Iordan Ganev

The quantum Frobenius for character varieties and multiplicative quiver varieties
With D. Jordan and P. Safronov. Journal of the European Mathematical Society (2024), p.1-62.

Abstract: We develop a general mechanism for constructing sheaves of Azumaya algebras on moduli spaces obtained by Hamiltonian reduction, via their quantizations at roots of unity. We achieve this by exploiting a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of Frobenius Poisson orders. We use these tools to construct canonical central subalgebras of quantum algebras, and explicitly compute the resulting Azumaya loci we encounter, using a natural nondegeneracy assumption. As our main applications, we prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces.

Wonderful asymptotics of matrix coefficients \(\mathcal{D}\)-modules
With D. Ben-Zvi. Advances in Mathematics, Volume 408, Part A (2022).

Abstract: The Beilinson-Bernstein localization realizes representations of complex reductive Lie algebras as monodromic \(\mathcal{D}\)-modules on the basic affine space \(G/N\), a torus bundle over the flag variety. The same space appears naturally (in a doubled version) as the horocycle space describing the geometry of the reductive group G at infinity, near the closed stratum of the wonderful compactification \(\overline{G}\), or equivalently in the special fiber of the Vinberg semigroup of \(G\). We show that Beilinson-Bernstein localization for arbitrary \(U\mathfrak{g}\)-bimodules arises naturally as the specialization at infinity in \(\overline{G}\) of the D-modules on \(G\) describing matrix coefficients of Lie algebra representations. More generally, the asymptotics of matrix coefficient D-modules along any stratum of \(\overline{G}\) are given by the matrix coefficient D-modules for parabolic restrictions. The result is an elementary consequence of the Rees construction relating \(\overline{G}\) and the Vinberg degeneration of \(G\). This suggests an analog of scattering for the multi-temporal wave equation, with times given by the Vinberg degeneration, and provides a simple algebraic interpretation of aspects of harmonic analysis on real reductive groups.

Quantum Weyl algebras and reflection equation algebras at a root of unity. With N. Cooney and D. Jordan. Journal of Pure and Applied Algebra, Volume 224, Issue 12 (2020).

Abstract: We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank \(N\), and quantum differential operators on the quantum group \(GL_2\). These examples illustrate in elementary terms much more general phenomena explored further in other works.

The wonderful compactification for quantum groups
Journal of the London Mathematical Society, Volume 99, Issue 3 (2019), Pages 778–806.

Abstract: In this paper, we introduce a quantum version of the wonderful compactification of a group as a certain noncommutative projective scheme. Our approach stems from the fact that the wonderful compactification encodes the asymptotics of matrix coefficients, and from its realization as a GIT quotient of the Vinberg semigroup. In order to define the wonderful compactification for a quantum group, we adopt a generalized formalism of \(\mathsf{Proj}\) categories in the spirit of Artin and Zhang. Key to our construction is a quantum version of the Vinberg semigroup, which we define as a q-deformation of a certain Rees algebra, compatible with a standard Poisson structure. Furthermore, we discuss quantum analogues of the stratification of the wonderful compactification by orbits for a certain group action, and provide explicit computations in the case of \(SL_2\).

Quantizations of multiplicative hypertoric varieties at a root of unity.
Journal of Algebra, Volume 506 (2018), Pages 92–128.

Abstract: We construct quantizations of multiplicative hypertoric varieties using an algebra of \(q\)-difference operators on affine space, where \(q\) is a root of unity in \(\mathbb{C}\). The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.

Groups of a square-free order.
Rose-Hulman Undergraduate Math. Journal, Vol. 11, Issue 1 (2010).

Abstract: Hölder's formula for the number of groups of a square-free order is an early advance in the enumeration of finite groups. This paper gives a structural proof of Hölder's result that is accessible to undergraduates. We introduce a number of group theoretic concepts such as nilpotency, the Fitting subgroup, and extensions. These topics, which are usually not covered in undergraduate group theory, feature in the proof of Hölder's result and have wide applicability in group theory. Finally, we remark on further results and conjectures in the enumeration of finite groups.

Order dimension of subgroups.
Rose-Hulman Undergraduate Math. Journal, Vol. 9, Issue 2 (2008).

Abstract: The number of different orders of nonidentity elements in a group is limited by the number divisors of the order of the group. This upper bound can be made more specific for proper subgroups, and can be calculated from the prime power factorization of the group's order. Some groups have subgroups with the highest possible number of different orders for nonidentity elements. This property can be characterized and general results exist for several families of groups.

The wonderful compactification for quantum groups.
University of Texas at Austin, 2016.

Abstract: This thesis studies the asymptotics of quantum groups using an approach centered on the wonderful compactification. The wonderful compactification of a semisimple group was introduced by De Concini and Procesi, and has become an important tool in geometric representation theory. We provide an exposition of several constructions of the wonderful compactification in order to illustrate how it links the geometry of the group to the geometry of its partial flag varieties, and how it encodes the asymptotics of matrix coefficients for the group. We then construct quantum group versions of the wonderful compactification, its associated Vinberg semigroup, its stratification by \(G × G\) orbits, and its algebra of differential operators. A key technical aspect of our approach is the notion of a noncommutative projective scheme associated to a ring graded by a lattice. We provide explicit descriptions of our constructions in the case of \(SL_2\), explain connections to previous work on the flag variety of a quantum group, and discuss conjectural applications of the newly-defined objects that appear in this thesis.