Centre-by-metabelian group algebras

In nonabelian groups, as well as in noncommutative associative algebras, one may measure the degree of noncommutativity with the help of commutators. Based on these, one defines in „both worlds" (the category of groups, and the category of associative algebras) analogous concepts, such as (Lie) solvability, (Lie) nilpotence, … - here the question arises whether one also obtains parallel properties in both categories. In this context we take as the object of study the group algebra of a (finite or infinite) group over some field. A „parallel property" would then be e.g. commutativity: A group is abelian, if and only if the associated group algebra is commutative. For other, more complex concepts such as the ones mentioned above, such a total correlation cannot be expected, although it should be clear that the commutator properties of the group algebra are derived from the commutator properties of the group we started with

Doctor of Philosophy

dissertationM-theory is the underlying theory of five di fferent string theories and 11D supergravity theory. While strings (1+1D) are fundamental objects in string theory, M2-branes (1+2D) are fundamental objects in M-theory. According to the Gauge/Gravity duality, a gravity theory is equivalent to a gauge theory. Extended (N ? 4) superconformal Chern-Simonsmatter (CSM) theories in 3D are natural candidates of the dual gauge theories of multi M2-branes. In the last two years, the N = 4; 5; 6 CSM theories were constructed by using ordinary Lie 2-algebras, and the N = 8 theory was constructed by using 3-algebra. However, it remains unclear whether these theories can be constructed in a uni ed 3-algebra approach or not. It is also natural to ask whether there are new examples of the extended superconformal CSM theories. In this thesis, we propose to solve these two problems. We de fine a 3-algebra with structure constants being symmetric in the fi rst two indices. We also introduce an invariant antisymmetric tensor into this 3-algebra and call it a symplectic 3-algebra. The D = 3;N = 4; 5; 6; 8 CSM theories are constructed in terms of this unifi ed 3-algebraic structure, and some new examples of the N = 4 quiver gauge theories are derived as well. In particular, in order to realize the 3-algebra used to construct the N = 4 quiver gauge theories, we `fuse' two simple super Lie algebras into a single new super Lie algebra, by requiring that the even parts of these two simple super Lie algebras share one simple factor. We demonstrate how to construct this class of new super Lie algebras by presenting an explicit example. Finally, a quantization scheme for the 3-brackets is proposed

Homogeneous nonrelativistic geometries as coset spaces

We generalize the coset procedure of homogeneous spacetimes in (pseudo-) Riemannian geometry to non-Lorentzian geometries. These are manifolds endowed with nowhere vanishing invertible vielbeins that transform under local non-Lorentzian tangent space transformations. In particular we focus on nonrelativistic symmetry algebras that give rise to (torsional) Newton-Cartan geometries, for which we demonstrate how the Newton-Cartan metric complex is determined by degenerate co- and contravariant symmetric bilinear forms on the coset. In specific cases we also show the connection of the resulting nonrelativistic coset spacetimes to pseudo-Riemannian cosets via Inonu-Wigner contraction of relativistic algebras as well as null reduction. Our construction is of use for example when considering limits of the AdS/CFT correspondence in which nonrelativistic spacetimes appear as gravitational backgrounds for nonrelativistic string or gravity theories

Zooming in on AdS3_3/CFT2_2 near a BPS Bound

Any $(d+1)$-dimensional CFT with a $U(1)$ flavor symmetry, a BPS bound and an exactly marginal coupling admits a decoupling limit in which one zooms in on the spectrum close to the bound. This limit is an In\"on\"u-Wigner contraction of $so(2,d+1)\oplus u(1)$ that leads to a relativistic algebra with a scaling generator but no conformal generators. In 2D CFTs, Lorentz boosts are abelian and by adding a second $u(1)$ we find a contraction of two copies of $sl(2,\mathbb{R})\oplus u(1)$ to two copies of $P_2^c$, the 2-dimensional centrally extended Poincar\'e algebra. We show that the bulk is described by a novel non-Lorentzian geometry that we refer to as pseudo-Newton-Cartan geometry. Both the Chern-Simons action on $sl(2,\mathbb{R})\oplus u(1)$ and the entire phase space of asymptotically AdS$_3$ spacetimes are well-behaved in the corresponding limit if we fix the radial component for the $u(1)$ connection. With this choice, the resulting Newton-Cartan foliation structure is now associated not with time, but with the emerging holographic direction. Since the leaves of this foliation do not mix, the emergence of the holographic direction is much simpler than in AdS$_3$ holography. Furthermore, we show that the asymptotic symmetry algebra of the limit theory consists of a left- and a right-moving warped Virasoro algebra.Comment: 38 pages, v2: references added and typos corrected, v3: references added, journal versio