Vertex Algebras and Combinatorial Identities

In the 1980's, J. Lepowsky and R. Wilson gave a Lie-theoretic interpretation of Rogers–Ramanujan identities in terms of level 3 representations of affine Lie algebra sl(2,C)~. When applied to other representations and affine Lie algebras, Lepowsky and Wilson's approach yielded a series of other combinatorial identities of the Rogers–Ramanujan type. At about the same time, R. Baxter rediscovered Rogers–Ramanujan identities within the context of statistical mechanics. The work of R. Baxter initiated another line of research which yielded numerous combinatorial and analytic generalizations of Rogers–Ramanujan identities. In this note, we describe some ideas and results related to Lepowsky and Wilson's approach and indicate the connections with some results in combinatorics and statistical physics.     

The Triangulated Hopf Category n +SL(2)

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.     

A Brauerian representation of split preorders

Split preorders are preordering relations on a domain whose composition is defined in a particular way by splitting the domain into two disjoint subsets. These relations and the associated composition arise in categorial proof theory in connection with coherence theorems. Here split preorders are represented isomorphically in the category whose arrows are binary relations and whose composition is defined in the usual way. This representation is related to a classical result of representation theory due to Richard Brauer. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sklyanin algebras and their symbols

Three-dimensional Sklyanin algebras are a new class of possible counterexamples to the cyclicity problem. In this note, we collect a few observations concerning this problem.     

The four-dimensional Sklyanin algebras

The four-dimensional Sklyanin algebras are certain noncommutative graded algebras having the same Hilbert series as the polynomial ring on four indeterminates. Their structure and representation theory is intimately connected with the geometry of an elliptic curve (and a fixed translation) embedded in ³. This is an account of the work done on these algebras over the past four years.Supported by NSF grant DMS-9100316.     

The first divisible sum

We consider the distribution of the first sum of a sequence of positive integer valued iid random variables which is divisible byd. This is known to converge, when divided byd, to a geometric distribution asd. We obtain results on the rate of convergence using two contrasting approaches. In the first, Stein's method is adapted to geometric limit distributions. The second method is based on the theory of Banach algebras. Each method is shown to have its merits.     

Classification of Cuntz-Krieger algebras

A classification is given of the simple Cuntz-Krieger algebras . It is proved that these algebras are classified up to stable isomorphism by their K₀-group. Thus the sign of the determinant of 1 —A is not an isomorphism invariant. The (non-stabilized) isomorphism type of is determined by K₀( ) together with the position of the class of the unit of .     

On the commutant of hyponormal operators

Let be a pure hyponormal operator with compact self-commutator. We show that the unit ball of the commutant of is compact in the strong operator topology.

     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Actions of Boolean rings on sets

LetB be a Boolean ring (with 1),S a sheaf of sets on the Stone space Spec(B), andS the set of global sections of S. For everya B ands, t S, leta(s, t) denote the element ofS which agrees withs on the support ofa, and witht elsewhere.We set down identities satisfied by this ternary operationB×S×SS (involving also the Boolean operations ofB). For a fixed Boolean ringB, we call a setS given with a ternary operation satisfying these identities aBset. The above construction is shown to give a functorial equivalence between sheaves of setsS on Spec(B) with nonempty sets of global sections, and nonemptyB-setsS. For any setA, the bounded Boolean powerA[B]^* is the freeB-set onA. The varieties ofB-sets, asB ranges over all Boolean rings, constitute (together with one trivial variety) the least nontrivial hypervariety of algebras, in the sense of W. Taylor.This work was done while the author was partly supported by NSF contract DMS 85-02330.Presented by R. S. Pierce.