Primitive living chitons: Morphological cladistic analysis as a model for character evaluation

Chitons are often referred to as “living fossils” in part because they are proposed as one of the earliest-diverging groups of living molluscs, but also because the gross morphology of the polyplacophoran shell has been conserved for hundreds of millions of years. As such, the analysis of evolution and radiation within polyplacophorans is of considerable interest not only for resolving the shape of pan-molluscan phylogeny but also as model organisms for the study of character evolution. This study presents a new, rigorous cladistic analysis of the morphological characters used in taxonomic descriptions for chitons in the living suborder Lepidopleurina Thiele, 1910 (the earliest-derived living group of chitons). Shell-based characters alone entirely fail to recover any recognized subdivisions within the group, which may raise serious questions about the application of fossil data (from isolated shell valves). New analysis including characters from girdle armature and gill arrangements recovers some genera within the group but also points to the lack of monophyly within the main genus Leptochiton Gray, 1847. Additional characters from molecular data and soft anatomy, used in combination, are clearly needed to resolve questions of chiton relationships. However, the data sets currently available already provide interesting insights into the analytical power of traditional morphology as well as some knowledge about the early evolution and radiation of this group.

Morphological cladistic analysis as a model for character evaluation in primitive living chitons (Polyplacophora, Lepidopleurina)

Chitons are often referred to as "living fossils" in part because they are proposed as one of the earliest-diverging groups of living molluscs, but also because the gross morphology of the polyplacophoran shell has been conserved for hundreds of millions of years. As such, the analysis of evolution and radiation within polyplacophorans is of considerable interest not only for resolving the shape of pan-molluscan phylogeny but also as model organisms for the study of character evolution. This study presents a new, rigorous cladistic analysis of the morphological characters used in taxonomic descriptions for chitons in the living suborder Lepidopleurina Thiele, 1910 (the earliest-derived living group of chitons). Shell-based characters alone entirely fail to recover any recognized subdivisions within the group, which may raise serious questions about the application of fossil data (from isolated shell valves). New analysis including characters from girdle armature and gill arrangements recovers some genera within the group but also points to the lack of monophyly within the main genus Leptochiton Gray, 1847. Additional characters from molecular data and soft anatomy, used in combination, are clearly needed to resolve questions of chiton relationships. However, the data sets currently available already provide interesting insights into the analytical power of traditional morphology as well as some knowledge about the early evolution and radiation of this group.

K1 of Chevalley groups are nilpotent

Abstract Let F be a reduced irreducible root system and R be a commutative ring. Further, let G(F,R) be a Chevalley group of type F over R and E(F,R) be its elementary subgroup. We prove that if the rank of F is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(F,R)/E(F,R) is nilpotent by abelian. In particular, when G(F,R) is simply connected the quotient K1(F,R)=G(F,R)/E(F,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.