Mathematics

The Jones-Wenzl idempotents of the Temperley-Lieb algebra are celebrated elements defined over characteristic zero and for generic loop parameter. Given pointed field (R,δ) , we extend the existing results of Burrull, Libedinsky and Sentinelli to determine a recursive form for the idempotents describing the projective cover of the trivial TL R n (δ) -module.

Let T=(A,M,0,B) be a triangular matrix algebra with its corner algebras A and B Artinian and A M B an A - B -bimodule. The 2-recollement structures for singularity categories and Gorenstein defect categories over T are studied. Under mild assumptions, we provide necessary and sufficient conditions for the existences of 2-recollements of singularity categories and Gorenstein defect categories over T relative to those of A and B . Parts of our results strengthen and unify the corresponding work in [27,28,34].

Let B be an one-point extension of a finite dimensional k -algebra A by a simple A -module at a source point i . In this paper, we classify the ? -tilting modules over B . Moreover, it is shown that there are equations |\tilt B|=|\tilt A|+|\tilt A/\langle e_i\rangle|\quad \text{and}\quad |\stilt B|=2|\stilt A|+|\stilt A/\langle e_i\rangle|. As a consequence, we can calculate the numbers of ? -tilting modules and support ? -tilting modules over linearly Dynkin type algebras whose square radical are zero.

We rule out a certain 9 -dimensional algebra over an algebraically closed field to be the basic algebra of a block of a finite group, thereby completing the classification of basic algebras of dimension at most 12 of blocks of finite group algebras.

A partial flag variety P K of a Kac-Moody group G has a natural stratification into projected Richardson varieties. When G is a connected reductive group, a Bruhat atlas for P K was constructed by He, Knutson and Lu: P K is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac-Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff-Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the J -Schubert varieties for a Birkhoff-Bruhat atlas. The notion of the J -Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac-Moody group). The main result of this paper is the construction of a Birkhoff-Bruhat atlas for any partial flag variety P K of a Kac-Moody group. We also construct a combinatorial atlas for the index set Q K of the projected Richardson varieties in P K . As a consequence, we show that Q K has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams in the case where the group G is a connected reductive group.

Building on classical invariant theory, it is observed that the polarised traces generate the centraliser Z L (sl(N)) of the diagonal embedding of U(sl(N)) in U(sl(N) ) ⊗L . The paper then focuses on sl(3) and the case L=2 . A Calabi--Yau algebra A with three generators is introduced and explicitly shown to possess a PBW basis and a certain central element. It is seen that Z 2 (sl(3)) is isomorphic to a quotient of the algebra A by a single explicit relation fixing the value of the central element. Upon concentrating on three highest weight representations occurring in the Clebsch--Gordan series of U(sl(3)) , a specialisation of A arises, involving the pairs of numbers characterising the three highest weights. In this realisation in U(sl(3))⊗U(sl(3)) , the coefficients in the defining relations and the value of the central element have degrees that correspond to the fundamental degrees of the Weyl group of type E 6 . With the correct association between the six parameters of the representations and some roots of E 6 , the symmetry under the full Weyl group of type E 6 is made manifest. The coefficients of the relations and the value of the central element in the realisation in U(sl(3))⊗U(sl(3)) are expressed in terms of the fundamental invariant polynomials associated to E 6 . It is also shown that the relations of the algebra A can be realised with Heun type operators in the Racah or Hahn algebra.

Let G=SL(2,5) be the special linear group of 2?2 -matrices with coefficients in the field with 5 elements. We show that the principal block over a splitting field K of characteristic two of the group algebra KG has a 3 -cluster tilting module. This gives the first example of a representation-infinite block of a group algebra having a cluster tilting module and answers a question by Erdmann and Holm.

In this paper, we calculate the dimension of root spaces g λ of a special type rank 3 Kac-Moody algebras g . We first introduce a special type of elements in g , which we call elements in standard form. Then, we prove that any root space is spanned by these elements. By calculating the number of linearly independent elements in standard form, we obtain a formula for the dimension of root spaces g λ , which depends on the root λ .

We establish a Drinfeld type new presentation for the ı quantum groups arising from quantum symmetric pairs of split affine ADE type, which includes the q -Onsager algebra as the rank 1 case. This presentation takes a form which can be viewed as a deformation of half an affine quantum group in the Drinfeld presentation.

Recently, Lu and Wang formulated a Drinfeld type presentation for ı quantum group U ? ı arising from quantum symmetric pairs of split affine ADE type. In this paper, we generalize their results by establishing a current presentation for U ? ı of arbitrary split affine type.