SOME Z/2-GRADED REPRESENTATION THEORY

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras $S(2, d)$ and their $q$ -analogues. Our second main result completely determines the relative dominant dimension of $S(2, d)$ with respect to $Q=V^{\otimes d}$ , the $d$ -th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of $q$ -Schur algebras $S(2,d)$ give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of the $q$ -Schur algebra $S(2, d)$ and precisely how far these two algebras are from being Morita equivalent, when they are not. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$ -Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley–Lieb algebra.

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

Koenig and Xi introduced {\em affine cellular algebras}. Kleshchev and Loubert showed that an important class of {\em infinite dimensional} algebras, the KLR algebras $R(\Gamma)$ of finite Lie type $\Gamma$, are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of {\em quasihereditary algebras} of Cline-Parshall-Scott in a {\em finite dimensional} situation. A fundamental result of Cline-Parshall-Scott says that a finite dimensional algebra $A$ is quasihereditary if and only if the category of finite dimensional $A$-modules is a {\em highest weight category}. On the other hand, S. Kato and Brundan-Kleshchev-McNamara proved that the category of {\em finitely generated graded} $R(\Gamma)$-modules has many features reminiscent of those of a highest weight category. The goal of this paper is to axiomatize and study the notions of an {\em affine quasihereditary algebra} and an {\em affine highest weight category}. In particular, we prove an affine analogue of the Cline-Parshall-Scott Theorem. We also develop {\em stratified} versions of these notions.

In their fundamental paper [14] Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan-Lusztig polynomials of W (see, e.g., [12], Chap. 7). These polynomials are intimately related to the Bruhat order of W and to the algebraic geometry of Schubert varieties, and have proven to be of fundamental importance in representation theory. Our aim in this work is to give a simple, nonrecursive, combinatorial formula for any Kazhdan-Lusztig polynomial of any Coxeter group (Theorem 4.1), and to study some consequences of it. The main idea involved in the proof and statement of this formula is that of extending the concept of the Rpolynomial (see, e.g., [12], w to any (finite) multichain of W (so that, for multichains of length 1, one obtains, apart from sign, the usual R-polynomials). Once this has been done, then the Kazhdan-Lusztig polynomial of a pair u, v turns out to be just the sum, over all multichains from u to v, of the corresponding (generalized) R-polynomials. The R-polynomial of a multichain can be readily defined, and computed from the ordinary R-polynomials (see (9), (10), and (11)). Since several combinatorial formulas and interpretations are known for these polynomials (see, e.g., [5, 10], and, for the case of symmetric groups, [3]) and simple recurrences exist for them, we feel that this formula is a significant step forward in the understanding of the KazhdanLusztig polynomials. Though combinatorial formulas for Kazhdan-Lusztig polynomials have appeared before in the literature (see, e.g., [16, 21, 8, 6]), none of them hold in complete generality.

At present there can be no doubt to say that the quantum group theory has been one of the most important, modern and rapidly developing directions of mathematics and mathematical physics in the end of the twentieth century. Although initially (in the early 1980’s) the quantum group theory was formulated for solving problems in the theory of integrable systems and statistical physics, later the surprising connection of this theory with many branches of mathematics, and the theoretical and mathematical physics was discovered. Today the quantum group theory is connected with such mathematical fields as special functions (especially, with (q-orthogonal polynomials and basic hypergeometric series), the theory of difference and differential equations, combinatorial analysis and representation theory, matrix and operator algebras, noncommutative geometry, knot theory, topology, category theory and so on. From point of view of the mathematical physics there exists interconnection of the quantum group theory with the quantum inverse scattering method, conformai and quantum fields theory and so on. It is expected that the quantum groups will provide deeper understanding of concept of symmetry in physics.KeywordsQuantum GroupSimple RootBasic Hypergeometric SeriesReduction ChainGeneral Projection OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Applications of Frobenius Algebras to Representation Theory of Schur Algebras

The authors present a new way to construct finite dimensional estimation algebra of nonmaximal rank. For a given finite dimensional estimation algebra E, one can construct a finite dimensional estimation algebra E' of nonmaximal rank which is isomorphic to E. The interesting fact is that even if E comes from a linear filtering system, there are arbitrary nonlinear filtering systems associated to E. Since every finite dimensional estimation algebra gives rise to a finite dimensional filter, we have constructed quite a large class of finite dimensional nonlinear filters.

This chapter is a brief introduction into the structure of algebras, mostly finite dimensional, over any field k. The main contents are the Wedderburn theorems for a finite dimensional algebras A over an algebraically closed field k. If A has no nilpotent ideals ≠ 0, then A is a finite product of total matrix algebras over k. In this case, the set d (A) of degrees of the total matrix algebras is a complete set of invariants of A. Thus, 13.7 two finite dimensional semiprime algebras A and B over k are isomorphic if and only if d (A) = d (B).

