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Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a "cellular algebra Clifford theory" for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(\ell,p,n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(\ell,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck's result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(\ell,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of "adjustment matrices" in positive characteristic. [ABSTRACT FROM AUTHOR]

Let \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of \omega-semistable properly supported sheaves on Y with fixed slope \mu and \zeta-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on \zeta depending on the polarization \omega and \mu. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of \omega-semistable properly supported sheaves on Y with fixed slope \mu: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit realization of all irreducible modules in Chari's category \widetilde {\mathcal O} for the affine Kac-Moody algebra A_{1}^{(1)} by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of \mathfrak {sl}_2(\mathbb {C}) with finite weight multiplicities, including the polynomial current algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t], the loop algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1}] and the three-point Lie algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1},(t-1)^{-1}] arisen in the work by Kazhdan-Lusztig. [ABSTRACT FROM AUTHOR]

In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if A is a quasi-hereditary algebra with a simple preserving duality and T is a faithful tilting A-module, then A has the double centralizer property with respect to T. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module T over A for which A = EndEndA(T)(T). As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra SKsy(m,n) and the Brauer algebra Bn(−2m) on the space of dual partially harmonic tensors under certain condition. [ABSTRACT FROM AUTHOR]

In this paper, we show that the finite subalgebra \mathcal {A}^\mathbb {R}(1), generated by \mathrm {Sq}^1 and \mathrm {Sq}^2, of the \mathbb {R}-motivic Steenrod algebra \mathcal {A}^\mathbb {R} can be given 128 different \mathcal {A}^\mathbb {R}-module structures. We also show that all of these \mathcal {A}-modules can be realized as the cohomology of a 2-local finite \mathbb {R}-motivic spectrum. The realization results are obtained using an \mathbb {R}-motivic analogue of the Toda realization theorem. We notice that each realization of \mathcal {A}^\mathbb {R}(1) can be expressed as a cofiber of an \mathbb {R}-motivic v_1-self-map. The {\mathrm {C}_2}-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the \mathrm {RO}({\mathrm {C}_2})-graded Steenrod operations on a {\mathrm {C}_2}-equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the {\mathrm {C}_2}-equivariant realizations of \mathcal {A}^{\mathrm {C}_2}(1). We find another application of the \mathbb {R}-motivic Toda realization theorem: we produce an \mathbb {R}-motivic, and consequently a {\mathrm {C}_2}-equivariant, analogue of the Bhattacharya-Egger spectrum \mathcal {Z}, which could be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra. [ABSTRACT FROM AUTHOR]

Let g be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D on the coordinate ring C[N] related to Brion's equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g is of type An or D4, the map D takes similar distinguished values on the set of all flag minors of C[N], raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D on flag minors belonging to the same standard seed, and we show that in any ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh's monoidal categorification of the cluster structure of C[N] via representations of quiver Hecke algebras. [ABSTRACT FROM AUTHOR]

Consider ideals I of the form \[ I=(x_1^2,\dots, x_n^2)+\operatorname {RLex}(x_ix_j) \] where \operatorname {RLex}(x_ix_j) is the ideal generated by all the square-free monomials which are greater than or equal to x_ix_j in the reverse lexicographic order. We will determine some interesting properties regarding the shape of the Hilbert series of I. Using a theorem of Lindsey [Proc. Amer. Math. Soc. 139 (2011), no. 1, 79–92], this allows for a short proof that any algebra defined by I has the strong Lefschetz property when the underlying field is of characteristic zero. Building on recent work by Phuong and Tran [Colloq. Math. 173 (2023), no. 1, 1–8], this result is then extended to fields of sufficiently high positive characteristic. As a consequence, this shows that for any possible number of minimal generators for an artinian quadratic ideal there exists such an ideal minimally generated by that many monomials and defining an algebra with the strong Lefschetz property. [ABSTRACT FROM AUTHOR]

For each odd integer n \geq 3, we construct a rank-3 graph \Lambda _n with involution \gamma _n whose real C^*-algebra C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n) is stably isomorphic to the exotic Cuntz algebra \mathcal E_n. This construction is optimal, as we prove that a rank-2 graph with involution (\Lambda,\gamma) can never satisfy C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda, \gamma)\sim _{ME} \mathcal E_n, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math. 10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution (\Lambda, \gamma) whose real C^*-algebra C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda, \gamma) is stably isomorphic to the suspension S \mathbb {R}. In the Appendix, we show that the i-fold suspension S^i \mathbb {R} is stably isomorphic to a graph algebra iff -2 \leq i \leq 1. [ABSTRACT FROM AUTHOR]

The \imathHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the \imathquantum loop algebra, which is a generalization of the \imathquantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The \imathHall algebra of the \imathquiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two \imathHall algebras, explaining the isomorphism of the \imathquantum group of split affine type A under the two presentations. [ABSTRACT FROM AUTHOR]

