Algebras and Representation Theory最新文献

In this note, we generalize the notion of quantum Berezinian to the double Yangian (textrm{DY}(mathfrak {gl}_{m|n})) of the Lie superalgebra ( mathfrak {gl}_{m|n} ). We show that its coefficients form a family of algebraically independent topological generators of the center of (textrm{DY}(mathfrak {gl}_{m|n})).

We consider the classification problem for finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras (mathfrak {osp}_{2n|2m}) with (ngeqslant 2). We give necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient and prove the conjecture for a class of representations with linear highest weights. The arguments are based on a new type of odd reflections for the Yangian associated with (mathfrak {osp}_{2|2}). In the Appendix, we construct an isomorphism between the Yangians associated with the Lie superalgebras (mathfrak {osp}_{2|2}) and (mathfrak {gl}_{1|2}).

We explain extremal weight crystals over affine Lie algebras of infinite rank using combinatorial models: a spinor model due to Kwon, and an infinite rank analogue of Kashiwara-Nakashima tableaux due to Lecouvey. In particular, we show that Lecouvey’s tableau model is isomorphic to an extremal weight crystal of level zero. Using these combinatorial models, we explain an algebra structure of the Grothendieck ring for a category consisting of some extremal weight crystals.

We construct an injective weight-preserving map (called the forgetful map) from the set of all admissible subsets in the quantum alcove model associated to an arbitrary weight. The image of this forgetful map can be explicitly described by introducing the notion of “interpolated quantum Lakshmibai-Seshadri (QLS for short) paths”, which can be thought of as a generalization of quantum Lakshmibai-Seshadri paths. As an application, we reformulate, in terms of interpolated QLS paths, an identity of Chevalley type for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra, which is a representation-theoretic analog of the Chevalley formula for the torus-equivariant K-group of a semi-infinite flag manifold.

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably quasi-compact, countably quasi-separated scheme. Moreover, for three categories of complexes of flat quasi-coherent sheaves, we show that all complexes in the category can be obtained as directed colimits of complexes of locally countably presentable flat quasi-coherent sheaves from the same category. In particular, on a quasi-compact semi-separated scheme, every flat quasi-coherent sheaf is a directed colimit of flat quasi-coherent sheaves of finite projective dimension. In the second part of the paper, we discuss cotorsion periodicity in category-theoretic context, generalizing an argument of Bazzoni, Cortés-Izurdiaga, and Estrada. As the main application, we deduce the assertion that any cotorsion-periodic quasi-coherent sheaf on a quasi-compact semi-separated scheme is cotorsion.

In this paper we use the double affine Hecke algebra to compute the Macdonald polynomial products (E_ell P_m) and (P_ell P_m) for type (SL_2) and type (GL_2) Macdonald polynomials. Our method follows the ideas of Martha Yip but executes a compression to reduce the sum from (2cdot 3^{ell -1}) signed terms to (2ell ) positive terms. We show that our rule for (P_ell P_m) is equivalent to a special case of the Pieri rule of Macdonald. Our method shows that computing (E_ell {textbf {1}}_0) and ({textbf {1}}_0 E_ell {textbf {1}}_0) in terms of a special basis of the double affine Hecke algebra provides universal compressed formulas for multiplication by (E_ell ) and (P_ell ). The formulas for a specific products (E_ell P_m) and (P_ell P_m) are obtained by evaluating the universal formulas at (t^{-frac{1}{2}}q^{-frac{m}{2}}).

Let (Lambda (f) = K[x][y; ffrac{d}{dx} ]) be an Ore extension of a polynomial algebra K[x] over an arbitrary field K of characteristic (p>0) where (fin K[x]). For each polynomial f, the automorphism group of the algebras (Lambda (f)) is explicitly described. The automorphism group (textrm{Aut}_K(Lambda (f))=mathbb {S}rtimes G_f) is a semidirect product of two explicit groups where (G_f) is the eigengroup of the polynomial f (the set of all automorphisms of K[x] such that f is their common eigenvector). For each polynomial f, the eigengroup (G_f) is explicitly described. It is proven that every subgroup of (textrm{Aut}_K(K[x])) is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra (Lambda (f)) are 2. The prime, completely prime, primitive and maximal ideals of the algebra (Lambda (f)) are classified.

In the theory of quantum automorphism groups, one constructs Hopf algebras acting on an algebra K from certain algebra morphisms ( sigma :K rightarrow textrm{M}_n(K)). This approach is applied to the field (K=k(t)) of rational functions, and it is investigated when these actions restrict to actions on the coordinate ring (B=k[t^2,t^3]) of the cusp. An explicit example is described in detail and shown to define a new quantum homogeneous space structure on the cusp.

We present a family of selfinjective algebras of type D, which arise from the 3-preprojective algebras of type A by taking a (mathbb {Z}_3)-quotient. We show that a subset of these are themselves 3-preprojective algebras, and that the associated 2-representation-finite algebras are fractional Calabi-Yau. In addition, we show our work is connected to modular invariants for SU(3).

In this paper, we present a generalization of well-established results regarding symmetries of (Bbbk )-algebras, where (Bbbk ) is a field. Traditionally, for a (Bbbk )-algebra A, the group of (Bbbk )-algebra automorphisms of A captures the symmetries of A via group actions. Similarly, the Lie algebra of derivations of A captures the symmetries of A via Lie algebra actions. In this paper, given a category (mathcal {C}) whose objects possess (Bbbk )-linear monoidal categories of modules, we introduce an objec (operatorname {Sym}_{mathcal {C}}(A)) that captures the symmetries of A via actions of objects in (mathcal {C}). Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected (Bbbk )-algebra A, some of its symmetries are naturally captured within the weak Hopf framework.