Déformation par quantification et rigidité des algèbres enveloppantes

In this paper, we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the Euler operator and under-determined partial differential equations. It is proved that if the estimation algebra contains a degree two polynomial, then the Wong Ω-matrix must be a constant matrix. Moreover, all functions in the estimation algebra must be linear functions.

R. W. Brockett (1983) proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow construction of all possible finite dimensional recursive filters from the Lie algebraic point of view. The concept of an estimation algebra with maximal rank was introduced by L. F. Tam et al. (1990). This is the most important general subclass of estimation algebras. S. S.-T. Yau and W.L. Chiou (1991) have already classified all maximal rank finite dimensional estimation algebras with state space dimension at most 2. Here, the case for state space dimension 3 is studied. >

Let A be a finite-dimensional algebra which is quasi-hereditary with re- spect to the poset (�,�), with standard modules �(�) for 2 �. Let F(�) be the category of A-modules which have filtrations where the quotients are standard modules. We determine some inductive results on the relative Auslander-Reiten quiver of F(�). Quasi-hereditary algebras were introduced by Cline, Parshall and Scott in the study of highest weight categories arising in the representation theory of semisimple Lie algebras and algebraic groups (2), and they were studied by Dlab and Ringel and others (4, 5). LetA be quasi-hereditary. Then the simple A-modules are labelled as L(�) forin some partially ordered set (�,≤). Of central importance are the standard modules, denoted by �(�), and the categoryF(�) of modules which have a filtration where the quotients are standard modules. In (12) it was proved that this category has relative Auslander-Reiten (AR) sequences. The aim of our paper is to present some inductive results on the relative Auslander-Reiten quiver ofF(�). For any subset of �, letF (�) be the category of modules inF(�) where all quotients are �(�) with �∈ . If � is an ideal in the poset (�,≤) then there is a subalgebra A of A which is quasi-hereditary with poset (, ≤). In this case, the categoryF (�) is equivalent toF(�A ). A main result in this paper relates, for a non-projective indecomposable mod- ule X inF (�), the two relative Auslander-Reiten sequences with cokernel X inF(�) and inF (�) (Theorem 2.5). By Ringel duality, there is an equivalent version for categories of modules filtered by costandard mod- ules. We describe this and some variations in Section 3. We continue with further results on relative Auslander-Reiten sequences. In particular, we determine certain relative Auslander-Reiten sequences where a projective- injective summand occurs; this generalizes the usual almost split sequence where a projective-injective module occurs.

Let Uζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra mathfrak {g} and a root of unity ζ. When L, L′ are irreducible Uζ-modules having regular highest weights, the dimension of text {Ext}^{n}_{U_{zeta }}(L,L^{prime }) can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of Uζ. This paper shows for L, L′ irreducible modules in a singular block that dim text {Ext}^{n}_{U_{zeta }}(L,L^{prime }) is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. This also computes the corresponding cohomology for q-Schur algebras and many generalized q-Schur algebras. The result depends on a certain parity vanishing property which we obtain from the Kazhdan-Lusztig correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the corresponding affine Lie algebra.

THE aim of this chapter is twofold. On the one hand, we aim to study the 4 convergence of the Dynkin series $$ \begin{array}{*{20}c} {uv: = } & {\sum\limits_{j = 1}^\infty {\left( {\sum\limits_{n = 1}^j {\frac{{\left( { - 1} \right)^{n + 1} }}{n}\,\sum\limits_{\begin{array}{*{20}c} {(h_1,k_1 ), \cdots (h_n,k_n ) \ne (0,0)} \\ {h_1 + k_1 + \cdots + h_n + k_n = j} \\ \end{array}} \times \frac{{(ad\,u)^{h_1 } (ad\,\upsilon )^{k_1 } \cdots (ad\,u)^{h_n } (ad\,\upsilon )^{k_n - 1} (\upsilon )}}{{h_1 ! \cdots h_n !k_1 ! \cdots k_n !(\sum\nolimits_{i = 1}^n {(h_i + k_i )} )}}} } \right),} } \\ \end{array} $$ in various contexts. For instance, this series can be investigated in any nilpotent Lie algebra (over a field of characteristic zero) where it is actually a finite sum, or in any finite dimensional real or complex Lie algebra and, more generally, its convergence can be studied in any normed Banach-Lie algebra (over R or C). For example, the case of the normed Banach algebras (becoming normed Banach-Lie algebras if equipped with the associated commutator) will be extensively considered here.