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Introduction to Finite and Infinite Dimensional Lie (Super)algebras

N. Sthanumoorthy
Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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When I was in graduate school, my office mate, the late Erek Behr, God rest his soul, was studying Lie superalgebras as part of his doctoral work. I, on the other hand, was doing number theory for my PhD, specifically modular forms and representation theory, and so generally regarded what Erek was doing as strict algebra proper and not likely to enter my orbit at any time — sooner or later. Well, how wrong I was, and how right my undergraduate professor, V. S. Varadarajan, was. But I am getting ahead of myself, as Bertie Wooster would say, so let me explain.

A few years earlier, during my undergraduate days, Varadarajan had raised me to believe that physics and number theory are, in his words, two sides of the same coin. Varadarajan’s own wonderful career has borne this out, given its connection with quantum theory as well as themes in number theory (see especially the bibliography of the Wikipedia article). But I was young and correspondingly shortsighted, and only occasionally paid lip-service to his sage observation. It wasn’t till decades later that I came to see how on target Varadarajan was: now my own work on number theory is hugely steeped in themes coming from physics. And most recently, in my reading of the titanic two-volume set Quantum Fields and Strings: A Course for Mathematicians (well, as the song goes, “We’ve only just begun …”), with Deligne and Ed Witten among its editors, I found that, yes, super algebras are prominently featured. And it’s once again Varadarajan who gets the last word: see, e.g., his book, Supersymmetry for Mathematicians.

The book under review manifestly focuses on the stuff Erek was doing back in the 1980s, and a lot of other things, of course: it’s a pretty big book. It features Lie superalgebras both of finite and of infinite dimension, Kac-Moody (and Borcherds-Kac-Moody) superalgebras, and spreads this material out over almost 500 pages. It’s accordingly rather an encyclopedic business, at the same time sharply focused, and this requires a sophisticated reading audience. The author does start with foundational material, but beware: it is in an opening interval of only 60 pages that he gets through the whole story of finite dimensional Lie algebras. The book begins with a sprint.

Then there are, again in pretty short order, two meaty and dense chapters on Kac-Moody algebras — for a compact definition, see, followed by proper super-stuff, if you’ll pardon the expression. Well, then what makes algebra superalgebra? It’s pretty much a doubling of things, i.e. a Z2-grading: see p. 204 of the book under review. The idea is that algebraic structures like algebras (or Lie algebras, specifically) split up into a direct sum (graded) of an “even” and an “odd” part (with indices accordingly given), replete with the operations respecting the grading. Again, Wikipedia (which has all the makings of becoming a vice, no?): . And the bulk of Sthanumoorthy’s book deals with these super themes. Physics is featured by way of a climactic finish to the sixth and last chapter of the book: it is the case that the fermionic/bosonic duality central to contemporary particle physics fits beautifully into this philosophy: see p. 460. It truly is remarkable stuff.

Again, it’s a dense book, not meant for neophytes, notwithstanding its many exercises. It is a very good book, I think, and remarkable for its sweep. But be prepared for a lot of work, and possible outside reading: Sthanumoorthy obviously sets a clip pace.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

1. Finite dimensional Lie Algebras

1.1 Basic definition of Lie algebras with examples and structure constants

1.2. Subalgebras of Lie algebras and different classes of subalgebras of gl(n;C)

1.3. Ideals, Quotient Lie algebras, derived sub Lie algebras and direct sum

1.4. Simple Lie algebras, Semisimple Lie algebras, Solvable and Nilpotent Lie algebras

1.5. Isomorphism theorems, Killing form and some basic theorems

1.6. Derivation of Lie algebras

1.7. Representations of Lie algebras and Representations of sl(2;C)

1.8. Rootspace decomposition of semisimple Lie algebras

1.9. Root system in Euclidean spaces and Root diagrams

1.10. Coxeter graphs and Dynkin Diagrams

1.11. Cartan Matrices, Ranks and dimensions of simple Lie algebras

1.12. Weyl groups and Structure of Weyl groups of simple Lie algebras

1.13. Root systems of Classical simple Lie algebras and Highest long and short roots

1.14. Universal enveloping algebras of Lie algebras

1.15. Representation theory of semisimple Lie algebras

1.16. Construction of semisimple Lie algebras by Generators and Relations

1.17. Cartan-Weyl basis

1.18. Character of a finite dimensional representation and Weyl dimension formula

1.19. Lie algebras of vector fields


2. Kac - Moody algebras

2.1. Basic concepts in Kac-Moody algebras

2.2. Classification of Finite, Affine, Hyperbolic and Extended-hyperbolic Kac-Moody algebras and their Dynkin diagrams

2.3. Invariant bilinear forms

2.4. Coxeter Groups and Weyl Groups

2.5. Real and imaginary roots of Kac-Moody algebras

2.6. Weyl Groups of affine Lie algebras

2.7. Realization of Affine Lie algebras

2.8. Different classes of imaginary roots(special imaginary roots, strictly imaginary roots, purely imaginary roots) in Kac-Moody algebras

