Automorphisms of groups

Let G be a finite nilpotent group and Autc(G) be the
group of all central automorphisms of G. Let C*(G) = C_Autc(G)(Z(G)) be the group of all central automorphisms of G fixing Z(G) elementwise. In this paper we give conditions on G such that Autc(G) =
C*(G). Also we mention to finitely generated p-group G such that
each central automorphism of G fixes the center of G elementwise.

We examine the Johnson filtration of the (outer) automorphism group of a finitely generated group. In the case of a free group, we find a surprising result: the first Betti number of the second subgroup in the Johnson filtration is finite. Moreover, the corresponding Alexander invariant is a non-trivial module over the Laurent polynomial ring. In the process, we show that the first resonance variety of the outer Torelli group of a free group is trivial. We also establish a general relationship between the Alexander invariant and its infinitesimal counterpart.

The automorphism group of a group $G$ comes endowed with a natural filtration: an automorphism belongs to the $k$-th term of this ``Johnson filtration" if it has the same $k$-jet as the identity, with respect to the lower central series of $G$. In this talk, I will discuss the Johnson filtration of the automorphism group of a finitely generated free group, and that of the mapping class group of a surface, with emphasis on the homological finiteness properties of the first few terms in these filtrations. A key ingredient in this approach is a rather surprising relationship between the classical representation theory of a complex, semisimple Lie algebra $\mathfrak{g}$ and the resonance varieties $R(V,K)\subset V^*$ attached to irreducible $\mathfrak{g}$-modules $V$ and submodules $K\subset V\wedge V$. In the case when $\mathfrak{g}= \mathfrak{sl}_2(\mathbb{C})$, this relationship sheds new light on certain modules studied by Weyman and Eisenbud in the context of Green's conjecture on free resolutions of canonical curves. This is joint work with Stefan Papadima (arXiv:1011.5292, 1207.2038).

We initiate the study of analogues of symmetric spaces for the family of finite dihedral groups. In particular, we investigate the structure of the automorphism group, characterize the involutions of the automorphism group, and determine the fixed-group and symmetric space of each automorphism.

We compute the automorphism group of the q-enveloping algebra U_q(sl_4^+) of the nilpotent Lie algebra of strictly upper triangular matrices of size 4. The result obtained gives a positive answer to a conjecture of Andruskiewitsch and Dumas. We also compute the derivations of this algebra and then show that the Hochschild cohomology group of degree 1 of this algebra is a free (left)-module of rank 3 (which is the rank of the Lie algebra sl(4)) over the center of U_q(sl_4^+).

We introduce a class of rings, namely the class of left or right $p$-nil rings, for which the adjoint groups behave regularly. Every $p$-ring is close to being left or right $p$-nil in the sense that it contains a large ideal belonging to this class. Also their adjoint groups occur naturally as groups of automorphisms of $p$-groups. These facts and some of their applications are investigated in this paper.

We study absolute valued algebras with involution, as defined in Urbanik (1961). We prove that these algebras are finite-dimensional whenever they satisfy the identity (x, x 2, x) = 0, where (·, ·, ·) means associator. We show that, in dimension different from two, isomorphisms between absolute valued algebras with involution are in fact *-isomorphisms. Finally, we give a classification up to isomorphisms of all finite-dimensional absolute valued algebras with involution. As in the case of a parallel situation considered in Rochdi (2003), the triviality of the group of automorphisms of such an algebra can happen in dimension 8, and is equivalent to the nonexistence of 4-dimensional subalgebras.

For any set X and any variety 𝒱 of algebras, let 𝒜 = 𝒜𝒱(X) be the free algebra in 𝒱 with set X of free generators, and let End(𝒜) be the monoid of endomorphisms of 𝒜. We provide a general approach to the description of the automorphism group of End(𝒜), and some subsemigroups of End(𝒜), provided that 𝒜 is 1-simple. We show

I will describe a relationship between the classical representation
theory of a complex, semisimple Lie algebra \g and the resonance
varieties R(V,K)\subset V^* attached to irreducible \g-modules V
and submodules K\subset V\wedge V. In the process, I will give a
precise roots-and-weights criterion insuring the vanishing of these
varieties  In the case when \g= \sl_n(\C) or \sp_{2g}(\C), our approach
yields a unified proof of two vanishing results for the resonance varieties
of the (outer) Torelli groups of finitely generated free groups and surface groups.  In turn, these vanishing results reveal certain homological finiteness properties of the Johnson filtration.  This is joint work with Stefan Papadima.