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Lie algebras. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Infinite-dimensional Lie (super)algebras. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Representations, algebraic theory (weights). --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Structure theory. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Virasoro and related algebras. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras.

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"We define and study cohomological tensor functors from the category Tn of finite-dimensional representations of the supergroup for the image of an arbitrary irreducible representation. In particular DS(L) is semisimple and multiplicity free. We derive a few applications of this theorem such as the degeneration of certain spectral sequences and a formula for the modified superdimension of an irreducible representation"--

Tensor algebra. --- Tensor products. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Representations, algebraic theory (weights). --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Simple, semisimple, reductive (super)algebras. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Homological methods in Lie (super)algebras. --- Category theory; homological algebra -- Categories with structure -- Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories. --- Group theory and generalizations -- Linear algebraic groups and related topics -- Representation theory.

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"We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim if and only if the size of the block is 2 and the eigenvalue is 1; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim. Consequently we present several new examples of Nichols algebras with finite GKdim, including two not in the class alluded to above. We determine which among these Nichols algebras are domains"--

Hopf algebras. --- Associative rings and algebras -- Hopf algebras, quantum groups and related topics -- Ring-theoretic aspects of quantum groups. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Quantum groups (quantized enveloping algebras) and related deformations.

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"We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R, S) in terms of ordinary Macdonald polynomials, are q, t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups. In the classical limit, our method implies that MacMahon's famous ex-conjecture for the generating function of symmetric plane partitions in a box follows from the identification of GL(n,R), O(n) as a Gelfand pair. As further applications, we obtain combinatorial formulas for characters of affine Lie algebras; Rogers-Ramanujan identities for affine Lie algebras, complementing recent results of Griffin et al.; and quadratic transformation formulas for Kaneko-Macdonald-type basic hypergeometric series"--

Orthogonal polynomials. --- Combinatorial identities. --- Combinatorics -- Algebraic combinatorics -- Symmetric functions and generalizations. --- Combinatorics -- Algebraic combinatorics -- Combinatorial aspects of representation theory. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. --- Special functions -- Basic hypergeometric functions -- Basic hypergeometric functions associated with root systems.

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"We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3- dimensional 3-framed local field theories. We also show that all finite tensor categories are 2-dualizable, and yield categorified 2-dimensional 3-framed local field theories. On the other hand, topological properties of 3-framed manifolds determine algebraic equations among functors of tensor categories. We show that the 1-dimensional loop bordism, which exhibits a single full rotation, acts as the double dual autofunctor of a tensor category. We prove that the 2-dimensional belt-trick bordism, which unravels a double rotation, operates on any finite tensor category, and therefore supplies a trivialization of the quadruple dual. This approach produces a quadruple-dual theorem for suitably dualizable objects in any symmetric monoidal 3-category. There is furthermore a correspondence between algebraic structures on tensor categories and homotopy fixed point structures, which in turn provide structured field theories; we describe the expected connection between pivotal tensor categories and combed fixed point structures, and between spherical tensor categories and oriented fixed point structures"--

Categories (Mathematics) --- Tensor fields. --- Duality theory (Mathematics) --- Category theory; homological algebra {For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for --- Algebraic topology -- Applied homological algebra and category theory [See also 18Gxx] -- Duality. --- Associative rings and algebras {For the commutative case, see 13-XX} -- Modules, bimodules and ideals -- Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morit --- Manifolds and cell complexes {For complex manifolds, see 32Qxx} -- Low-dimensional topology -- Invariants of knots and 3-manifolds. --- Nonassociative rings and algebras -- Lie algebras and Lie superalgebras {For Lie groups, see 22Exx} -- Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 8