Topics in Algebraic Deformation Theory

We give a selective survey of topics in algebraic deformation theory ranging from its inception to current times. Throughout, the numerous contributions of Murray Gerstenhaber are emphasized, especially the common themes of cohomology, infinitesimal methods, and explicit global deformation formulas.Comment: To appear in "Higher Structures in Geometry and Physics", papers in honor of M. Gerstenhaber and J. Stasheff, A. Cattanneo, A. Giaquinto, and P. Xu, eds. Birkhauser Progress in Mathematics, 201

Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl(2)

We obtain a presentation of quantum Schur algebras (over the field Q(v)) by generators and relations. This presentation is compatible with the usual presentation of the quantized universal enveloping algebra of the Lie algebra gl(2). We also locate the ``integral'' form of the quantum Schur algebra within the presented algebra and show it has a basis which is closely related to Lusztig's basis of the integral form of the quantized enveloping algebra.Comment: 22 pages, submitted to the Journal of Algebra, (proof of Theorem 2.6 revised

Boundary solutions of the quantum Yang-Baxter equation and solutions in three dimensions

Boundary solutions to the quantum Yang-Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the ``modified'' qYB equation, the latter being analogous to the modified classical Yang-Baxter (cYB) equation. We construct, for a large class of solutions $r$ to the modified cYB equation, explicit ``boundary quantizations'', i.e., boundary solutions to the qYB equation of the form $I+tr+ t^2r_{2} + ...$. In the last section we list and give quantizations for all classical r-matrices in $sl(3) \wedge sl(3)$.Comment: 9 pages, AMS-LaTe

Generators and relations for Schur algebras

We obtain a presentation of Schur algebras (and q-Schur algebras) by generators and relations which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra gl(n) of n x n matrices. We also find several new bases of Schur algebras and their corresponding integral forms.Comment: 9 page

Cellular bases of generalized q-Schur algebras

We show that cellular bases of generalized $q$-Schur algebras can be constructed by gluing arbitrary bases of the cell modules and their dual basis (with respect to the anti-involution giving the cell structure) along defining idempotents. For the rational form, over the field $\mathbb{Q}(v)$ of rational functions in an indeterminate $v$, our proof of this fact is self-contained and independent of the theory of quantum groups. In the general case, over a commutative ring $\Bbbk$ regarded as a $\mathbb{Z}[v,v^{-1}]$-algebra via specialization $v \mapsto q$ for some chosen invertible $q \in \Bbbk$, our argument depends on the existence of the canonical basis.Comment: 33 page

Graphs, Frobenius functionals, and the classical Yang-Baxter equation

A Lie algebra is Frobenius if it admits a linear functional F such that the Kirillov form F([x,y]) is non-degenerate. If g is the m-th maximal parabolic subalgebra P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a "cyclic" functional F on P(n,m) and prove it is non-degenerate using properties of certain graphs associated to F. These graphs also provide in some cases readily computable associated solutions of the classical Yang-Baxter equation. We also define a local ring associated to each connected loopless graph from which we show that the graph can be reconstructed. Finally, we examine the seaweed Lie algebras of Dergachev and Kirillov from our perspective

Boundary Solutions of the Classical Yang-Baxter Equation

We define a new class of unitary solutions to the classical Yang-Baxter equation (CYBE). These ``boundary solutions'' are those which lie in the closure of the space of unitary solutions to the modified classical Yang-Baxter equation (MCYBE). Using the Belavin-Drinfel'd classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer-Gervais solution to the MCYBE, we explicitly construct for all n > 2 a boundary solution based on the maximal parabolic subalgebra of sl(n) obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to $n$. We also give examples of non-boundary solutions for the classical simple Lie algebras.Comment: 21 pages, AMSTEX, A version of this report will appear in Lett. Math. Physic

Presenting Schur algebras as quotients of the universal enveloping algebra of gl(2)

We give a presentation of Schur algebras (over the rational number field) by generators and relations, in fact a presentation which is compatible with Serre's presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for Schur algebras, a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant's Z-form, and show that it has an integral basis which is a truncated version of Kostant's basis.Comment: 22 pages; submitted to Algebras and Representation Theor

Nonstandard solutions of the Yang-Baxter equation

Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5).Comment: 8 page

Bialgebra actions, twists, and universal deformation formulas

We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation produced from a universal deformation formula (UDF) is a special case of a twist. The most familiar example of a deformation produced from a UDF is perhaps the "Moyal product" which (locally) is the canonical quantization of the algebra of functions on a symplectic manifold in the direction of the Poisson bracket. In this case, the derivations comprising the Poisson bracket mutually commute and so this quantization is essentially obtained by exponentiating this bracket. For more general Poisson manifolds, this formula is not applicable since the associated derivations may no longer commute. We provide here generalizations of the Moyal formula which (locally) give canonical quantizations of various Poisson manifolds. Specifically, whenever a certain central extension of a Heisenberg Lie group acts on a manifold, we obtain a quantization of its algebra of functions in the direction of a suitable Poisson bracket obtained from noncommuting derivations.Comment: 21 pages, AMSTEX, no figure