Algebras and Representation Theory最新文献

In (Davydov et al. Selecta Mathematica (N.S.) 19, 237–269 2013, Rem. 3.4) the authors asked the question if any étale subalgebra of an étale algebra in a braided fusion category is also étale. We give a positive answer to this question if the braided fusion category (mathcal {C}) is pseudo-unitary and non-degenerate. In the case of a pseudo-unitary fusion category we also give a new description of the lattice correspondence from (Davydov et al. J. für die reine und angewandte Mathematik. 677, 135–177 2013, Theorem 4.10). This new description enables us to describe the two binary operations on the lattice of fusion subcategories.

This article develops a practical technique for studying representations of (Bbbk )-linear categories arising in the categorification of quantum groups. We work in terms of locally unital algebras which are (mathbb {Z})-graded with graded pieces that are finite-dimensional and bounded below, developing a theory of graded triangular bases for such algebras. The definition is a graded extension of the notion of triangular basis as formulated in Brundan and Stroppel (Mem. Amer. Math. Soc. 293(1459), vii+152 2024). However, in the general graded setting, finitely generated projective modules often fail to be Noetherian, so that existing results from the study of highest weight categories are not directly applicable. Nevertheless, we show that there is still a good theory of standard modules. In motivating examples arising from Kac-Moody 2-categories, these modules categorify the PBW bases for the modified forms of quantum groups constructed by Wang.

We define categories (varvec{mathcal {O}}^{varvec{w}}) of representations of Borel subalgebras (varvec{mathcal {U}}_{!varvec{q}}varvec{mathfrak {b}}) of quantum affine algebras (varvec{mathcal {U}}_{!varvec{q}}hat{varvec{mathfrak {g}}}), which come from the category (varvec{mathcal {O}}) twisted by Weyl group elements (varvec{w}). We construct inductive systems of finite-dimensional (varvec{mathcal {U}}_{varvec{q}}varvec{mathfrak {b}})-modules twisted by (varvec{w}), which provide representations in the category (varvec{mathcal {O}}^{varvec{w}}). We also establish a classification of simple modules in these categories (varvec{mathcal {O}}^{varvec{w}}). We explore convergent phenomenon of (varvec{q})-characters of representations of quantum affine algebras, which conjecturally give the (varvec{q})-characters of representations in (varvec{mathcal {O}}^{varvec{w}}). Furthermore, we propose a conjecture concerning the relationship between the category (varvec{mathcal {O}}) and the twisted category (varvec{mathcal {O}}^{varvec{w}}), and we propose a possible connection with shifted quantum affine algebras.

We give simple procedures to obtain the left and right keys of a semi-standard Young tableau. Keys derive their interest from the fact that they encode the characters of Demazure and opposite Demazure modules for the general and special linear groups. Given the importance of keys, there are indeed several procedures available in the literature to determine them. In comparison, our procedures are new (to the best of our knowledge) and especially simple. Having said that, we hasten to add that there is nothing new in any individual ingredient that goes into our procedures. These ingredients are all routine, straight forward, and (in any case) occur in the literature. But they never quite seem to have been put together as done here. Our procedures end up repeatedly performing the “Deodhar lifts”, maximal lifts for the left key and minimal lifts for right key. Together with the well known fact that keys can be obtained by such repeated lifts, this justifies the procedures. The relevance of Deodhar lifts to combinatorial models for Demazure characters is well known in Standard Monomial Theory. Right and left keys appear respectively as initial and final directions of Lakshmibai-Seshadri paths in Littelmann’s Path Model Theory.

In this paper, we define and study the universal enveloping algebra of a Poisson superalgebra. In particular, a new PBW Theorem for Lie-Rinehart superalgebras is proved leading to a PBW Theorem for Poisson superalgebras, we show the universal enveloping algebra of a Poisson Hopf superalgebra (resp. Poisson-Ore extension) is a Hopf superalgebra (resp. iterated Ore extension), and we study the universal enveloping algebra for interesting classes of Poisson superalgebras such as Poisson symplectic superalgebras.

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.

This paper uses Lusztig varieties to give central elements of the Iwahori-Hecke algebra corresponding to unipotent conjugacy classes in the finite Chevalley group (GL_n(mathbb {F}_q)). We explain how these central elements are related to Macdonald polynomials and how this provides a framework for generalizing integral form and modified Macdonald polynomials to Lie types other than (GL_n). The key steps are to recognize (a) that counting points in Lusztig varieties is equivalent to computing traces on the Hecke algebras, (b) that traces on the Hecke algebra determine elements of the center of the Hecke algebra, (c) that the Geck-Rouquier basis elements of the center of the Hecke algebra produce an ‘expansion matrix’, (d) that the parabolic subalgebras of the Hecke algebra produce a ‘contraction matrix’ and (e) that the combination ‘expansion-contraction’ is the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials.

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.