Diagonal Algebras Research Articles

Diagonal tensor algebra and naïve liftings

The notion of naïve lifting of differential graded (DG) modules along certain DG algebra extensions was introduced by the authors for the purpose of studying problems in homological commutative algebra that involve self-vanishing of Ext. Our goal in this paper is to deeply study naïve lifting property using the diagonal tensor algebra. Our main result provides several characterizations of naïve liftability of DG modules under certain Ext vanishing conditions. As an application, we affirmatively answer a question posed by the authors regarding a specific characterization of naïve liftability.

Generalized F-depth and graded nilpotent singularities

In this article, we provide explicit and effective bounds for Frobenius test exponents, integers crucial for computing Frobenius closures of parameter ideals, in new settings and articulate their explicit computation along common graded constructions. We do this via the study of variants of F-nilpotent singularities. In particular, we explore how (generalized) weakly F-nilpotent singularities behave under Segre products, Veronese subrings, and the formation of diagonal hypersurface algebras. To accomplish these tasks, we introduce the generalized F-depth in analogy to Lyubeznik’s F-depth. These depth-like invariants track (generalized) weakly F-nilpotent singularities in a similar fashion as (generalized) depth tracks (generalized) Cohen-Macaulay singularities.

Deformations of modified $r$-matrices and cohomologies of related algebraic structures

Modified r -matrices are solutions of the modified classical Yang–Baxter equation, introduced by Semenov-Tian-Shansky, and play important roles in mathematical physics. In this paper, first we introduce a cohomology theory for modified r -matrices. Then we study three kinds of deformations of modified r -matrices using the established cohomology theory, including algebraic deformations, geometric deformations and linear deformations. We give the differential graded Lie algebra that governs algebraic deformations of modified r -matrices. For geometric deformations, we prove the rigidity theorem and study when is a neighborhood of a modified r -matrix smooth in the space of all modified r -matrix structures. In the study of trivial linear deformations, we introduce the notion of a Nijenhuis element for a modified r -matrix. Finally, applications are given to study deformations of the complement of the diagonal Lie algebra and compatible Poisson structures.

Operator algebra generalization of a theorem of Watrous and mixed unitary quantum channels

We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.

Regularity and stable ranges of FI-modules

We give refined bounds for the regularity of FI-modules and the stable ranges of FI-modules for various forms of their stabilization studied in the representation stability literature. We show that our bounds are sharp in several cases. We apply these to get explicit stable ranges for diagonal coinvariant algebras, and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.

Quotient gradings and the intrinsic fundamental group

Quotient grading classes are essential participants in the computation of the intrinsic fundamental group π1(A) of an algebra A. In order to study quotient gradings of a finite-dimensional semisimple complex algebra A it is sufficient to understand the quotient gradings of twisted group algebra gradings. We establish the graded structure of such quotients using Mackey's obstruction class. Then, for matrix algebras A=Mn(C) we tie up the concepts of braces, group-theoretic Lagrangians and elementary crossed products. We also manage to compute the intrinsic fundamental group of the diagonal algebras A=C4 and A=C5.

On a [formula omitted]-Mixing property for Entangled Markov Chains

We study a mixing condition for entangled Markov chains, so-called ψ-mixing property. We prove that every entangled Markov chain, whose Markov operators satisfy the Markov–Dobrushin condition, is ψ-mixing. Moreover, we show the restriction of the underlying QMC to the diagonal algebra gives rise to a classical mixing Markov chains with the same exponential rate of convergence.

ON SHAVRUKOV’S NON-ISOMORPHISM THEOREM FOR DIAGONALIZABLE ALGEBRAS

AbstractWe prove a strengthened version of Shavrukov’s result on the non-isomorphism of diagonalizable algebras of two $\Sigma _1$ -sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms from the algebra $(\mathfrak {L}_T, \Box _T\Box _T)$ onto the algebra $(\mathfrak {L}_T, \Box _T)$ . The case of bimodal diagonalizable algebras is also considered. We give several examples of pairs of theories with isomorphic diagonalizable algebras but non-isomorphic bimodal diagonalizable algebras.

