### A comparison of deformations and geometric study of varieties of associative algebras.

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Additive deformations of bialgebras in the sense of J. Wirth [PhD thesis, Université Paris VI, 2002], i.e. deformations of the multiplication map fulfilling a certain compatibility condition with respect to the coalgebra structure, can be generalized to braided bialgebras. The theorems for additive deformations of Hopf algebras can also be carried over to that case. We consider *-structures and prove a general Schoenberg correspondence in this context. Finally we give some examples.

Let 𝒯 be the Itô Hopf algebra over an associative algebra 𝓛 into which the universal enveloping algebra 𝓤 of the commutator Lie algebra 𝓛 is embedded as the subalgebra of symmetric tensors. We show that there is a one-to-one correspondence between deformations Δ[h] of the coproduct in 𝒯 and pairs (d⃗[h],d⃖[h]) of right and left differential maps which are deformations of the differential maps for 𝒯 [Hudson and Pulmannová, J. Math. Phys. 45 (2004)]. Corresponding to the multiplicativity and...

We prove that deformations of tame Krull-Schmidt bimodule problems with trivial differential are again tame. Here we understand deformations via the structure constants of the projective realizations which may be considered as elements of a suitable variety. We also present some applications to the representation theory of vector space categories which are a special case of the above bimodule problems.

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras...

We consider the socle deformations arising from formal deformations of a class of Koszul self-injective special biserial algebras which occur in the study of the Drinfeld double of the generalized Taft algebras. We show, for these deformations, that the Hochschild cohomology ring modulo nilpotence is a finitely generated commutative algebra of Krull dimension 2.

We study equivalences for category ${\mathcal{O}}_{p}$ of the rational Cherednik algebras ${\mathbf{H}}_{p}$ of type ${G}_{\ell}\left(n\right)={\left({\mu}_{\ell}\right)}^{n}\u22ca{\U0001d516}_{n}$: a highest weight equivalence between ${\mathcal{O}}_{p}$ and ${\mathcal{O}}_{\sigma \left(p\right)}$ for $\sigma \in {\U0001d516}_{\ell}$ and an action of ${\U0001d516}_{\ell}$ on an explicit non-empty Zariski open set of parameters $p$; a derived equivalence between ${\mathcal{O}}_{p}$ and ${\mathcal{O}}_{{p}^{\text{'}}}$ whenever $p$ and ${p}^{\text{'}}$ have integral difference; a highest weight equivalence between ${\mathcal{O}}_{p}$ and a parabolic category $\mathcal{O}$ for the general linear group, under a non-rationality assumption on the parameter $p$. As a consequence, we confirm special cases of conjectures...