24 2. Basic notions of representation theory Example 2.9.8. Some examples of representations of Lie algebras are: (1) V = 0. (2) Any vector space V with ρ = 0 (the trivial representation). (3) The adjoint representation V = g with ρ(a)(b) := [a, b]. That this is a representation follows from equation (2.9.1). Thus, the meaning of the Jacobi identity is that it is equiv- alent to the existence of the adjoint representation. It turns out that a representation of a Lie algebra g is the same thing as a representation of a certain associative algebra U(g). Thus, as with quivers, we can view the theory of representations of Lie alge- bras as a part of the theory of representations of associative algebras. Definition 2.9.9. Let g be a Lie algebra with basis xi and [ , ] defined by [xi,xj] = k ck ij xk. The universal enveloping algebra U(g) is the associative algebra generated by the xi’s with the defining relations xixj xjxi = k ck ij xk. Remark 2.9.10. This is not a very good definition since it depends on the choice of a basis. Later we will give an equivalent definition which will be basis-independent. Exercise 2.9.11. Explain why a representation of a Lie algebra is the same thing as a representation of its universal enveloping algebra. Example 2.9.12. The associative algebra U(sl(2)) is the algebra generated by e, f, h, with relations he eh = 2e, hf fh = −2f, ef fe = h. Example 2.9.13. The algebra U(H), where H is the Heisenberg Lie algebra, is the algebra generated by x, y, c with the relations yx xy = c, yc cy = 0, xc cx = 0. Note that the Weyl algebra is the quotient of U(H) by the relation c = 1. Remark 2.9.14. Lie algebras were introduced by Sophus Lie (see Section 2.10) as an infinitesimal version of Lie groups (in early texts
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