Twisted partial coactions of Hopf algebras

Recent developments around partial actions

“…Furthermore, Birget [56] applied the partial action definition of Thompson's groups to study algorithmic problems for them. Since then the algebraic approach is being developed in diverse directions in various levels of generality, including partial actions of Hopf (or, more generally, weak Hopf) algebras [15,[17][18][19][20][21][22]48,69,72,[78][79][80][81][82][83][86][87][88]165,250,282], semigroups [68,97,132,[197][198][199]209,213,220,227,233,234,242,255], inductive constellations [198], groupoids [37,[40][41][42][43]178], and, more generally, categories [244]. In particular, further algebraic applications have been found to graded algebras [117,121], to Hecke algebras…”

“…Recent works around partial actions include also the study of the category of the partial Doi-Hopf modules in [86], of partial actions on power sets in [34], of the category of partial G-sets for a fixed group G in [28], of partial orbits and n-transitivity in [31], of partial group entwining structures and partial group (co)actions of a Hopf group coalgebra on a family of algebras in [87], of generalized partial smash products in [143], and of twisted partial Hopf coactions and corresponding partial crossed coproducts in [88], as well as a note on sums of ideals [33]. More information around partial actions may be found in the surveys [45,115,116,170,250,251,260].…”

Enveloping actions and duality theorems for partial twisted smash products

“…Alves, Batista, Dokuchaev and Paques introduced the notion of a twisted partial Hopf action as a unified approach for twisted partial group actions, partial Hopf actions and twisted actions of Hopf algebras, they established the conditions on partial cocycles in order to construct partial crossed products, and explored the relations between partial crossed products with partial cleft extensions of algebras in [4]. Chen and Wang introduced the twisted partial coactions of Hopf algebras and studied their properties in [7]. Recently, the first author introduced the notion of partial representation of partial twisted smash products and explored its relationship with partial actions of Hopf algebras in [13].…”

A linearly convergent algorithm for sparse signal reconstruction

“…A similar algorithm based on conjugate gradient technique is proposed by Xiao and Zhu [61]. For more detail, see [4,5,[9][10][11][12]17,20,24,26,31,34,36,39,41,42,56,58,59,62,68]. Due to the high computing cost of the line search procedure, we propose a new type of projection algorithm for problem (1.1) without line search at each iteration in this paper which marginally decrease the computing cost of the algorithm.…”