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Register a new userA note on stable equivalence between nonstandard RFS algebras
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Let k be an algebraically closed field. It is known that any stable equivalence between standard representation-finite self-injective k-algebras (without blocks of Lowey length 2) lifts to a standard derived equivalence, in particular, it is of Morita type. In this note, we show that the same holds for any stable equivalence between nonstandard representation-finite self-injective k-algebras. This settles an open question raised by H. Asashiba about twenty years ago.
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We review Morita equivalence for finite type $k$-algebras $A$ and also a weakening of Morita equivalence which we call stratified equivalence. The spectrum of $A$ is the set of equivalence classes of irreducible $A$-modules. For any finite type $k$-algebra $A$, the spectrum of $A$ is in bijection with the set of primitive ideals of $A$. The stratified equivalence relation preserves the spectrum of $A$ and also preserves the periodic cyclic homology of $A$. However, the stratified equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. A key example illustrating the distinction between Morita equivalence and stratified equivalence is provided by affine Hecke algebras associated to extended affine Weyl groups.
We give a short proof based on Lusztigs generalized Springer correspondence of some results of [BrCh,BaCr,P].
In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description. We do so by analysing the Fourier transform of the nearby cycle sheaves constructed in [GVX2].
We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an irreducible algebraic representation of $G$.
We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $Delta $-equivalent if they have completely isometric representations $alpha $ and $beta $ respectively and there exists a ternary ring of operators $M$ such that $alpha (A)$ (resp. $beta (B)$) is equal to the norm closure of the linear span of the set $M^*beta (B)M, $ (resp. $Malpha (A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $Delta $-equivalent, then the operator algebras $Aotimes cl K$ and $Botimes cl K$ are isomorphic. Here $cl K$ is the set of compact operators on an infinite dimensional separable Hilbert space and $otimes $ is the spatial tensor product. Conversely, if $Aotimes cl K$ and $Botimes cl K$ are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $Delta $-equivalent.
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