Stable invariance of the restricted Lie algebra structure of Hochschild cohomology

We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the $p$-toral rank of ${\rm HH}^1(A,A)$. We also prove a more general result concerning Iwanaga Gorenstein algebras, using a more general notion of stable equivalences of Morita type. We provide several applications to commutative algebra and modular representation theory. The proof exploits in an essential way the $B_\infty$-structure of the Hochschild cochain complex. In the appendix we explain how the well-definedness of the $p$-power operation on Hochschild cohomology follows from some (originally topological) results of May and Cohen, and (on the algebraic side) Turchin. We give complete proofs, using the language of operads.Comment: 20 pages, v2: inaccurate statement about restriction functors correcte

On hochschild cohomology and modular representation theory

The aim of this thesis is to study local and global invariants in representation theory of finite groups using the (restricted) Lie algebra structure of the first degree of Hochschild cohomology of a block algebra B as a main tool. This lead to two directions: In the first part we investigate the global approach. In particular, we prove the compatibility of the p-power map under stable equivalence of Morita type of subclasses of the first Hochschild cohomology represented by integrable derivations. Further results in this aspect include an example showing that the p-power map cannot generally be expressed in terms of the BV operator. We also study some properties of r-integrable derivations and we provide a family of examples given by the quantum complete intersections where all the derivations are r-integrable. In the second part our attention is focused on the local invariants. More precisely, we fully characterise blocks B with unique isomorphism class of simple modules such that the first degree Hochschild cohomology HH1(B) is a simple as Lie algebra. In this case we prove that B is a nilpotent block with an elementary abelian defect group P of order at least 3 and HH1(B) is isomorphic to the Witt algebra HH1(kP)

Block algebras with HH1 a simple Lie algebra

The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH1(B) is a simple Lie algebra and such that B has a unique isomorphism class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH1(B) is in that case isomorphic to the Witt algebra HH1(kP)⁠. In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single isomorphism class of simple modules

On blocks of defect two and one simple module, and Lie algebra structure of HH¹

Let k be a field of odd prime characteristic p. We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over k. As a consequence, we prove that if B is a defect 2-block of a finite group algebra $kG$ whose Brauer correspondent C has a unique isomorphism class of simple modules, then a basic algebra of B is a local algebra which can be generated by at most 2√I elements, where I is the inertial index of B, and where we assume that k is a splitting field for B and C

Higgs field in cosmology

The accelerated expansion of the early universe is an integral part of modern cosmology and dynamically realized by the mechanism of inflation. The simplest theoretical description of the inflationary paradigm is based on the assumption of an additional propagating scalar degree of freedom which drives inflation - the inflaton. In most models of inflation the fundamental nature of the inflaton remains unexplained. In the model of Higgs inflation, the inflaton is identified with the Standard Model Higgs boson and connects cosmology with elementary particle physics. A characteristic feature of this model is a non-minimal coupling of the Higgs boson to gravity. I review and discuss several phenomenological and fundamental aspects of this model, including the impact of quantum corrections and the renormalization group, the derivation of initial conditions for Higgs inflation in a quantum cosmological framework and the classical and quantum equivalence of different field parametrizations.Comment: 36 pages, 9 figures; references added, typos corrected. Invited contribution to the Heraeus-Seminar "Hundred Years of Gauge Theory", 30 July - 3 August 2018, Physikzentrum Bad Honnef, organized by Silvia De Bianchi and Claus Kiefer. To appear in the proceedings "100 Years of Gauge Theory. Past, present and future perspectives" in the series `Fundamental Theories of Physics' (Springer

The first Hochschild cohomology as a Lie algebra

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections.Fil: Rubio y Degrassi, Lleonard. Universita Degli Studi Di Verona; ItaliaFil: Schroll, Sibylle. Universitat zu Köln; AlemaniaFil: Solotar, Andrea Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

The first Hochschild cohomology as a Lie algebra

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections