## Casimir invariants

For some time I have wanted to learn about the concept of Casimir invariants and I was not very satisfied with the information I found. Now I have made new efforts to learn about this topic and I will record some of what I learned here. Let $g$ be a finite-dimensional Lie algebra and let $G$ be the corresponding connected and simply connected Lie group.The Lie algebra $g$ can be identified with $T_e G$, the tangent space to $G$ at the identity. There is a one-to-one correspondence between elements of this tangent space and left-invariant vector fields on $G$. Let $S(g)$ be the algebra of symmetric tensors over $g$. Let $U(g)$ denote the associative algebra which is the quotient of $S(g)$ by elements of the form $x\otimes y-y\otimes x-[x,y]$. This is called the universal enveloping algebra of $g$. There is a natural embedding $i$ of $g$ into $U(g)$ and it is a Lie algebra homomorphism into $L(U(g))$, the Lie algebra obtained from $U(g)$ by using the commutator to define a Lie bracket. Given an associative algebra $A$ and a Lie algebra homomorphism $\phi$ from $g$ to $L(A)$ there exists an algebra homomorphism $\psi$ from $U(g)$ to$A$ such that $\phi=\psi\circ i$. This is the universal property which appears in the name of the object. Here the fact has been used implicitly that $A$ and $L(A)$ can be identified as sets. An important example is given by a representation $\rho$ of the Lie algebra $g$ on a vector space $V$, which can be thought of a Lie algebra homomorphism from $g$ to $gl(V)=L(GL(V))$.

A Casimir invariant, or Casimir element or Casimir operator of $g$ is an element of the centre of $U(g)$. What remains unclear to me is whether these three concepts are supposed to be equivalent, or just related. I am also not sure whether (in any of these cases) any element of the centre is allowed, or just a particular one or a particular type. One definition I have found is the following. Suppose that $G$ semisimple. Then it has a Killing form $K$ which is a non-degenerate bilinear form. Let $X^i$ be a basis of $g$ and let $X_i$ be the basis of one-forms associated to it via $K$. (I.e. for any vector $Y$ we have $X_i(Y)=K(X^i,Y)$. Then the Casimir invariant is defined to be the element of $U(g)$ given by $C=\sum _i X^i X_i$. This is independent of the basis chosen. Since $K$ is an invariant bilinear form it follows that $C$ commutes with all elements of $g$ and in fact lies in the centre of $U(g)$. As a consequence of the universal property it is possible to define an object $\rho (C)$. This is an operator on $V$ which commutes with all elements of the image of $\rho$. If the representation is irreducible then this implies that $\rho(C)$ is a multiple of the identity. The factor of proportionality is a real number which is an invariant of the representation.

There is a theorem on the structure of the centre of the universal enveloping algebra of a semisimple Lie algebra which is associated with the term ‘Harish-Chandra homomorphism’. It can be used to list the number of elements required to generate the centre and their orders as polynomials in basis vectors of the Lie algebra. The number of these generators is the rank of the algebra. For instance for $SL(2,R)$ there is one generator of order two. For $SU(3)$ there are generators of order two and three and the rank is two. The same will be true for $SL(3,R)$ or $SU(2,1)$. My aim at the moment is not to learn the abstract theory in depth but rather to understand enough to do some calculations for a specific application. I plan to say more about the application in a later post.

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