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Anthony Giaquinto

Publications and source records attributed to Anthony Giaquinto.

27 records · Page 2Linked to original sources
arXiv2002

Presenting Schur Algebras

Motivated by work of R.M. Green, we obtain a presentation of Schur algebras (both the classical and quantized versions) in terms of generators and relations. The presentation is compatible with the usual presentation of the (quantized or classical) enveloping algebra of gl(n). As a result, we obtain a new ``integral'' basis for Schur algebras which is a subset of Kostant's basis of the integral form of the enveloping algebra (or its q-analogue). Projection onto an appropriate component gives a new "integral" basis and a presentation for the Hecke algebra, compatible with the basis and presentation for the Schur algebra. Finally, we find a second presentation of Schur algebras which is similar to Luzstig's modified form of the quantized enveloping algebra.

math.RT↗
arXiv2001

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.

math.RT↗
arXiv2001

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.

math.RT↗
arXiv2000

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.

math.RT↗
arXiv2000

The Donald-Flanigan problem for finite reflection groups

The Donald--Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and for the classical Weyl groups (whose rational group algebras are also computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras then there is also a solution for the wreath product of H with any symmetric group, abelian group, or dihedral group. The theorems suggested by the Donald-Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.

math.QA↗
arXiv1997

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 $sl(3) \wedge sl(3)$.

q-alg↗
arXiv1996

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.

q-alg↗
arXiv1994

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.

hep-th↗