Classical Lie Algebras at Infinity

“The book is self-contained in an introductory way, it would be better for the reader to know the basics of Lie algebras and their representations. … the book is most suitable for advanced graduate students and research mathematicians.” (Chengkang Xu, Mathematical Reviews, October, 2022)“Classical Lie Algebras at Infinity is a very interesting book by I. Penkov and C. Hoyt. … Plenty of exercises of various levels of difficulty and open problems are given throughout the book. This exposition is an invitation to a wide open area of research, aiming to invoke a complex combination of advanced mathematical ideas.” (Mee Seong Im, zbMATH 1490.17025, 2022)“The book is designed to not only serve as a reference but also to introduce well-prepared people to the subject and its current areas of inquiry. … it seems fair to say that this book will appeal to a somewhat narrower subset of readers of this column than do many of the books reviewed here. But the people in this subset should find this a useful, in the authors’ words, ‘invitation to a wide open research area’.” (Mark Hunacek, MAA Reviews, August 1, 2022)

• Preface.-  Notation and Terminology. - I. Structure of Locally Reductive Lie Algebras.- 1. Finite-dimensional Lie algebras.- 2. Finite-dimensional Lie superalgebras.- 3. Root-reductive Lie algebras.- 4. Two generalizations.- 5. Splitting Borel subalgebras of sl(infinity), frak o (infinity), sp(infinity) and generalized flags.- 6. General  Cartan, Borel and parabolic subalgebras of gl(infinity) and sl(infinity).- II. Modules over Locally Reductive Lie Algebras.- 7. Tensor modules of sl(infinity), frak o(infinity), sp (infinity).- 8. Weight modules.- 9.Generalized Harish-Chandra modules.-  III. Geometric aspects. - 10.The Bott-Borel-Weil Theorem.- References.- Index of Notation.- Index.