This volume contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held from August 10-19, 2016, at Syracuse University, Syracuse, NY. ...Included are three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories. Other articles represent contributions to areas in and related to representation theory, such as noncommutative resolutions, twisted commutative algebras, and upper cluster algebras.
The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust ...structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory.This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory.The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.
The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust ...structures capable of describing more complex phenomena. Categorification is a powerful tool for relating various branches of mathematics and exploiting the commonalities between fields. It provides a language emphasizing essential features and allowing precise relationships between vastly different fields.This volume focuses on the role categorification plays in geometry, topology, and physics. These articles illustrate many important trends for the field including geometric representation theory, homotopical methods in link homology, interactions between higher representation theory and gauge theory, and double affine Hecke algebra approaches to link homology.The companion volume (Contemporary Mathematics, Volume 683) is devoted to categorification and higher representation theory.
For the vanishing deformation parameter Formula omitted, the full structure of the (anti)commutator relations in the Formula omitted supersymmetric linear Formula omitted algebra is obtained for ...arbitrary weights Formula omitted and Formula omitted of the currents appearing on the left hand sides in these (anti)commutators. The Formula omitted algebra can be seen from this by taking the vanishing limit of other deformation parameter q with the proper contractions of the currents. For the nonzero Formula omitted, the complete structure of the Formula omitted supersymmetric linear Formula omitted algebra is determined for the arbitrary weight Formula omitted together with the constraint Formula omitted. The additional structures on the right hand sides in the (anti)commutators, compared to the above Formula omitted case, arise for the arbitrary weights Formula omitted and Formula omitted where the weight Formula omitted is outside of above region.
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In this paper, we study the Aggarwal, Ciambelli, Detournay, and Somerhausen (ACDS) boundary conditions (Aggarwal et al. in JHEP 22:013, 2020) for Warped AdS Formula omitted (WAdS Formula omitted) in ...the framework of General Massive Gravity (GMG) in the quadratic ensemble. We construct the phase space, the asymptotic structure, and the asymptotic symmetry algebra. We show that the global surface charges are finite, but not integrable, and also we find the conditions to make them integrable. In addition, to confirm that the phase space has the same symmetries as that of a Warped Conformal Field Theory (WCFT), we compare the bulk entropy of Warped BTZ (WBTZ) black holes with the number of states belonging to a WCFT.
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By using the K-free complex bosons and the K-free complex fermions, we construct the Formula omitted supersymmetric Formula omitted algebra which is the matrix generalization of previous Formula ...omitted supersymmetric Formula omitted algebra. By twisting this Formula omitted supersymmetric Formula omitted algebra, we obtain the Formula omitted supersymmetric Formula omitted algebra which is the matrix generalization of known Formula omitted supersymmetric topological Formula omitted algebra. From this two-dimensional symmetry algebra, we propose the operator product expansion (OPE) between the soft graviton and gravitino (as a first example), at nonzero deformation parameter, in the supersymmetric Einstein-Yang-Mills theory explicitly. Other six OPEs between the graviton, gravitino, gluon and gluino can be determined completely. At vanishing deformation parameter, we reproduce the known result of Fotopoulos, Stieberger, Taylor and Zhu on the above OPEs via celestial holography.
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We present the first example of Formula omitted formulation for the extended higher-spin Formula omitted supergravity with the most general boundary conditions as an extension of the Formula omitted ...work, discovered recently by us (Özer and Filiz in Eur Phys J C 80(11):1072, 2020). Using the method proposed by Grumiller and Riegler, we restrict a consistent class of the most general boundary conditions to extend it. An important consequence of our method is that, for the loosest set of boundary conditions it ensures that their asymptotic symmetry algebras consist of two copies of the Formula omitted. Moreover, we impose some restrictions on the gauge fields for the most general boundary conditions, leading to the supersymmetric extensions of the Brown and Henneaux boundary conditions. Based on these results, we finally find out that the asymptotic symmetry algebras are two copies of the super Formula omitted algebra for Formula omitted extended higher-spin supergravity theory in Formula omitted.
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This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real ...numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices.
We obtain the Courant bracket twisted simultaneously by a 2-form B and a bi-vector Formula omitted by calculating the Poisson bracket algebra of the symmetry generator in the basis obtained acting ...with the relevant twisting matrix. It is the extension of the Courant bracket that contains well known Schouten-Nijenhuis and Koszul bracket, as well as some new star brackets. We give interpretation to the star brackets as projections on isotropic subspaces.
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Vasiliev generating system of higher-spin equations allowing to reconstruct nonlinear vertices of field equations for higher-spin gauge fields contains a free complex parameter Formula omitted. ...Solving the generating system order by order one obtains physical vertices proportional to various powers of Formula omitted and Formula omitted. Recently Formula omitted and Formula omitted vertices in the zero-form sector were presented in Didenko et al. (JHEP 2012:184, 2020) in the Z-dominated form implying their spin-locality by virtue of Z-dominance Lemma of Gelfond and Vasiliev (Phys. Lett. B 786:180, 2018). However the vertex of Didenko et al. (2020) had the form of a sum of spin-local terms dependent on the auxiliary spinor variable Z in the theory modulo so-called Z-dominated terms, providing a sort of existence theorem rather than explicit form of the vertex. The aim of this paper is to elaborate an approach allowing to systematically account for the effect of Z-dominated terms on the final Z-independent form of the vertex needed for any practical analysis. Namely, in this paper we obtain explicit Z-independent spin-local form for the vertex Formula omitted for its Formula omitted-ordered part where Formula omitted and C denote gauge one-form and field strength zero-form higher-spin fields valued in an arbitrary associative algebra in which case the order of product factors in the vertex matters. The developed formalism is based on the Generalized Triangle identity derived in the paper and is applicable to all other orderings of the fields in the vertex.
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