2.9. Representations of Kac-Moody algebras, integrable Highest Weight Modules, Verma modules and Character formulas

2.10. Graded Lie algebras and Root multiplicities


3. Generalized Kac-Moody algebras

3.1. Borcherds Cartan Matrix(BCM), Generalized Generalized Cartan Matrix (GGCM), Borcherds Kac-Moody(BKM)algebras and Generalized Kac-Moody (GKM) algebras

3.2. Dynkin Diagrams of GKM algebras

3.3. Root systems and Weyl groups of GKM algebras

3.4. Special imaginary roots in GKM algebras and their complete classifications

3.5. Strictly imaginary roots in GKM algebras and their complete classifications

3.6. Purely imaginary roots in GKM algebras and their complete classifications

3.7. Representations of GKM algebras

3.8. Homology modules and root multiplicities in GKM algebras


4. Lie superalgebras

4.1. Basic concepts in Lie superalgebras with examples

1.2. Coloring matrices, ??colored Lie superalgebras and examples

1.3. Subsuperalgebras, ideals of Lie superalgebras, abelian Lie superalgebras, solvable and nilpotent Lie superalgebras

4.4. General linear Lie superalgebras

4.5. Simple and semisimple Lie superalgebras and bilinear forms

4.6. Representations of Lie superalgebras

4.7. Different classes of classical Lie superalgebras

4.8. Universal enveloping algebras of Lie superalgebras and ??colored Lie superalgebras

4.9. Cartan subalgebras and root systems of Lie superalgebras

4.10. Killing forms on Lie superalgebras

4.11. Dynkin diagrams of Lie superalgebras

4.12. Lie superalgebras over an algebraically closed field of characteristic zero

4.13. Classification of non-classical Lie superalgebras

4.14. Lie superalgebras of vector fields


5. Borcherds Kac-Moody Lie superalgebras

5.1. BKM supermatrices and BKM Lie superalgebras

5.2. Dynkin diagrams of BKM Lie superalgebras and in paticular Kac Moody Lie superalgebras and some examples

5.3. Domestic type and Alien type imaginary roots in BKM Lie superalgebras

5.4. Special imaginary roots in BKM Lie superalgebras and their complete classifications

5.5. Strictly and Purely imaginary roots in BKM Lie superalgebras and complete classification of BKM Lie superalgebras possessing purely imaginary roots

5.6. BKM Lie superalgebras possessing purely imaginary property but not strictly imaginary property

5.7. Complete classification of BKM Lie superalgebras possessing strictly imaginary property

5.8. Borcherds superalgebras and root supermultiplicities

5.9. Root supermultiplicities of Borcherds superalgebras which are extensions of Kac-Moody Algebras and some combinatorial identities

5.10. Description of all finite and infinite dimensional Lie algebras and Lie superalgebras:


6. Lie algebras of Lie groups, Kac-Moody groups, supergroups and some specialized topics in finite and infinite dimensional Lie algebras

6.1. Lie groups and Lie algebras of Lie groups

6.2. Kac-Moody groups, supergroups and some applications

6.3. Homogeneous spaces, corresponding Lie algebras and spectra of some differential operators on homogeneous spaces

6.4. Spectral invariants of zeta function of the Laplace Beltrami operator

6.5. Generalization of McDonald’s identities for some Kac-Moody algebras

6.6. Some special infinite dimensional Lie algebras

6.7. Hirota bilinear differential operators and soliton solutions for KdV equation

6.8. Principal vertex operator construction of basic representation and homogeneous vertex operator construction of the basic representation

6.9. Principal vertex operators for A(1) 2 ; A(2) 2 ;C(1) 2 and B(1) 3

6.10. Fermionic Fock space, Clifford algebra, Bosonic Fock space and Boson-Fermion correspondence