In mathematics [4, 5], a collection, or a set of elements is often studied by associating it with a certain operation. This is the primary notion of group theory that distinguishes it from set theory.
Definition 1.A group is defined as a setGalong with a binary operation◦ such that all the following conditions hold:
1. Closure: For anya, b∈G,a◦b=c∈G;
2. Associativity: For everya, b, c∈G, (a◦b)◦c=a◦(b◦c);
3. Identity: There is an identity elementι∈Gsuch thatι◦g=g◦ι=g for allg∈G;
4. Inverse: For eachg ∈G, there exists an elementh∈Gsuch thatg◦h= h◦g=ι.
All the real integers associated with addition form an additive group, but not under multiplication because of violating the inverse condition. All the real (complex) numbers under either addition or multiplication constitute an additive or a multiplicative group, respectively, and they are qualified to form a field, called the real field (complex field). If a collection along with a certain operation satisfies every condition but the inverse, even though the identity condition can still hold, then, the collection just forms a semigroup.
In contrast, the set of all 3D real vectors in R3 under the cross product is
neither a group, nor a semigroup, because it violates the associativity, identity and inverse conditions.
A group can be classified into finite and infinite groups, and discrete and continuous groups, depending on the number of elements and the property of element variations. A Lie group is a typical infinite and continuous group. As examples, all the non-singularnbynreal matrices with multiplication form a general linear Lie groupGL(n). All thenbynreal orthogonal matrices with multiplication form an orthogonal Lie groupO(n), and by further imposing the determinant of each member of O(n) to be +1, it becomes a special orthogonal groupSO(n). Every rotation matrix belongs to theSO(3) group.
However, many useful sets under certain binary operations violate either one or more conditions of the group definition, and they cannot be a group, though they are quite useful. In order to keep them for further study and application, we have to relax the conditions. Lie algebra is one of the most typical and important approaches to rescuing the useful sets that have been ruled out by the group definition.
Definition 2.A Lie algebra over the real field R or the complex field Cis a vector space G, in which a bilinear map (X, Y) →[X, Y] is defined from G × G → G such that
1. [X, Y] =−[Y, X], for allX, Y ∈ G, and
2. [X,[Y, Z]] + [Y,[Z, X]] + [Z,[X, Y]] = 0 for allX, Y, Z∈ G.
The second equation in the above definition is called the Jacobi Identity.
Now, all the 3-dimensional real vectors along with the cross product operation form a Lie algebra, although they cannot be a Lie group. Since fora×b = [a, b] = c, one can rewrite it as S(a)b =c, whereS(a) is the 3 by 3 skew- symmetric matrix of the vectora∈R3. In other words, [a,ã] =S(a)ã=aìis treated as an operator. Such a Lie algebra is often denoted asso(3) in lower case. For example, if two vectors
a=
⎛
⎝ 3 2
−1
⎞
⎠ and b=
⎛
⎝−1 0 2
⎞
⎠
are given, to find c= [a, b] =a×b, let us first construct a skew-symmetric matrixS(a) for the vector aby
S(a) =a×=
⎛
⎝ 0 1 2
−1 0 −3
−2 3 0
⎞
⎠∈so(3). (2.3)
Then, it is easily seen that
c= [a, b] =a×b=S(a)b=
⎛
⎝ 4
−5 2
⎞
⎠,
which agrees with the result using the conventional determinant method in calculus and physics as follows:
c=a×b=
i j k
3 2 −1
−1 0 2
= 4i−5j+ 2k.
More typical examples of Lie algebra include the matrix commutations and vector field derivatives. All the n byn square real matrices along with the commutator [A, B] = AB−BA constitute a Lie algebra. One can also verify that all then-dimensional real vector fields, each of which is a smooth function of pointx∈Rn under the following Lie bracket
[f, g] = ∂g
∂xf−∂f
∂xg (2.4)
also constitute a Lie algebra.
As an interesting property, let us look at two vectorsaandband their skew- symmetric matricesS(a) =a×andS(b) =b×as the cross-product operators ofaandb, respectively. We wish to know what the commutator [S(a), S(b)] = S(a)S(b)−S(b)S(a) is supposed to be. First, by taking transpose on both sides, we find that the result of the skew-symmetric matrices commutator [S(a), S(b)] is still a skew-symmetric matrix, and thus, satisfies the closure condition. Letv∈R3 be an arbitrary vector. Then,
[S(a), S(b)]v=S(a)S(b)v−S(b)S(a)v=a×(b×v)−b×(a×v), i.e., the difference between two triple vector products. By applying the triple vector product formula in (2.1), we obtain
[S(a), S(b)]v= (aTv)b−(aTb)v−(bTv)a+ (bTa)v
= (aTv)b−(bTv)a= (a×b)×v.
Sincev is arbitrary, the above equation implies that
[S(a), S(b)] = (a×b)×=S(a×b), (2.5) i.e. the cross-product operator ofa×b.
Let us now continue the previous numerical example, i.e., S(a) has been found in (2.3), while
S(b) =b×=
⎛
⎝0 −2 0
2 0 1
0 −1 0
⎞
⎠.
Thus,
[S(a), S(b)] =S(a)S(b)−S(b)S(a)
=
⎛
⎝2 −2 1
0 5 0
6 4 3
⎞
⎠−
⎛
⎝ 2 0 6
−2 5 4
1 0 3
⎞
⎠=
⎛
⎝0 −2 −5
2 0 −4
5 4 0
⎞
⎠,
which exactly meetsS(c) =c×=S(a×b).
In mathematical history, one of the most significant discoveries in the the- ory of Lie group and Lie algebra is the following exponential mapping:
EXP : so(3)−→SO(3). (2.6)
This mapping means that for each 3 by 3 skew-symmetric matrixS∈so(3), its exponential function exp(S) =R∈SO(3) is always a rotation matrix. In more general words, the exponential mapping can convert a Lie algebra to a Lie group in any finite dimension. This mapping property will be very useful and underlie the robotic kinematics to seek a new representation for rotation and orientation of coordinate frames [4, 6, 8].