We give an informal exposition of pushforwards and orientations in generalized cohomology theories in the language of spectra. The whole note can be seen as an attempt at convincing the reader that Todd classes in Grothendieck–Hirzebruch–Riemann–Roch type formulas are not Devil’s appearances but rather that things just go in the most natural possible way.
Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this article, we use Hodge Laplacian to study the cohomology of these Lie algebras. The “total rank conjecture” and “b2-conjecture” for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler–Poincaré principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras. 相似文献
Let G be the simple, simply connected algebraic group SL3 defined over an algebraically closed field K of characteristic p > 0. In this article, we find H2(G, V) for any irreducible G-module V. When p > 7, we also find H2(G(q), V) for any irreducible G(q)-module V for the finite Chevalley groups G(q) = SL(3, q) where q is a power of p. 相似文献
In this article, we introduce Adem-Cartan operads and prove that the cohomology of any algebra over such an operad is an unstable level algebra over the extended Steenrod algebra. Moreover, we prove that this cohomology is endowed with secondary cohomology operations. 相似文献
We prove that for q∈C? not a nontrivial root of unity any symmetric invariant 2-cocycle for a completion of Uqg is the coboundary of a central element. Equivalently, a Drinfeld twist relating the coproducts on completions of Uqg and Ug is unique up to coboundary of a central element. As an application we show that the spectral triple we defined in an earlier paper for the q-deformation of a simply connected semisimple compact Lie group G does not depend on any choices up to unitary equivalence. 相似文献
We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is Pn, these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points. 相似文献
Using Barnes-Wall lattices and 1-cocycles on finite groups of monomial matrices, we give a procedure to construct tricosine spherical codes. This was inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and Morris discovered in studies of the universally optimal property. Their code has 64 vectors and cosines . We construct the Optimism Code, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are . Its automorphism group has shape 21+8⋅GL(4,2). The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters (16,256,6), a Nordstrom-Robinson code, and gives a context for determining its automorphism group, which has form . 相似文献