Symmetry, Representations, and Invariants (Graduate Texts in by Roe Goodman, Nolan R. Wallach

By Roe Goodman, Nolan R. Wallach

Symmetry is a key aspect in lots of mathematical, actual, and organic theories. utilizing illustration thought and invariant idea to research the symmetries that come up from workforce activities, and with powerful emphasis at the geometry and easy conception of Lie teams and Lie algebras, Symmetry, Representations, and Invariants is an important remodeling of an previous highly-acclaimed paintings through the authors. the result's a entire creation to Lie conception, illustration idea, invariant thought, and algebraic teams, in a brand new presentation that's extra available to scholars and encompasses a broader diversity of applications.

The philosophy of the sooner e-book is retained, i.e., offering the central theorems of illustration idea for the classical matrix teams as motivation for the overall idea of reductive teams. The wealth of examples and dialogue prepares the reader for the whole arguments now given within the basic case.

Key positive factors of Symmetry, Representations, and Invariants:
• Early chapters appropriate for honors undergraduate or starting graduate classes, requiring basically linear algebra, simple summary algebra, and complex calculus
• functions to geometry (curvature tensors), topology (Jones polynomial through symmetry), and combinatorics (symmetric team and younger tableaux)
• Self-contained chapters, appendices, finished bibliography
• greater than 350 routines (most with specified tricks for recommendations) extra discover major concepts
• Serves as a superb major textual content for a one-year path in Lie workforce theory
• merits physicists in addition to mathematicians as a reference paintings

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6. There exist e > 0 and an analytic matrix-valued function R(X,Y ) on Be (0) ⇥ Be (0) such that ⇣ ⌘ 1 exp X expY = exp X +Y + [X,Y ] + R(X,Y ) 2 24 1 Lie Groups and Algebraic Groups when X,Y 2 Be (0). Furthermore, kR(X,Y )k  C(kXk + kY k)3 for some constant C and all X,Y 2 Be (0). 6 we now obtain the fundamental identities relating the Lie algebra structure of gl(n, R) to the group structure of GL(n, R). 7. 14) ⇣ 1 ⌘ X exp k 1 ⌘⌘k2 Y . 15) k Proof. 6 implies that exp ⇣1 ⌘ ⇣1 ⌘ ⇣1 ⌘ X exp Y = exp (X +Y ) + O(1/k2 ) , k k k where O(r) denotes a matrix function of r whose norm is bounded by Cr for some constant C (depending only on kXk + kY k) and all small r.

Identify Hn with C2n as a vector space over C by the map a + jb 7! ab , where a, b 2 Cn . Let T = A + jB 2 Mn (H) with A, B 2 Mn (C). (a) Show left multiplication by T on Hn corresponds to multiplication by the ⇥ Athat ¯⇤ B matrix B A¯ 2 M2n (C) on C2n . (b) Show that multiplication by i on Mn (H) becomes the transformation   A B¯ iA iB¯ 7! B A¯ iB iA¯ 11. 6) as an H-valued function on C2n ⇥ C2n . 2 The Classical Lie Algebras 13 (a) Show that B(x, y) = B0 (x, y) + jB1 (x, y), where B0 is a C-Hermitian form on C2n of signature (2p, 2q) and B1 is a nondegenerate skew-symmetric C-bilinear form on C2n .

Then H is a closed subgroup of GL(n, R), and Lie(H) = {X 2 Lie(G) : exp(tX) 2 H for all t 2 R} . Proof. It is obvious that H is a closed subgroup of GL(n, R). If X 2 Lie(H) then exp(tX) 2 H ⇢ G for all t 2 R. Thus X 2 Lie(G). t u We consider Lie(Sp(n, C)). Since Sp(n, C) ⇢ GL(2n, C) ⇢ GL(2n, R), we can look upon Lie(Sp(n, C)) as the set of X 2 M2n (C) such that exptX 2 Sp(n, C) for all t 2 R. This condition can be expressed as exp(tX t )J exp(tX) = J for all t 2 R . 19) Differentiating this equation at t = 0, we find that X t J + JX = 0 for all X 2 Lie(Sp(n, C)).

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