Hardcover ISBN:  9781470437114 
Product Code:  SURV/226 
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Product Code:  SURV/226.E 
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Hardcover ISBN:  9781470437114 
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Product Code:  SURV/226.B 
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Hardcover ISBN:  9781470437114 
Product Code:  SURV/226 
List Price:  $169.00 
MAA Member Price:  $152.10 
AMS Member Price:  $135.20 
eBook ISBN:  9781470442545 
Product Code:  SURV/226.E 
List Price:  $159.00 
MAA Member Price:  $143.10 
AMS Member Price:  $127.20 
Hardcover ISBN:  9781470437114 
eBook ISBN:  9781470442545 
Product Code:  SURV/226.B 
List Price:  $328.00 $248.50 
MAA Member Price:  $295.20 $223.65 
AMS Member Price:  $262.40 $198.80 

Book DetailsMathematical Surveys and MonographsVolume: 226; 2017; 611 ppMSC: Primary 05; 06; 16; 17; 20; 52
This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material.
Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples.
Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic halfspaces which generalize the classical Dynkin idempotent. Another important object is the luneincidence algebra which marks the beginning of noncommutative Möbius theory. These ideas are also brought upon the study of the Solomon descent algebra.
The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.
ReadershipGraduate students and researchers interested in hyperplane arrangements (of interest in several areas of mathematics).

Table of Contents

Part I

Hyperplane arrangements

Cones

Lunes

Category of lunes

Reflection arrangements

Braid arrangement and related examples

Descent and lune equations

Distance functions and Varchenko matrix

Part II

Birkhoff algebra and Tits algebra

Lie and Zie elements

Eulerian idempotents

Diagonalizability and characteristic elements

Loewy series and Peirce decompositions

Dynkin idempotents

Incidence algebras

Invariant Birkhoff algebra and invariant Tits algebra

Appendices

Regular cell complexes

Posets

Incidence algebras of posets

Algebras and modules

Bands

References


Additional Material

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This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material.
Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples.
Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic halfspaces which generalize the classical Dynkin idempotent. Another important object is the luneincidence algebra which marks the beginning of noncommutative Möbius theory. These ideas are also brought upon the study of the Solomon descent algebra.
The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.
Graduate students and researchers interested in hyperplane arrangements (of interest in several areas of mathematics).

Part I

Hyperplane arrangements

Cones

Lunes

Category of lunes

Reflection arrangements

Braid arrangement and related examples

Descent and lune equations

Distance functions and Varchenko matrix

Part II

Birkhoff algebra and Tits algebra

Lie and Zie elements

Eulerian idempotents

Diagonalizability and characteristic elements

Loewy series and Peirce decompositions

Dynkin idempotents

Incidence algebras

Invariant Birkhoff algebra and invariant Tits algebra

Appendices

Regular cell complexes

Posets

Incidence algebras of posets

Algebras and modules

Bands

References