2 Lie algebrasLie groups such as GLn(R) are quite complicated nonlinear objects. A Liealgebra is a way of linearizing a Lie group, which is often easier to handle.Roughly speaking, the addition and Lie bracket of the Lie algebra are given bythe lowest order terms in the product and commutator of the Lie group. Bya minor miracle (the Campbell-Baker-Hausdorff formula) we do not need anyhigher order terms: the Lie algebra is enough to determine the group productlocally. We first recall some background about vector fields and differentialoperators on a manifold. We will then define the Lie algebra of a Lie group tobe the left invariant vector fields on the group.For any algebra over a ring we define the Lie bracket [a, b] to be ab − ba. Itsatisfies the identities• [a, b] is bilinear• [a, b] = −[b, a]• [[a, b], c] + [[b, c], a] + [[c, a], b] = 0 (Jacobi identity)Definition 19 A Lie algebra over a ring is a module with a bracket satisfyingthe conditions above, in other words it is bilinear, skey symmetric, and satisfiesthe Jacobi idenity.These conditions make sense in any additive tensor category, so for examplewe can define Lie algebras of sheaves, or graded Lie algebras. An interstingvariation is Lie superalgebras, where we use the tensor category of supermodulesover a ring or field. Some authors add the non-linear condition that [a, a] = 0.Example 20 The basic example of a Lie algebra is given by taking V to be anassociative algebra and defining [a, b] to be [ab − ba].The Lie algebra of a Lie group can be defined as its tangent space at theidentity, with the Lie bracket given by the lowest order part of the commutator.The lowest-order terms of the group law are just given by addition on the Liealgebra, as can be seen in GLn(R): the product of 1 + ǫA and 1 + ǫB is 1 + ǫ(A + B) to first order. Howeverdefining the Lie bracket in terms of the commutator is a little messy, and it istechnically more convenient to define the Lie algebra as the left invariant vectorfields on the manifold.There are several different ways to think of vector fields:• Informally, a vector field is a little tangent vector at each point.• A vector field is informally an infinitesimal diffeomorphism, where we getan infinitesimal diffeomorphism from a vector field by pushing each pointslightly in the direction of the vector field.• More formally, a vector field is a section of the tangent bundle or sheaf.• A vector field is a normalized differential operator of order at most 1• A vector field is a derivation of the ring of smooth functions.16The last two seem less intuitive but turn out to b e the easiest definitions towork with.Suppose we have a manifold M, with its ring R of smooth functions. Adifferential operator on M should be something that in local coordinates lookslike a partial differential operator times a smooth function. It is easier to forgetabout local coordinates, and just use the following key property of differentialoperators: the commutator of an nth order operator with a smooth function is adifferential operator of smaller order. This is really just a form of Leibniz’s rulefor differentiating a product. We will use this to DEFINE differential operatorsas follows.Definition 21 A differential operator of order less than 0 is 0. A differentialoperator of order at most n ≥ 0 is an operator on R whose commutator withelements of R is a differential operator of order at most n − 1.Differential operators on R form a filtered ring D0⊂ D1⊂ D2· · · , where Dnisthe differential operators of order at most n. The differential operators of orderat most 0 can be identified with the ring R (look at their action on 1), andany differential operator can be normalized by adding a function so that it kills1. So a differential operator can be written canonically as a function (order 0operator) plus a normalized differential operator.The product of differential operators of orders at most m, n has order at mostm + n. Differential operators do not quite commute with each other; howeverthe commutator or Lie bracket [D1, D2] of operators of orders at most m, n hasorder at most m + n − 1; in other words differential operators commute “up tolower order terms”. This means that the associated graded ring D0⊕ D1/D0⊕D2/D1⊕· · · is a commutative graded ring (whose elements are sometimes calledsymbols).We will call a differential operator normalized if it kills the function 1. Dif-ferential operators of order at most 1 can be written canonically as the sum of anorder 0 differential operator and a normalized differential operator. (Howeverthere is no canonical way to write an operator of order n > 1 as an operator oforder less than n and something “homogeneous” of order n.) A vector field ona manifold is the same as a normalized differential operator of order at most 1.Vector fields are closed under the Lie bracket, and in particular form a Lie alge-bra. It is useful to think of a vector field as a sort of infinitesimal diffeomorphismof the manifold: each point is moved an infinitesimal distance in the directionof the vector at that point. Since the Lie algebra of a group can be thought ofas the “infinitesimal” elements of the group, this means that the vector fieldson a manifold are more or less the Lie algebra of the group of diffeomorphisms.The Lie algebra of vector fields is an infinite dimensional Lie algebra, whichis too big for this course, so we cut it down.Definition 22 The Lie algebra of a Lie group is the Lie algebra of left-invariantvector fields on the group.We explain what this means. The group is a manifold, so we have the Liealgebra of all vector fields on it forming an infinite dimensional lie algebra. Thegroup acts on itself by left translation, and so acts on everything constructedfrom the manifold, such as vector fields. We just take the vector fields fixed by17this action of left translation. It is automatically a subalgebra of the lie algebraof all vector fields, as the group action preserves the Lie bracket.We can also identify the Lie algebra of the group with the tangent space atthe origin. The reason is that if we pick a tangent vector at the origin, thereis a unique vector field on G given by left translating this vector everywhere.We could have defined the Lie algebra to be the tangent space at the origin,but then it would not have been so clear how (or why) we can define the Liebracket.Now we will calculate the left invariant vector fields