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PhD / None.

PhD / None.

Indiana University-Purdue University Indianapolis (IUPUI) / We introduce the quantum toroidal superalgebra E(m|n) associated with the Lie superalgebra gl(m|n) and initiate its study. For each choice of parity "s" of gl(m|n), a corresponding quantum toroidal superalgebra E(s) is defined. To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed. The superalgebra E(s) contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra Uq sl̂(m|n) with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E(s), which exchanges the vertical and horizontal subalgebras. If m and n are different and "s" is standard, we give a construction of level 1 E(m|n)-modules through vertex operators. We also construct an evaluation map from E(m|n)(q1,q2,q3) to the quantum affine algebra Uq gl̂(m|n) at level c=q3^(m-n)/2.

A partir de um grupo H e um elemento h em H, nós definimos uma representação : \'B IND. n\' Aut(\'H POT. n\' ), onde \'B IND. n\' denota o grupo de trança de n cordas, e \'H POT. n\' denota o produto livre de n cópias de H. Chamamos a a representação de tipo Artin associada ao par (H, h). Nós também estudamos varios aspectos de tal representação. Primeiramente, associamos a cada trança um grupo \' IND. (H,h)\' () e provamos que o operador \' IND. (H,h)\' determina um grupo invariante de enlaçamentos orientados. Então damos uma construção topológica da representação de tipo Artin e do invariante de enlaçamentos \' IND.(H,h)\' , e provamos que a representação é fiel se, e somente se, h é não trivial / From a group H and a element h H, we define a representation : \' B IND. n\' Aut(\'H POT. n\'), where \'B IND. n\' denotes the braid group on n strands, and \'H POT. n\' denotes the free product of n copies of H. We call the Artin type representation associated to the pair (H, h). Here we study various aspects of such representations. Firstly, we associate to each braid a group \' IND. (H,h)\' () and prove that the operator \' IND. (H,h)\' determines a group invariant of oriented links. We then give a topological construction of the Artin type representations and of the link invariant \' iND. (H,h)\' , and we prove that the Artin type representations are faithful if and only if h is nontrivial

In this thesis, we classify finite orbits of the action of the pure braid group over a certain large open subset of the SL(2,C) character variety of the Riemann sphere with five boundary components, i.e. Σ5. This problem arises in the context of classifying algebraic solutions of the Garnier system G2, that is the two variable analogue of the famous sixth Painleve equation PVI. The structure of the analytic continuation of these solutions is described in terms of the action of the pure braid group on the fundamental group of Σ5. To deal with this problem, we introduce a system of co-adjoint coordinates on a big open subset of the SL(2,C) character variety of Σ5. Our classifica- tion method is based on the definition of four restrictions of the action of the pure braid group such that they act on some of the co-adjoint coordi- nates of Σ5 as the pure braid group acts on the co-adjoint coordinates of the character variety of the Riemann sphere with four boundary components, i.e. Σ4, for which the classification of all finite orbits is known. In order to avoid redundant elements in our final list, a group of symmetries G of the large open subset is introduced and the final classification is achieved modulo the action of G. We present a final list of 54 finite orbits.

A partir de um grupo H e um elemento h em H, nós definimos uma representação : \'B IND. n\' Aut(\'H POT. n\' ), onde \'B IND. n\' denota o grupo de trança de n cordas, e \'H POT. n\' denota o produto livre de n cópias de H. Chamamos a a representação de tipo Artin associada ao par (H, h). Nós também estudamos varios aspectos de tal representação. Primeiramente, associamos a cada trança um grupo \' IND. (H,h)\' () e provamos que o operador \' IND. (H,h)\' determina um grupo invariante de enlaçamentos orientados. Então damos uma construção topológica da representação de tipo Artin e do invariante de enlaçamentos \' IND.(H,h)\' , e provamos que a representação é fiel se, e somente se, h é não trivial / From a group H and a element h H, we define a representation : \' B IND. n\' Aut(\'H POT. n\'), where \'B IND. n\' denotes the braid group on n strands, and \'H POT. n\' denotes the free product of n copies of H. We call the Artin type representation associated to the pair (H, h). Here we study various aspects of such representations. Firstly, we associate to each braid a group \' IND. (H,h)\' () and prove that the operator \' IND. (H,h)\' determines a group invariant of oriented links. We then give a topological construction of the Artin type representations and of the link invariant \' iND. (H,h)\' , and we prove that the Artin type representations are faithful if and only if h is nontrivial

Crystallographic complex reflection groups are generated by reflections about affine hyperplanes in complex space and stabilize a full rank lattice. These analogs of affine Weyl groups have infinite order and were classified by V.L. Popov in 1982. The classical Braid theorem (first established by E. Artin and E. Brieskorn) asserts that the Artin group of a reflection group (finite or affine Weyl) gives the fundamental group of regular orbits. In other words, the fundamental group of the space with reflecting hyperplanes removed has a presentation mimicking that of the Coxeter presentation; one need only remove relations giving generators finite order. N.V Dung used a semi-cell construction to prove the Braid theorem for affine Weyl groups. Malle conjectured that the Braid theorem holds for all crystallographic complex reflection groups after constructing Coxeter-like reflection presentations. We show how to extend Dung's ideas to crystallographic complex reflection groups and then extend the Braid theorem to some groups in the infinite family [G(r,p,n)]. The proof requires a new classification of crystallographic groups in the infinite family that fail the Steinberg theorem.

<div> We introduce the quantum toroidal superalgebra E_m|nassociated with the Lie superalgebra gl_m|n and initiate its study. For each choice of parity "s" of gl_m|n, a corresponding quantum toroidal superalgebra E_s is defined. </div><div> </div><div><br></div><div>To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed. </div><div><br></div><div>The superalgebra E_s contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra U_q sl̂_m|n with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E_s, which exchanges the vertical and horizontal subalgebras.</div><div><br></div><div>If <i>m</i> and <i>n</i> are different and "s" is standard, we give a construction of level 1 E_m|n-modules through vertex operators. We also construct an evaluation map from E_m|n(q₁,q₂,q₃) to the quantum affine algebra U_q gl̂_m|n at level c=q₃^(m-n)/2.</div>

We prove that the non-k-equal configuration space of a graph has a discretized model, analogous to the discretized model for configurations on graphs. We apply discrete Morse theory to the latter to give an explicit combinatorial formula for the ranks of homology and cohomology of configurations of two points on a tree. We give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on a few simple trees, and show that the first homology group of ordered and unordered configurations of two points in any tree is generated by the first homology groups of configurations of two points in three particular graphs, K_{1,3}, K_{1,4}, and the trivalent tree with 6 vertices and 2 vertices of degree 3, via graph embeddings.