Group graded endomorphism algebras and Morita equivalences

Abstract

We prove a group graded Morita equivalences version of the "butterfly theorem" on character triples. This gives a method to construct an equivalence between block extensions from another related equivalence.

Character triples and equivalences over a group graded G-algebra

“…This property is required in Section 3, where we show that Ḡ-graded Morita equivalences over C can be induced from certain equivariant Morita equivalences between B and B ′ . We also prove in Theorem 3.9 an analogue of the Butterfly theorem [15,Theorem 2.16], generalizing the main result of [12]. In Section 4 we show how to obtain Ḡ-graded Morita equivalences over C from the Morita equivalences induced by the Scott module Sc(N × N ′ , ∆Q) of Koshitani and Lassueur [7], [8].…”

“…We already know that equivalences induced by Ḡ-graded bimodules preserve many Clifford theoretical invariants (see [10,Chapter 5], [11] and [12])). The relation ≤ c between character triples leads us to the consideration of Ḡ-graded (A, A ′ )-bimodules M satisfying m ḡ c = ḡ cm ḡ for all ḡ ∈ Ḡ, c ∈ C and m ḡ ∈ Mḡ .…”

“…According to[12, Proposition 3.4. ], the upper square of this diagram is commutative, where between C A (B) and C A ′ (B ′ ), we have the following construction: By Theorem Morita II [6, Theorem 12.12], we can choose the ∆ O -linear Ḡ-graded bimodule isomorphism:…”