Periodic automorphisms of takiff algebras, contractions, and θ-groups

Let be an algebraic Lie algebra and a (generalised) Takiff algebra. Any finite-order automorphism θ of induces an automorphism of of the same order, denoted . We study invariant-theoretic properties of representations of the fixed point subalgebra of on other eigenspaces of in . We use the observation that, for special values of m, the fixed point subalgebra, , turns out to be a contraction of a certain Lie algebra associated with and θ. To my teacher Supported in part by R.F.B.R. grant 06-01-72550.     

The automorphisms of Petit’s algebras

Let σ be an automorphism of a field K with fixed field F. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t;σ]∕fK[t;σ] obtained when the twisted polynomial f∈K[t;σ] is invariant, and were first defined by Petit. We compute all their automorphisms if σ commutes with all automorphisms in Aut_F(K) and n≥m?1, where n is the order of σ and m the degree of f, and obtain partial results for n<m?1. In the case where K∕F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F. We also briefly investigate when two such algebras are isomorphic.     

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive(generalized)ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive(generalized)ξ-Lie derivation with ξ = 1 if and only if it is an additive(generalized)derivation satisfying L(ξA)= ξL(A)for all A. These results are then used to characterize additive(generalized)ξ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.     

Theorems of Main Boundedness Type, Derivations and Module Derivations

The study of automatic continuity for Banach algebras rests heavily on threetechnical results, namely the main boundedness theorem (cf. [1] and [2]),stability of separating spaces and finiteness of discontinuous points (Lemma1. 6 and Theorem 2.3 in [3]. It has long been asked whether the last two results     

Central invariants of ρ-Lie algebras in Yetter-Drinfeld categories

The central invariants of the derived ρ-Lie algebras in Yetter-Drinfeld categories are studied.     

Completely splittable representations of affine Hecke-Clifford algebras

We classify and construct irreducible completely splittable representations of affine and finite Hecke-Clifford algebras over an algebraically closed field of characteristic not equal to 2.     

Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras,Calogero–Moser systems,and KZB equations

We construct twisted Calogero–Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D’Hoker–Phong and Bordner–Corrigan–Sasaki–Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik–Zamolodchikov–Bernard equations related to the automorphisms of Lie algebras.     

Cuspidal representations of rational Cherednik algebras at t?=?0

We study those finite dimensional quotients of the rational Cherednik algebra at t?=?0 that are supported at a point of the centre. It is shown that each such quotient is Morita equivalent to a certain ??cuspidal?? quotient of a rational Cherednik algebra associated to a parabolic subgroup of W.     

Localization of IC-complexes on Kashiwara’s flag scheme and representations of Kac–Moody algebras

We study equivariant localization of intersection cohomology complexes on Schubert varieties in Kashiwara’s flag manifold. Using moment graph techniques we establish a link to the representation theory of Kac–Moody algebras and give a new proof of the Kazhdan–Lusztig conjecture for blocks containing an antidominant element.     

Norm Attainability of Elementary Operators and Derivations

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Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.     

Conjugacy classes of affine automorphisms of and linear automorphisms of ℙ n in the Cremona groups

We describe the conjugacy classes of affine automorphisms in the group Aut(n,) (respectively Bir()) of automorphisms (respectively of birational maps) of . From this we deduce also the classification of conjugacy classes of automorphisms of ℙⁿ in the Cremona group Bir().     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.     

Koszulness,Krull dimension,and other properties of graph-related algebras

The algebra of basic covers of a graph G, denoted by [`(A)](G)\bar{A}(G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of [`(A)](G)\bar{A}(G) in terms of the combinatorics of G. As a consequence, we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Furthermore, we show that if the graph is bipartite, then [`(A)](G)\bar{A}(G) is a homogeneous algebra with straightening laws, and thus it is Koszul. Finally, we characterize the Cohen–Macaulay property and the Castelnuovo–Mumford regularity of the edge ideal of a certain class of graphs.