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated categories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category \underline {\mathscr {C}} of an extriangulated category \mathscr {C} is a \tau-category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if \mathscr {C} has enough projectives, almost split extensions and source morphisms. This gives various consequences on \underline {\mathscr {C}}, including Igusa–Todorov's Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of \underline {\mathscr {C}} via the complete mesh category of the Auslander–Reiten species of \underline {\mathscr {C}}. Finally we prove that any locally finite symmetrizable \tau-quiver (=valued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms. [ABSTRACT FROM AUTHOR]

We show that R is regular if Tor_{i}^{R}(R^{+},k) = 0 for some i\geq 1 assuming further that R is a \mathbb {N}-graded ring of dimension 2 finitely generated over an algebraically closed equicharacteristic zero field k. This answers a question of Bhatt, Iyengar, and Ma [Comm. Algebra 47 (2019), pp. 2367–2383]. We use almost mathematics over R^{+} to deduce properties of the noetherian ring R and rational surface singularities. Moreover we observe that R^{+} in equicharacteristic zero has a rich module-theoretic structure; it is m-adically ideal(wise) separated, (weakly) intersection flat, and Ohm-Rush. As an application we show that the hypothesis can be astonishingly vacuous for i \ll dim(R). We show that a positive answer to an old question of Aberbach and Hochster [J. Pure Appl. Algebra 122 (1997), pp. 171–184] also answers this question and we use our techniques to study a question of André and Fiorot [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), pp. 81–114] regarding 'fpqc analogues' of splinters. [ABSTRACT FROM AUTHOR]

This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology. [ABSTRACT FROM AUTHOR]

The equivariant \mathcal W-algebra of a simple Lie algebra \mathfrak {g} is a BRST reduction of the algebra of chiral differential operators on the Lie group of \mathfrak {g}. We construct a family of vertex algebras A[\mathfrak {g}, \kappa, n] as subalgebras of the equivariant \mathcal W-algebra of \mathfrak {g} tensored with the integrable affine vertex algebra L_n(\check {\mathfrak {g}}) of the Langlands dual Lie algebra \check {\mathfrak {g}} at level n\in \mathbb {Z}_{>0}. They are conformal extensions of the tensor product of an affine vertex algebra and the principal \mathcal {W}-algebra whose levels satisfy a specific relation. When \mathfrak {g} is of type ADE, we identify A[\mathfrak {g}, \kappa, 1] with the affine vertex algebra V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g}), giving a new and efficient proof of the coset realization of the principal \mathcal W-algebras of type ADE. [ABSTRACT FROM AUTHOR]

Motivated by the theory of Cuntz-Krieger algebras we define and study C^\ast-algebras associated to directed quantum graphs. For classical graphs the C^\ast-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to KK-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C^\ast-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of KK-theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general. [ABSTRACT FROM AUTHOR]

In this article, we define an algebraic version of the Knizhnik–Zamolodchikov (KZ) functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo–Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases. [ABSTRACT FROM AUTHOR]

Let \mathfrak {g} be an affine Kac-Moody Lie algebra and let \lambda, \mu be two dominant integral weights for \mathfrak {g}. We prove that under some mild restriction, for any positive root \beta, V(\lambda)\otimes V(\mu) contains V(\lambda +\mu -\beta) as a component, where V(\lambda) denotes the integrable highest weight (irreducible) \mathfrak {g}-module with highest weight \lambda. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V(\lambda)\otimes V(\mu). Then, we prove the corresponding geometric results including the higher cohomology vanishing on the \mathcal {G}-Schubert varieties in the product partial flag variety \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of \mathcal {G}-sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map. [ABSTRACT FROM AUTHOR]

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H endowed with a universal K-matrix, i.e. , a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U_qL\mathfrak {sl}_{2}. [ABSTRACT FROM AUTHOR]

Green's general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over an infinite field K modulo a general linear form. We strengthen Green's result by showing that the linear forms that do not satisfy such estimate belong to a finite union of proper linear spaces. As an application we give a method to derive variations of the Eakin-Sathaye theorem on reductions. In particular, we recover and extend results by O'Carroll on the Eakin-Sathaye theorem for complete and joint reductions. [ABSTRACT FROM AUTHOR]

Let X= \{x_1, x_2, \cdots, x_n\} be a finite alphabet, and let K be a field. We study classes \mathfrak {C}(X, W) of graded K-algebras A = K\langle X\rangle / I, generated by X and with a fixed set of obstructions W. Initially we do not impose restrictions on W and investigate the case when the algebras in \mathfrak {C} (X, W) have polynomial growth and finite global dimension d. Next we consider classes \mathfrak {C} (X, W) of algebras whose sets of obstructions W are antichains of Lyndon words. The central question is " when a class \mathfrak {C} (X, W) contains Artin-Schelter regular algebras? " Each class \mathfrak {C} (X, W) defines a Lyndon pair (N,W), which, if N is finite, determines uniquely the global dimension, gl\,dimA, and the Gelfand-Kirillov dimension, GK dimA, for every A \in \mathfrak {C}(X, W). We find a combinatorial condition in terms of (N,W), so that the class \mathfrak {C}(X, W) contains the enveloping algebra U\mathfrak {g}, of a Lie algebra \mathfrak {g}. We introduce monomial Lie algebras defined by Lyndon words, and prove results on Gröbner-Shirshov bases of Lie ideals generated by Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter regular algebras of global dimension 6 and 7 occurring as enveloping U = U\mathfrak {g} of standard monomial Lie algebras. The classification is made in terms of their Lyndon pairs (N, W), each of which determines also the explicit relations of U. [ABSTRACT FROM AUTHOR]

