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Representations of Lie Algebras

Book description

This bold and refreshing approach to Lie algebras assumes only modest prerequisites (linear algebra up to the Jordan canonical form and a basic familiarity with groups and rings), yet it reaches a major result in representation theory: the highest-weight classification of irreducible modules of the general linear Lie algebra. The author's exposition is focused on this goal rather than aiming at the widest generality and emphasis is placed on explicit calculations with bases and matrices. The book begins with a motivating chapter explaining the context and relevance of Lie algebras and their representations and concludes with a guide to further reading. Numerous examples and exercises with full solutions are included. Based on the author's own introductory course on Lie algebras, this book has been thoroughly road-tested by advanced undergraduate and beginning graduate students and it is also suited to individual readers wanting an introduction to this important area of mathematics.

Reviews

'This short book develops the standard tools (special bases, duality, tensor decompositions, Killing forms, Casimir operators …) and aims for a single result: the classification of integral modules over gln. Requiring only basic linear algebra, this book can serve as an interesting alternative platform (to basic group theory) for introducing abstract algebra … Recommended.'

D. V. Feldman Source: Choice

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References
References
[1] N., Bourbaki, Lie Groups and Lie Algebras, 3 vols., Springer-Verlag, 1989 (Chapters 1–3), 2002 (Chapters 4–6), 2005 (Chapters 7–9).
[2] R.W., Carter, Lie Algebras of Finite and Affine Type, vol. 96 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2005.
[3] R. W., Carter, G., Segal, and I. G., Macdonald, Lectures on Lie Groups and Lie Algebras, vol. 32 of London Mathematical Society Student Texts, Cambridge University Press, 1995.
[4] K. E, Rdmann and M. J., Wildon, Introduction to Lie Algebras, Springer Undergraduate Mathematics Series, Springer-Verlag, 2006.
[5] W., Fulton, Young Tableaux, vol. 35 of London Mathematical Society Student Texts, Cambridge University Press, 1997.
[6] W., Fulton and J., Harris, Representation Theory: A First Course, vol. 129 of Graduate Texts in Mathematics, Springer-Verlag, 1991.
[7] R., Goodman and N. R., Wallach, Representations and Invariants of the Classical Groups, vol. 68 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1998.
[8] B. C., Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, vol. 222 of Graduate Texts in Mathematics, Springer-Verlag, 2003.
[9] J., Hong and S.–J., Kang, Introduction to Quantum Groups and Crystal Bases, vol.42 of Graduate Studies in Mathematics, American Mathematical Society, 2002.
[10] J. E., Humphreys, Introduction to Lie Algebras and Representation Theory, vol. 9 of Graduate Texts in Mathematics, Springer-Verlag, 1972.
[11] J. C., Jantzen, Lectures on Quantum Groups, vol. 6 of Graduate Studies in Mathematics, American Mathematical Society, 1996.
[12] V. G., Kac, Infinite-Dimensional Lie Algebras, 3rd edn, Cambridge University Press, 1990.
[13] A. W., Knapp, Lie Groups: Beyond an Introduction, 2nd edn, vol. 140 of Progress in Mathematics, Birkhäuser, 2002.
[14] I. G., Macdonald, Symmetric Functions and Hall Polynomials, 2nd edn, Oxford University Press, 1995.
[15] W., Rossmann, Lie Groups: An Introduction Through Linear Groups, vol. 5 of Oxford Graduate Texts in Mathematics, Oxford University Press, 2002.

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