In this paper we prove that every automorphism of a (elementary) Chevalley group of type A l , D l , or E l , l 2, over a commutative local ring with 1/2 is standard, i. e., is the composition of inner, ring, graph and central automorphisms.

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In the paper we prove that every automorphism of any adjoint Chevalley group of types B 2 or G 2 is standard, i.e., it is a composition of the "inner" automorphism, ring automorphism and central automorphism.Introduction.An associative ring R with a unit is called local, if it has exactly one maximal ideal (which coincides with the Jacobson radical of R). Equivalently, all non-invertible elements of R form an ideal. In this paper all rings under consideration are commutative.Let G ad be a Chevalley-Demazure group scheme associated with an irreducible root system Φ of type B 2 or G 2 (see detailed definitions in the next section); G ad (R, Φ) be a set of points G with values in R; E ad (R, Φ) be the elementary subgroup of G ad (R, Φ), where R is a local commutative ring with 1. In this paper we describe automorphisms of G ad (R, Φ) and E ad (R, Φ) (for the root systems A l , D l and E l the automorphisms were described in the paper [1]). Suppose that R is a local ring with 1/2 and 1/3. Then every automorphism of G ad (R, Φ) (E ad (R, Φ)) is standard (see below definitions of standard automorphisms). These results for Chevalley groups over fields were proved by R. Steinberg [2] for finite case and by J. E. Humphreys [3] for infinite case. K. Suzuki [4] studied automorphisms of Chevalley groups over rings of p-adic numbers. E. Abe [5] proved this result for Noetherian rings, but the class of all local rings is not completely contained in the class of Noetherian rings, and the proof of [5] can not be extended to the case of arbitrary local rings. From the other side, automorphisms of classical groups over rings were discussed by many authors. This field of research was open by Schreier and Van der Varden, who described automorphisms of the group P SL n (n ≥ 3) over arbitrary field. Then J. Diedonne [6], L. Hua and I. Reiner [7], O'Meara [8], B. R. McDonald [9], I. Z. Golubchik and A. V. Mikhalev [10] and others studied this problem for groups over more general rings. To prove our theorem we generalize some methods from the paper of V. M. Petechuk [11].Every Chevalley group under consideration is embedded into the group GL N (R) for some N ∈ N. Therefore we can consider Chevalley groups as matrix groups and use linear algebraic group technique: invertible coordinate changes in local rings, uniqueness of a solution of systems of linear equations over local rings with the condition, that determinant of a corresponding matrix is invertible, and so on. As the result we come to the fact that every automorphism of Chevalley group is induced by automorphism of the corresponding matrix ring.The author would like to thank A.Yu. Golubkov, A.A. Klyachko and A.V. Mikhalev for valuable comments and attention to the work.

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A new method of determining surface electronic structure is described in which the surface atomic layers are embedded on to a semi-infinite substrate. An embedding potential, derived from the bulk Green function, is added to the Hamiltonian for the surface region, which can then be solved by conventional surface band-structure techniques such as the LAPW method used here. Results are presented for AI(001) and Ni (001), illustrating the way that the embedding potential produces the true surface density of states rather than the discrete states of the usual slab calculations.