Let G be a connected and simply connected nilpotent Lie group, K an analytic subgroup of G and \pi an irreducible unitary representation of G whose coadjoint orbit of G is denoted by \Omega (\pi). Let \mathscr U(\mathfrak g) be the enveloping algebra of {\mathfrak g}_{\mathbb C}, \mathfrak g designating the Lie algebra of G. We consider the algebra D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi)\right)^K of the K-invariant elements of \mathscr U(\mathfrak g)/\operatorname {ker}(\pi). It turns out that this algebra is commutative if and only if the restriction \pi |_K of \pi to K has finite multiplicities (cf. Baklouti and Fujiwara [J. Math. Pures Appl. (9) 83 (2004), pp. 137-161]). In this article we suppose this eventuality and we provide a proof of the polynomial conjecture asserting that D_{\pi }(G)^K is isomorphic to the algebra \mathbb C[\Omega (\pi)]^K of K-invariant polynomial functions on \Omega (\pi). The conjecture was partially solved in our previous works (Baklouti, Fujiwara, and Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]; J. Lie Theory 29 (2019), pp. 311-341). [ABSTRACT FROM AUTHOR]

We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras, which are given a priori as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or period-index bound) and the cyclic length of a general henselian-valued field of finite rank and finite residue field. [ABSTRACT FROM AUTHOR]

We prove that the kernel of the action of the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary commutative ring and proves that the centraliser algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras. [ABSTRACT FROM AUTHOR]

For a Dynkin quiver Q (of type \mathrm {ADE}), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category \mathcal {C}_{Q} of modules over the quantum loop algebra U_{q}(L\mathfrak {g}) via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with Q and Hernandez-Leclerc's category \mathcal {C}_{Q}, assuming the simpleness of some poles of normalized R-matrices for type \mathrm {E}. [ABSTRACT FROM AUTHOR]

We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p ≥ 3, starting with a definition of a π-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k[t,τ]/(tp−τ2), where t has even degree and τ has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme. [ABSTRACT FROM AUTHOR]

We give a graphical theory of integral indefinite binary Hamiltonian forms ƒ analogous to the one of Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order O in a definite quaternion algebra over Q, we define the waterworld of ƒ, analogous to Conway's river and Bestvina-Savin's ocean , and use it to give a combinatorial description of the values of ƒ on O × O. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), the SL2(O)-equivariant Ford-Voronoi cellulation of the real hyperbolic 5-space, and the conformal action of SL2(O) on the Hamilton quaternions. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive group defined over a locally compact non-archimedean field F, let P be a parabolic subgroup with Levi M and compatible with a pro-p Iwahori subgroup of G ≔ G(F). Let R be a commutative unital ring. We introduce the parabolic pro-p Iwahori–Hecke R-algebra HR(P) of P ≔ P(F) and construct two R-algebra morphisms ΘPM : HR(P) → HR(M) and &#926PG : HR(P) → HR(G) into the pro-p Iwahori–Hecke R-algebra of M ≔ M(F) and G, respectively. We prove that the resulting functor Mod − HR(M) → Mod − HR(G) from the category of right HRR(M)-modules to the category of right HR(G)-modules (obtained by pulling back via ΘPM and extension of scalars along ΞPG) coincides with the parabolic induction due to Ollivier–Vignéras. The maps ΘPM and ΞPG factor through a common subalgebra HR(M,G) of HR(G) which is very similar to HR(M). Studying these algebras HR(M,G) for varying (M,G) we prove a transitivity property for tensor products. As an application we give a new proof of the transitivity of parabolic induction. [ABSTRACT FROM AUTHOR]

We generalize Contou-Carrère symbols to higher dimensions. To an (n + 1)-tuple ƒ0, . . . ,ƒn ∈ A((t1))⋅⋅⋅((tn))×, where A denotes an algebra over a field k, we associate an element (ƒ0, . . . ,ƒn) ∈ A×, extending the higher tame symbol for k = A, and earlier constructions for n = 1 by Contou-Carrère, and n = 2 by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic K-theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols. [ABSTRACT FROM AUTHOR]

This is a contribution to the structure theory of finite pointed quasi-quantum groups. We classify all finite-dimensional connected graded pointed Majid algebras of rank two which are not twist equivalent to ordinary pointed Hopf algebras. [ABSTRACT FROM AUTHOR]