Algebras and Representation Theory最新文献

Let A be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to generalize the results proven in [8]. Precisely, we determine which vertices of (Q_A) are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of a monomial algebra and a toupie algebra A, when the Auslander-Reiten quiver is not necessarily a component with length.

We propose a mutation formula for the general rank from a principal component ({{,textrm{PC},}}(delta )) of representations to another one ({{,textrm{PC},}}({epsilon })) for a quiver with potential. We give sufficient conditions for the formula to hold. In particular, the formula holds when any of (delta ) and ({epsilon }) is reachable. We discover several related mutation invariants.

In this paper we study algebras acted on by a finite group G and the corresponding G-identities. Let (M_2( mathbb {C})) be the (2times 2) matrix algebra over the field of complex numbers ( mathbb {C}) and let (sl_2( mathbb {C})) be the Lie algebra of traceless matrices in (M_2( mathbb {C})). Assume that G is a finite group acting as a group of automorphisms on (M_2( mathbb {C})). These groups were described in the Nineteenth century, they consist of the finite subgroups of (PGL_2( mathbb {C})), which are, up to conjugacy, the cyclic groups ( mathbb {Z}_n), the dihedral groups (D_n) (of order 2n), the alternating groups ( A_4) and (A_5), and the symmetric group (S_4). The G-identities for (M_2( mathbb {C})) were described by Berele. The finite groups acting on (sl_2( mathbb {C})) are the same as those acting on (M_2( mathbb {C})). The G-identities for the Lie algebra of the traceless (sl_2( mathbb {C})) were obtained by Mortari and by the second author. We study the weak G-identities of the pair ((M_2( mathbb {C}), sl_2( mathbb {C}))), when G is a finite group. Since every automorphism of the pair is an automorphism for (M_2( mathbb {C})), it follows from this that G is one of the groups above. In this paper we obtain bases of the weak G-identities for the pair ((M_2( mathbb {C}), sl_2( mathbb {C}))) when G is a finite group acting as a group of automorphisms.

We consider bicrossed products obtained by twisting compact semi-direct products by a suitable finite subgroup. We give a practical criterion for the rapid decay property and polynomial growth of the dual of such bicrossed products under a mild restriction. Using this theory, we construct concrete new examples of discrete quantum groups possessing the rapid decay property but not growing polynomially.

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}}).