Klaus Behrndt, Renata Kallosh, Joachim Rahmfeld, Marina Shmakova, and Wing Kai Wong ∗ Humboldt-Universität, Institut für Physik, Invalidenstraße 110, 10115 Berlin, Germany Physics Department, Stanford University, Stanford, CA 94305-4060 Department of Physics, Texas A&M University , College Station, TX 77843 University of Tennessee, Knoxville, TN 37996 Stanford Linear Accelerator Center, Stanfor...

We determine which simple algebraic groups of type D4 over arbitrary fields of characteristic different from 2 admit outer automorphisms of order 3, and classify these automorphisms up to conjugation. The criterion is formulated in terms of a representation of the group by automorphisms of a trialitarian algebra: outer automorphisms of order 3 exist if and only if the algebra is the endomorphis...

The Cayley 4-form ωCa is of fundamental importance both in the calibration theory of R. Harvey and H. B. Lawson [HL82], who calculated its comass ‖ωCa‖ using the non-associative algebra of the Cayley numbers, and in exceptional holonomy of type Spin(7) studied by R. Bryant [Br87] and D. Joyce [Jo00]. The Cayley form is self-dual and the triality property of SO(8) relates the representation on s...

We prove the conjecture of Gaiotto and Rap\v{c}\'ak that $Y$-algebras $Y_{L,M,N}[\psi]$ with one parameters $L,M,N$ zero, are simple one-parameter quotients universal two-parameter $\mathcal{W}_{1+\infty}$-algebra, satisfy a symmetry known as triality. These defined cosets certain non-principal $\mathcal{W}$-algebras $\mathcal{W}$-superalgebras by their affine vertex subalgebras, triality is an...

We prove that H( n 2)(SLn Z;Q) = 0, where ( n 2 ) is the cohomological dimension of SLn Z, and similarly for GLn Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GLn. These theorems are derived from a presentation of the Steinberg module for SLn Z whose generators are integral apartment classes, generalizing Manin...