2.9. Lie algebras 23 This provides a motivation for the notion of a Lie algebra. Namely, we see that Lie algebras arise as spaces of infinitesimal automorphisms (= derivations) of associative algebras. In fact, they similarly arise as spaces of derivations of any kind of linear algebraic structures, such as Lie algebras, Hopf algebras, etc., and for this reason play a very important role in algebra. Here are a few more concrete examples of Lie algebras: (1) R3 with [u, v] = u × v, the cross-product of u and v. (2) sl(n), the set of n × n matrices with trace 0. For example, sl(2) has the basis e = 0 1 0 0 , f = 0 0 1 0 , h = 1 0 0 −1 with relations [h, e] = 2e, [h, f] = −2f, [e, f] = h. (3) The Heisenberg Lie algebra H of matrices 0 ∗ ∗ 0 0 ∗ 0 0 0 . It has the basis x = ⎛ ⎝0 0 0 0 0 1⎠ 0 0 0 ⎞ , y = ⎛ ⎝0 0 1 0 0 0⎠ 0 0 0 ⎞ , c = ⎛ ⎝0 0 0 1 0 0⎠ 0 0 0 ⎞ with relations [y, x] = c and [y, c] = [x, c] = 0. (4) The algebra aff(1) of matrices ( ∗ ∗ 0 0 ). Its basis consists of X = ( 1 0 0 0 ) and Y = ( 0 1 0 0 ), with [X, Y ] = Y . (5) so(n), the space of skew-symmetric n × n matrices, with [a, b] = ab − ba. Exercise 2.9.5. Show that example (1) is a special case of example (5) (for n = 3). Definition 2.9.6. Let g1, g2 be Lie algebras. A homomorphism of Lie algebras ϕ : g1 −→ g2 is a linear map such that ϕ([a, b]) = [ϕ(a),ϕ(b)]. Definition 2.9.7. A representation of a Lie algebra g is a vector space V with a homomorphism of Lie algebras ρ : g −→ End V .

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