# Tensor products of representations

math101
I'm a mathematician, and I have trouble understanding the physics notation. I'm glad if someone could help me out.

Here's my question:
Let g be a Lie algebra and r_1: g -> End(V_1), r_2: g -> End(V_2) be two representations.
Then there is a representation r3:=r_1 \otimes r_2: g -> End (V_1 \otimes V_2)
given by r3(x)(v_1 \otimes v_2):=r1(x)(v_1) \otimes v_2 + v_1\otimes r_2(x)(v_2),
which is called the tensor product of the representations.

An example is given by the vector-spinor representation of so(2k): Take the standard representation of so(2k) on V_1=R^{2k} and the spin-representation on V_2=R^{2^(k)}, then the tensor product is a representation on a 2k*2^k dimensional vector space V_1 \otimes V_2

Now, say in Palmkvist's Ph.D. thesis (http://front.math.ucdavis.edu/0912.1612), p. 41,
he tries to extend the vector-spinor representation of so(10) to k(E_10).
He writes

J^{abc} \psi^d=....
(on the bottom of p.41).

Here J^{abc} is an element of k(E_10) - but what is \psi^d? It must be a vector in V_1 \otimes V_2, but which one? Is it one of the form e_1 \otimes v_k, where (e_i) is the standard basis of R^10 and (v_k) is a basis of R^32?

-- If this is a too specialized question, I'd be happy if someone could me give a link to vector-spinor representations in physics, maybe I can learn enough from the literature to answer it then myself. Thank you!

## Answers and Replies

It's any one. $\psi^d$ is any vector in $V_1 \otimes V_2$. Note that there is a suppressed spinor index as well.