Matrix Algebra: Info on M^{-1}_{i} M_{j} + M^{-1}_{j} M_{i} = 2\delta_{ij} I

[JustMeDK]

I would like to know where, if possible, I could find some information on the (matrix) algebra

$M^{-1}_{i} M_{j} + M^{-1}_{j} M_{i} = 2\delta_{ij} I$.

I expect this algebra to be among the very many different algebras that mathematicians have studied, but I have been unable to Google my way to any relevant reference regarding it.

 

[chiro]

Hey JustMeDK and welcome to the forums.

This looks a lot like the commutator relationship for operators in Quantum mechanics with the exception that there is an imaginary coeffecient there. I'd imagine though that you can get a similar result if you used standard non-complex rotation matrices (i.e. unitary as opposed to special unitary).

Here's the wiki for commutators (Look at the section on ring theory for definition):

http://en.wikipedia.org/wiki/Commutator

Here is the article on the quantum mechanical aspect I referred to above:

http://en.wikipedia.org/wiki/Canonical_commutation_relation

The thing though that we need to know is what these subscripts i and j refer to. What does M_i and M_j refer to? How do you define them?

 

[JustMeDK]

Thanks for your reply, chiro.

But, unless I am fundamentally mistaken, the algebra I mention is generally not (due to the presence also of inverses, $M_{i}^{-1}$) equivalent to some (anti)commutator algebra. Of course, in special cases equivalence is present: for instance, if $M_{i}$ are taken to be the Pauli matrices, $\sigma_{i} = \sigma_{i}^{-1}$, then the algebra reduces to an anticommutator algebra, the signature (3,0) Clifford algebra specifically. But I am unable to see that in general there should be such equivalences.

PS: The subscripts $i,j$ are just indices running from $1$ to the number $n$, say, of elements $M_{i}$.

 

[chiro]

Thanks for your reply, chiro.

But, unless I am fundamentally mistaken, the algebra I mention is generally not (due to the presence also of inverses, $M_{i}^{-1}$) equivalent to some (anti)commutator algebra. Of course, in special cases equivalence is present: for instance, if $M_{i}$ are taken to be the Pauli matrices, $\sigma_{i} = \sigma_{i}^{-1}$, then the algebra reduces to an anticommutator algebra, the signature (3,0) Clifford algebra specifically. But I am unable to see that in general there should be such equivalences.

PS: The subscripts $i,j$ are just indices running from $1$ to the number $n$, say, of elements $M_{i}$.

So can you outline any constraints imposed on the operators in explicit detail?

 

[JustMeDK]

I do not think that it should be necessary to specify any constraints on the operators $M_{i}$, other than they are taken to be matrices of some dimension (i.e., focusing on representations only of the algebra, rather than realizations generally). What I would like to know is the classification of the representations of such an algebra, in the same way that for instance Clifford algebras have been classified. In order to do so, it would, of course, be very helpful for me to know the mathematical name, if any, of such an algebra.