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Consider a semidirect product group G=H⋉V, where H is reductive and V is a vector group. Two irreps π1 and π2 of H can be “assembled” into a representation of G if it is possible to construct an indecomposable representation Π of G whose restriction to H is π1⊕π2. It is shown that this is equivalent to the existence of a tensor operator from π2 to π1 carrying a representation of H which is equivalent to a nontrivial quotient of the representation which defines the semidirect product. This provides a systematic method for deciding whether two irreps can be assembled, and, if so, in how many inequivalent ways. The method is applied in many of the standard examples that arise in physical questions.

We define a Dirac–Ramond operator over the quotient (LG)/H of a loop group by a subgroup H of the compact Lie group G. We state the conjecture that its equivariant Index with respect of the natural circle action over the quotient of the loop group is equal to the Witten genus of the homogeneous manifold G/H. We motivate the conjecture by a short time argument which allows to come back to a Dirac–Ramond operator at the manner of Taubes on a limit model where all the computations can be done by hand.

The case when there exists a homomorphism σ of a group G into Aut(K) of a non‐Abelian group K[σ having at most one image in every coset of Aut(K) with respect to I(K)] is investigated. It is shown that any extension E ∈ extσ(G, K) can be obtained as a generalized semidirect product (GSP):E=(KτH)/C′, where H belongs to extσ(G, C) (the group C being the center of K), the semidirect product of K and H is based on τ which equals σ∘n (n being the homomorphism of H onto G), and C′ is the antidiagonal of C⊗C. The GSP is a natural generalization of the central extensions, it is applicable to most groups in theoretical physics, and it has a suitable form for the derivation of the irreducible representations of E.

This note is a short comment on central extensions of Lie groups of the form Map(X, G) when X=Sn and G is a compact simple Lie group. The cohomology of Map(X, G) is computed and cases where the Lie algebra is extended but the corresponding Lie group extension is trivial are considered.

The direct generalization of the isoparity (or G‐parity), with the defining property that it is commutable with the referring internal symmetry group, is investigated on the basis of the theory of Lie algebra. This is one special problem of the group extension of a simple Lie group by an involution. It is shown that the isoparity of this type can be defined for the simple Lie groups SU(2)(A1 type), SO(2l + 1)(Bl, l ≥ 2), Sp(2l)(Cl, l ≥ 2), SO(2l)(Dl, l ≥ 3), G2, F4, E7, and E8, but not for the SU(l + 1)(Al, l ≥ 2). The relation between the inner automorphism group and the Weyl group of the simple Lie algebra concerned is available to construct the isoparity operator explicitly. Some illustrative examples are presented.