Schur algebras of Brauer algebras,II

A classical problem of invariant theory and of Lie theory is to determine endomorphism rings of representations of classical groups, for instance of tensor powers of the natural module (Schur–Weyl duality) or of full direct sums of tensor products of exterior powers (Ringel duality). In this article, the endomorphism rings of full direct sums of tensor products of symmetric powers over symplectic and orthogonal groups are determined. These are shown to be isomorphic to Schur algebras of Brauer algebras as defined in Henke and Koenig (Math Z 272(3–4):729–759, 2012). This implies structural properties of the endomorphism rings, such as double centraliser properties, quasi-hereditary, and a universal property, as well as a classification of simple modules.     

Graded Brauer tree algebras

In this paper we construct non-negative gradings on a basic Brauer tree algebra A_Γ corresponding to an arbitrary Brauer tree Γ of type (m,e). We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra A_S, whose tree is a star with the exceptional vertex in the middle, to A_Γ. The grading on A_S comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green’s walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras.We also compute , the group of outer automorphisms that fix the isomorphism classes of simple A_Γ-modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on A_Γ up to graded Morita equivalence and rescaling.     

Brauer algebras of type C

For each $n\ge 2$, we define an algebra satisfying many properties that one might expect to hold for a Brauer algebra of type $C_n$. The monomials of this algebra correspond to scalar multiples of symmetric Brauer diagrams on $2n$ strands. The algebra is shown to be free of rank the number of such diagrams and cellular, in the sense of Graham and Lehrer.     

Quantum Schur algebras revisited

A geometric construction of quantum Schur algebras was given by Beilinson, Lusztig and MacPherson in terms of pairs of flags in a vector space. By viewing such pairs of flags as representations of a poset, we give a recursive formula for the structure constants of quantum Schur algebras which is related to certain Hall polynomials. As an application, we provide a direct proof of the fundamental multiplication formulas which play a key role in the Beilinson-Lusztig-MacPherson realization of quantum gl_n. In the appendix we show how to groupoidify quantum Schur algebras in the sense of Baez, Hoffnung and Walker.     

On the radical of Brauer algebras

The radical of the Brauer algebra is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of . The proof is by direct approach for x  =  0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable -modules. We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some more precise results for the module of pointed chord diagrams, and for the Temperley–Lieb algebra—realised inside —acting on it.

“Ahi quanto a dir che sia è cosa dura lo radical dell’algebra di Brauer pur se’l pensier già muove a congettura” N. Barbecue, “Scholia”

Partially supported by the European RTN “LieGrits”, contract no. MRTN-CT-2003-505078, and by the Italian PRIN 2005 “Moduli e teorie di Lie”.     

Representation type of Schur algebras

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Received October 17, 1997; in final form March 5, 1998     

Brauer algebras of simply laced type

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A _{n − 1} on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A _{n − 1}, D _n , E₆, E₇, E₈). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.     

Integral Schur algebras forGL 2

The structure of Schur algebrasS(2,r) over the integral domainZ is intensively studied from the quasi-hereditary algebra point of view. We introduce certain new bases forS(2,r) and show that the Schur algebraS(2,r) modulo any ideal in the defining sequence is still such a Schur algebra of lower degree inr. A Wedderburn-Artin decomposition ofS _K (2,r) over a fieldK of characteristic 0 is described. Finally, we investigate the extension groups between two Weyl modules and classify the indecomposable Weyl-filtered modules for the Schur algebrasS _Zp(2,r) withr<p ² . Research supported by ARC Large Grant L20.24210     

The Schur Lie-multiplier of Leibniz algebras

Abstract

For a free presentation 0 → τ → → → 0 of a Leibniz algebra , the Baer invariant is called the Schur multiplier of relative to the Liezation functor or Schur Lie-multiplier. For a two-sided ideal of a Leibniz algebra , we construct a four-term exact sequence relating the Schur Lie-multipliers of and /, which is applied to study and characterize Lie-nilpotency, Lie-stem covers and Lie-capability of Leibniz algebras.     

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.     

A criterion on the semisimple Brauer algebras

In this paper, we give a necessary and sufficient condition for a Brauer algebra to be semisimple.