Topic
Idempotence
About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.
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13 citations
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13 citations
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TL;DR: In this paper, it was shown that real Clifford algebras CL(V,Q) posses a unique transposition anti-involution, which reduces to reversion for any Euclidean (resp. anti-Euclidean) signature.
Abstract: In the first article of this work [... I: The transposition map] we showed that real Clifford algebras CL(V,Q) posses a unique transposition anti-involution \tp. There it was shown that the map reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the associated matrix of that element in the left regular representation of the algebra. In this paper we show that, depending on the value of (p-q) mod 8, where \ve=(p,q) is the signature of Q, the anti-involution gives rise to transposition, Hermitian complex, and Hermitian quaternionic conjugation of representation matrices in spinor representations. We realize spinors in minimal left ideals S=CL_{p,q}f generated by a primitive idempotent f. The map \tp allows us to define a dual spinor space S^\ast, and a new spinor norm on S, which is different, in general, from two spinor norms known to exist. We study a transitive action of generalized Salingaros' multiplicative vee groups G_{p,q} on complete sets of mutually annihilating primitive idempotents. Using the normal stabilizer subgroup G_{p,q}(f) we construct left transversals, spinor bases, and maps between spinor spaces for different orthogonal idempotents f_i summing up to 1. We classify the stabilizer groups according to the signature in simple and semisimple cases.
13 citations
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TL;DR: In this paper, the theory of categorical diagonalization is studied for Hecke algebras of type A. In particular, the authors construct a Young symmetrizer by simultaneously diagonalizing certain functors associated to the full twist braids.
Abstract: This paper lays the groundwork for the theory of categorical diagonalization Given a diagonalizable operator, tools in linear algebra (such as Lagrange interpolation) allow one to construct a collection of idempotents which project to each eigenspace These idempotents are mutually orthogonal and sum to the identity We categorify these tools At the categorical level, one has not only eigenobjects and eigenvalues but also eigenmaps, which relate an endofunctor to its eigenvalues Given an invertible endofunctor of a triangulated category with a sufficiently nice collection of eigenmaps, we construct idempotent functors which project to eigencategories These idempotent functors are mutually orthogonal, and a convolution thereof is isomorphic to the identity functor In several sequels to this paper, we will use this technology to study the categorical representation theory of Hecke algebras In particular, for Hecke algebras of type A, we will construct categorified Young symmetrizers by simultaneously diagonalizing certain functors associated to the full twist braids
13 citations
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TL;DR: The dependences between algebraic properties of the operation * and the induced sup - * composition are discussed and consequences of these results for compositions based on triangular norms, triangular conorms and uninorms are presented.
Abstract: We examine compositions of fuzzy relations based on a binary operation *. We discuss the dependences between algebraic properties of the operation * and the induced sup - * composition. It is examined independently for monotone operations, for operations with idempotent, zero or identity element, for distributive and associative operations. Finally, we present consequences of these results for compositions based on triangular norms, triangular conorms and uninorms.
13 citations