Calculation of the a-function a and the algebra £

     §24. An algebra of trace polynomials. Main notions

     §25. An auxiliary trace algebra

          25.1. The algebra 9,,.

          25.2. The algebra g(y).

          25.3. Extension of the domain of the cr-function a : B - K[y].

          25.4. A bilinear pairing b: G; R,,G; - K[y].

          25.5. The closure operations 0 and i& and their relationships with 0a and q,,,.

          25.6. Properties of the bilinear form bE: TI RTI -* K[y].

          25.7. Endowing of T1 with the structure of a group algebra

     §26. Classification of q,5-closed ideals and ideals of trace identities V such that v n T1 is a two-sided ideal in T1 over fields of characteristic zero

     §27. Description of trace identities in the full matrix algebras M and in the matrix superalgebras Mn,k

          27.1 The Hamilton-Cayley trace identity and trace identities of the algebra M

          27.2. Model algebras for ideals of trace identities VD

     §28. Three lemmas

          28.1. A ramification lemma

          28.2. Full matrix algebras over a field in the variety var M,,.

          28.3. Matrix superalgebras in the variety var Mn,k

     §29. c-dual sets in the algebra Mn,k

     §30. Trace identities of the superalgebra M,,.k

          30.1 The algebra Gn

          30.2. The proof of Theorem 27.2

     §31. Central polynomials in the algebras Mn and Mn,k

          31.1. Weak identities in the algebras M,,.k and an existence criterion of central polynomials for the algebras M,,,k.

          31.2. Existence of multilinear essentially weak identities for the algebras M,,.k over fields of positive characteristic

          31.3. The construction of polynomial c-dual sets and central polynomials for the algebra M,,.k over fields of positive characteristic

          31.4. Existence of multilinear essentially weak identities for the algebras M,,.k over fields of characteristic zero

          31.5. Construction of polynomial c-dual sets and central polynomials over fields of characteristic zero

     §32. A description of the lattice of q,,-closed ideals in K[y]

     §33. Consequences of the classification of B&-closed ideals of trace identities related to varieties of associative nil-algebras

          33.1. Varieties of nil-algebras over fields of characteristic zero.

          33.2. Examples of nonsolvable (p - 1)th Engel varieties of associative algebras over fields of characteristic p > 5.

     Comments

CHAPTER V  THE a-FUNCTION RELATED TO REPRESENTATIONS OF THE SIMPLE THREE-DIMENSIONAL LIE ALGEBRAAND ITS APPLICATIONS TO VARIETIES OF GROUPS AND ASSOCIATIVE ALGEBRAS

     §34. Preliminaries

          34.1. A summary of the results of the chapter

          34.2. Irreducible representations and primitive ideals of the algebra U(9).

     §35. Computation of the a-function a : B -* E and of the algebra \Epsilon

          35.1. Some identities of the algebra { K, g, U(g) } with three supports and of the pair (U(g), g

          35.2. The auxiliary algebra with three supports

          35.3. Auxiliary associative algebras A (y) and \Bar A (y)

          35.4. The bilinear pairing b : 'UA2 R ,A2 -> F

          35.5. q,,-closed ideals of the algebra \Epsilon.

     §36. A basis of identities for the Lie algebra g

     §37. Finite basis property for the identities of subvarieties of pairs in var( U(g), g) over fields of characteristic zero

          37.1. A minimality condition for subvarieties of var( U(g), g).

          37.2. Finite basis property for the identities of the pair (U(2), g).

     §38. Bases of identities for irreducible representations of the Lie algebra g

          38.1. Identities of the pair (U, g) in two variables

          38.2. The main lemma

          38.3. Certain identities in three variables can be remo

          38.4. The construction of a finite basis of identities for the pair (U, g).

     §39. Examples of minimal nonsolvable Engel varieties of pairs over fields of characteristic p > 5

     §40. Nonsolvability of varieties of locally finite groups of exponent 4 and of prime exponent p for p > 5

     §41. A basis of identities for the full matrix algebra of order two

     Comments

CHAPTER VI  VARIETIES GENERATED BY LIE ALGEBRAS OF CARTAN TYPE

     §42. Summary of results

     §43. Identities in Lie algebras of Cartan type

     §44. Embedding of algebras of Cartan type in W,, (K)

          44.1. Completed divided power algebra.

          44.2. A representation of a Lie algebra by derivations of the divided power algebra.

     §45. Recovery of the algebra of regular functions of a smooth irreducible affine variety from the Lie algebra of its vector fields

     §46. Simple Lie algebras satisfying the standard Lie identity of degree 5

          46.1. Some identities of the Lie algebra W, (K)

          46.2. Characteristic properties of simple algebras with identity (46.1).

          46.3. Relationships with simple differential algebras

          46.4. Division algebras in the variety var W

     §47. Criteria for existence of proper subalgebras of finite codimension in Lie algebras

     §48. Simple Lie algebras in varieties of exponential type

          48.1. Identities in varieties of exponential type.

          48.2. Criteria for a simple algebra to be of Cartan type in a variety 91 of Lie algebras given by identity (48.4

          48.3. The proof of Theorem 42.1.

          48.4. The proof of Theorem 42.2

     Comments

CHAPTER VII  ALGEBRAIC SUPPLEMENTS

     §49. Generalities of the theory of universal algebras and algebras with several supports

          49.1. Binary algebras

          49.2. Algebras of arbitrary signature

          49.3. Algebras with several supports

     §50. Partially ordered sets

          50.1. Basic concepts and Zorn's lemma

          50.2. The Dilworth theorem.

     §51. Theorems on homomorphisms of commutative algebras

          51.1. Basic notions

          51.2. Hilbert's Nullstellensatz

          51.3. An analog of Hilbert's Nullstellensatz for differential algebras

     §52. Universal enveloping algebras

          52.1. The Poincare-Birkhoff-Witt theorem.

          52.2. Shirshov basis in a free Lie algebra

     §53. Modules over associative algebras

          53.1. Basic notions.

          53.2. Injective. modules

     §54. Properties of enveloping algebras

          54.1. Centralizers of irreducible representations of finite-dimensional Lie algebras

          54.2. Noether property of enveloping algebras

HISTORICAL SURVEY

REFERENCES

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