UNT and TAMU Workshop on Algebras

Saturday, April 23, 2011
University of North Texas, Denton, Texas

We will meet for coffee/tea in General Academic Building (GAB) Room 472 at 8:45 AM on Saturday, April 23rd. Talks will be held in GAB Room 406.

Parking

Program

• 9:00-9:20    Jeanette Shakalli (TAMU): Universal Deformation Formulas
• In 1998, Giaquinto and Zhang developed the theory of universal deformation formulas based on a bialgebra B. Such a formula is universal in the sense that it applies to any B-module algebra to yield a formal deformation. In this short talk, I will define the formal deformation of an algebra and the universal deformation formula. Then I will present a deformation formula derived from the action of a Hopf algebra, which was obtained by Witherspoon in 2006, and illustrate this concept by giving an example.
• 9:40-10:00  Philip Puente (UNT):  Restricted Lie Algebras in Positive Characteristic
• In this talk, we consider deformations of polynomial rings in positive characteristic with a view toward their centers.  We define a restricted Lie Algebra which gives rise to a special type of deformation, in which the commutator of two elements in a vector space is replaced with a linear term.  We give some concrete examples and examine centers that arise.
• 10:20-10:40 Christine Uhl (UNT): Groups Acting on Restricted Lie Algebras and Centers of Deformations
• In this talk, we will look at skew group algebras formed by a finite group acting on a restricted Lie Algebra.  We give some concrete examples with an eye toward computing the centers of these algebras.
• 11:00-11:20  Deepak Naidu (TAMU): Quantum Drinfeld Hecke Algebras
  
• Let G be a group acting on the quantum symmetric algebra S_q(V). I will introduce the quantum Drinfeld Hecke Algebras. These objects manifest themselves as certain types of deformations of the skew group algebras S_q(V)#G. I will present examples arising from the symmetric groups.
  
• 11:40-12:00   Briana Foster-Greenwood (UNT): Invariant Theory and Hochschild Cohomology of Skew Group Algebras
• Hochschild cohomology abstractly measures when and how one can deform the multiplication in an algebra while maintaining associativity.  In the context of skew group algebras S(V)#G (combining a group G with a polynomial ring S(V) on which the group acts), Shepler and Witherspoon give an invariant theoretic formulation of the Hochschild cohomology groups.  With an eye towards deformations of S(V)#G for G a tetrahedral complex reflection group, I will highlight some tools from invariant theory (e.g. characters and Molien series) useful in computing Hochschild cohomology of skew group algebras.

• 2:00-2:20    Van Nguyen (TAMU): Cohomology of Hopf Algebras
• Conjecture: "The cohomology of a finite dimensional Hopf Algebra is finitely generated." This statement is still an open question. However, there are several modified cases when it is known true. In this talk, I will introduce some basic concepts in homological algebra, mainly focus on examples of cohomology of an augmented algebra, cohomology of finite groups (as we recall kG is a Hopf Algebra). After that, we will look at some conditions for a Hopf Algebra A such that H*(A) is finitely generated.
• 2:40-3:00  Piyush Shroff (TAMU): Cohomology of Quotients of Quantum Symmetric Algebra
• Let A=B/(x_1^{N_1},...,x_r^{N_r}) where B is PBW algebra. My goal is to show that Ext_A^{*}(k,k) = H^{*}(A,k) is finitely generated. In this talk I will give some basic definitions and show that H^{*}(S,k) is finitely generated with the help of techniques of Mastnak, Pevtsova, Schauenburg and Witherspoon where S is quotient of quantum symmetric algebra.
• 3:20-3:40   Rabin Dahal (UNT): Representations of sl(2)
• We will give an introduction to finite dimensional representations of the Lie algebra sl(2), including their semisimplicity and the classification of the irreducible representations.  Time permitting, we will discuss tensor products and the universal enveloping algebra.
  
• 4:00-4:20   Jeannette Larsen (UNT): Tensor Density Representations and Length Two Representations of Vec R 
• In this talk we will discuss the Lie algebra Vec R, its subalgebra a, and the tensor density representations of VecR of the type (π_λ , F (λ)) for λ a complex number.   I will show when such representations are irreducible and when they are equivalent (as both VecR and a representations).  We will then extend to VecR representations of length two, built out of the above length-one representations, and examine the cohomology of these length-two representations, particularly in the case when they are a-relative.
• 4:40-5:00   Jose Franco (Baylor): Global SL(2,R) representations of the Schrodinger equation with time-dependent potentials.
• We study the representation theory of the solution space of the one-dimensional Schrodinger equation with time-dependent potentials that posses sl(2,R)-symmetry. We give explicit local intertwining maps to multiplier representations and show that the study of the solution space for potentials of the form V(t,x)=g_2(t)x^2+g_1(t)x+g_0(t) reduces to the study of the potential free case. We also show that the study of the time-dependent potentials of the form V(t,x)= c x^(-2)+g_2(t)x^2+g_0(t) reduces to the study of the potential V(t,x)=c x^(-2). Therefore, we study the representation theory associated to solutions of the Schrodinger equation with this potential. The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category.

Restaurants

• Dinner  Oriental Garden
• Lunch  Cool Beans

• La Quinta
• Holiday Inn Express
• Courtyard Marriott