PBW deformations of quadratic monomial algebras

A result of Braverman and Gaitsgory from 1996 gives necessary and sufficient conditions for a filtered algebra to be a Poincaré-Birkhoff-Witt (PBW) deformation of a Koszul algebra. The main theorem in this paper establishes conditions equivalent to the Braverman-Gaitsgory Theorem to efficiently determine PBW deformations of quadratic monomial algebras. In particular, a graphical interpretation is presented for this result, and we discuss circumstances under which some of the conditions of this theorem need not be checked. Several examples are also provided. Finally, with these tools, we show that each quadratic monomial algebra admits a nontrivial PBW deformation.     

Hopf algebra deformations of binary polyhedral groups

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a two-dimensional irreducible comodule in the finite-dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras.     

Hochschild homology criteria for trivial algebra structures

We prove two similar results by quite different methods. The first one deals with augmented artinian algebras over a field: we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the Hochschild homology (or cyclic homology) groups. For the second result, let be a 1-connected finite CW-complex. We characterize the trivial algebra structure on the cohomology algebra of with coefficients in a fixed field in terms of the maximality of the Betti numbers of the free loop space.

     

Weak flatness criteria for codimension 2 spheres in codimension 1 manifolds

The “Kiee problem” is to determine which submanifolds of Sⁿ have the property that all their subspheres and subcells are flat. In this paper we study the related problem of determining submanifolds of Sⁿ having all subspheres weakly flat and all subcells cellular.     

Existence criteria for cycles in two-dimensional quadratic systems

The existence of a bifurcation of simultaneously arising cycles in a neighborhood of two equilibrium states of two-dimensional quadratic systems under a variation of one scalar parameter is proved.     

Finite criteria for conditional definiteness of quadratic forms

A real symmetric n × n matrix Q is A-conditionally positivesemidefinite, where A is a given m × n matrix, if x′Qx?0 whenever Ax?0, and is A-conditionally positive definite if strict inequality holds except when x=0. When A is the identity matrix these notions reduce to the well-studied notions of copositivity and strict copositivity respectively. This paper presents finite criteria, involving only the solution of sets of linear equations constructed from the matrices Q,A, for testing both types of conditional definiteness. These criteria generalize known facts about copositive matrices and, when Q is invertible and all row submatrices of A have maximal rank, can be very elegantly stated in terms of Schur complements of the matrix AQ^-1A′.     

Flat grafting deformations of quadratic differentials on surfaces

Geometriae Dedicata - In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces....     

Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras. Translated fromMatemalicheskie Zametki, Vol. 63, No. 1, pp. 9–20, January, 1998. The author gladly expresses his sincere gratitude to A. Ya. Khelemskii and Yu. V. Selivanov for their assistance in this work. In particular, A. Ya. Khelemskii indicated how to simplify the proof of Theorem 1. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00156, by the International Science Foundation under grant No. M95000, and by the INTAS fund under grant No. 93-1376.     

Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras.     

Minimum deformations of commutative algebra and linear groupGL(n)

In the algebra of formal seriesM _q (x ⁱ ), the relations of generalized commutativity that preserve the tensorI _q grading and depend on parametersq(i, k) are considered. A norm of the differential calculus onM _q consistent with theI _q grading is chosen. A new construction of a symmetrized tensor product of algebras of the typeM _q (x ⁱ ) and a corresponding definition of the minimally deformed linear groupQGL(n) and Lie algebraqgl(n) are proposed. A study is made of the connection ofQGL(n) andqgl(n) with the special matrix algebra Mat(n, Q), which consists of matrices with noncommuting elements. The deformed determinant in the algebra Mat(n, Q) is defined. The exponential mapping in the algebra Mat(n, Q) is considered on the basis of the Campbell-Hausdorff formula.Institute of Applied Physics, Tashkent State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 403–417, June, 1993.     

An affine prime non-semiprimitive monomial algebra with quadratic growth

We give a simple construction of a prime monomial algebra with quadratic growth, which is neither primitive nor PI.     

An affine prime non-semiprimitive monomial algebra with quadratic growth

We give a simple construction of a prime monomial algebra with quadratic growth, which is neither primitive nor PI.     

The alternative Clifford algebra of a ternary quadratic form

We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion ring whose center is the ring of polynomials in one variable over the field of definition.