What is a good reference or how can one check to given a set of possible structure constants? In particular for algebras for three generators. How does one know of given the structure constants, these correspond either to

  1. An algebra: here I know that verifying the jacobi identity is a crucial to be a valid algebra. But if so, does someone know a good reference where I can read off what algebra these then correspond to?

  2. What about a the algebra of a semi-direct product of groups? I know that for a direct product of groups, at the level of the algebra, the structure constant should show that their is a separation into two sets of generators, that do not mix under the commutator. But for a semi-direct product:

    1. Do the structure constant still need to satisfy the Jacobi identity?
    2. Do the generators also separate into two mutually disjoint sets, as it does for the direct sum of algebras?

e.g. given the structure constants $$f_{312}=-1 \quad \text{and}\quad f_{121}=1\,, $$ does not satisfy the Jacobi identity and cannot be a direct sum of algebra but does or does not correspond to the algebra of a semi-direct product of groups?

As you might remark, this question shows my lack of knowledge about the algebra of semi-direct products. Consequently, a good reference (preferably physics-oriented) about this would be more than welcome!


First we need to clarify what "an algebra of a group" is. One reasonable answer is that we are talking about a Lie group and its associated Lie algebra. Since you are mentioning the Jacobi identity, I suppose that this is the case. Concerning the first part of the questions, there is an algorithm, based on Groebner bases, to determine whether two given Lie algebras are isomorphic, or not. However, it is much more complicated than just "reading off" what algebra it is. Concerning semidirect products:
Question 1: The Jacobi identity is also satisfied for the Lie algebra of a semi-direct product. Of course, we need to say, how the Lie bracket of a semi-direct product is defined. This is defined in these MSE-questions: Lie bracket of a semidirect product; Semi-direct product Lie algebra
Question 2: Yes, the underlying set is still the cartesian product for a semidirect product, as in the case of a direct product.

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