# Tensor products of representations

• math101
In summary, the conversation discusses the concept of tensor product representations in mathematics, specifically in the context of Lie algebra and spinor representations. The example of a vector-spinor representation of so(2k) is given, and the conversation then moves on to discussing the extension of this representation to k(E_10) in Palmkvist's Ph.D. thesis. A question is raised about the notation used in the thesis, particularly about the vector \psi^d and its relation to the tensor product representation. The conversation ends with a request for further reading on vector-spinor representations in physics.

#### 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!

math101 said:
what is \psi^d? It must be a vector in V_1 \otimes V_2, but which one?
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.

This paper seems to me to have poor notation in general.

## What is a tensor product of representations?

A tensor product of representations is a mathematical operation that combines two representations of a group into a new representation. It is often used in the study of group theory and is closely related to the concept of direct sum of vector spaces.

## How is a tensor product of representations calculated?

A tensor product of representations is calculated by taking the direct product of the two representations and then applying a specific rule to the resulting vectors. This rule is known as the tensor product rule and involves multiplying the basis elements of the two representations and then taking the sum of these products.

## What is the significance of tensor products of representations?

Tensor products of representations are important in the study of group theory because they allow us to decompose a representation into simpler representations. This can provide insight into the structure and properties of the original group and its representations.

## Can tensor products of representations be applied to non-group structures?

Yes, tensor products of representations can also be applied to other algebraic structures such as Lie algebras and associative algebras. In these cases, the tensor product may have different properties and uses, but the basic concept remains the same.

## Are there any applications of tensor products of representations in physics?

Yes, tensor products of representations have many applications in physics, particularly in the study of symmetries and quantum mechanics. In quantum mechanics, tensor products of representations are used to describe composite systems and their symmetries. They are also used in the study of quantum entanglement.

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