No abstract

We study the Grassmann T -space, S 3 , generated by the commutator [x 1 , x 2 , x 3 ] in the free unital associative algebra K x 1 , x 2 , . . . over a field of characteristic zero. We prove that S 3 = S 2 ∩ T 3 , where S 2 is the commutator T -space generated by [x 1 , x 2 ] and T 3 is the Grassmann T -ideal generated by S 3 . We also construct an explicit basis for each vector space S 3 ∩ Pn, where Pn represents the space of all multilinear polynomials of degree n in x 1 , . . . , xn, and deduce the recursive vector space decomposition T 3 ∩ Pn = (S 3 ∩ Pn) ⊕ (T 3 ∩ P n−1 )xn.

This study combines microstructural observations with Raman spectroscopy on carbonaceous material (RSCM), phase equilibria modelling and U-Pb dating of titanite to delineate the metamorphic history of a well-exposed section through the South Tibetan Detachment System (STDS) in the Dzakaa Chu valley of Southern Tibet. In the hanging wall of the STDS, undeformed Tibetan Sedimentary Series rocks consistently record peak metamorphic temperatures of 340°C. Temperatures increase down-section, reaching 650°C at the base of the shear zone, defining an apparent metamorphic field gradient of 310°C km )1 across the entire structure. U-Th-Pb geochronological data indicate that metamorphism and deformation at high temperatures occurred over a protracted period from at least 20 to 13 Ma. Deformation within this 1-km-thick zone of distributed top-down-to-the-northeast ductile shear included a strong component of vertical shortening and was responsible for significant condensing of palaeo-isotherms along the upper margin of the Greater Himalayan Series (GHS). We interpret the preservation of such a high metamorphic gradient to be the result of a progressive up-section migration in the locus of deformation within the zone. This segment of the STDS provides a detailed thermal and kinematic record of the exhumation of footwall GHS rocks from beneath the southern margin of the Tibetan plateau.

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ∼ = U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H.