Homology of étale groupoids a graded approach

We introduce a graded homology theory for graded étale groupoids. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to the homology of its ε-th component with ε the identity of the grade group. For Z-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.

Algebras of Non-Local Screenings and Diagonal Nichols Algebras

In a vertex algebra setting, we consider non-local screening operators associated to the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated to a diagonal braiding, which encodes the non-locality and non-integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as classified by Heckenberger, and find all lattice realizations of the braiding that are compatible with reflections. Usually, the realizations are unique or come as one- or two-parameter families. Examples include realizations of Lie superalgebras. We then study the associated algebra of screenings with improved methods. Typically, for positive definite lattices we obtain the Nichols algebra, such as the positive part of the quantum group, and for negative definite lattices we obtain a certain extension of the Nichols algebra generalizing the infinite quantum group with a large center.

Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $$( \mathfrak{G} , \mathfrak{g} )$$ has previously been considered by Khoroshkin and Ogievetsky in the case of the diagonal reduction algebra for $$\mathfrak{gl}(n)$$ . In this paper, we consider the diagonal reduction algebra of the pair of Lie superalgebras $$( \mathfrak{osp}(1|2) \times \mathfrak{osp}(1|2) , \mathfrak{osp}(1|2) )$$ as a double coset space having an associative $$ \mathrel{\scriptstyle\lozenge} $$ -product and give a complete presentation in terms of generators and relations. We also provide a PBW basis for this reduction algebra along with Casimir-like elements and a subgroup of automorphisms.

Diagonal decompositions of shapes and their algebras

AbstractThe formal approach to shapes and their algebras, as it appears in shape grammar theory, has been reviewed. It starts with geometric elements and their partial algebras, continues to shapes, their algebras, and boundaries, as well as algebras that calculate with shapes and their boundaries. There is a number of new concepts introduced along the way. These include diagonal decompositions and their algebras which simplify calculations with shapes, b-paired diagonal decompositions which extend calculations with shapes and their boundaries from diagonal shapes only to all shapes, andm-order boundaries which extend the concept of shape boundaries and allow for calculations with multiple representations of shapes. It also shows that algebras of shapes are infinite direct sums of diagonal algebras.

Diagonalizability of Quantum Markov States on Trees

We introduce quantum Markov states (QMS) in a general tree graph $$G= (V, E)$$ , extending the Cayley tree’s case. We investigate the Markov property w.r.t. the finer structure of the considered tree. The main result of the present paper concerns the diagonalizability of a locally faithful QMS $$\varphi $$ on a UHF-algebra $${\mathcal {A}}_V$$ over the considered tree by means of a suitable conditional expectation into a maximal abelian subalgebra. Namely, we prove the existence of a Umegaki conditional expectation $${\mathfrak {E}} : {\mathcal {A}}_V \rightarrow {\mathcal {D}}_V$$ such that $$\varphi =\varphi _{\lceil {\mathcal {D}}_V}\circ {\mathfrak {E}}$$ . Moreover, it is clarified the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra $${\mathcal {D}}_V$$ .

On real algebras generated by positive and nonnegative matrices

Algebras generated by strictly positive matrices are described up to similarity, including the commutative, simple, and semisimple cases. We provide sufficient conditions for some block diagonal matrix algebras to be generated by a set of nonnegative matrices up to similarity. Also we find all realizable dimensions of algebras generated by two nonnegative semi-commuting matrices. The last result provides the solution to the problem posed by M. Kandić, K. Šivic (2017) [13].

Representation stability of the cohomology of Springer varieties and some combinatorial consequences

A sequence of $$S_n$$ -representations $$\{V_n\}$$ is said to be uniformly representation stable if the decomposition of $$V_n = \bigoplus _{\mu } c_{\mu ,n} V(\mu )_n$$ into irreducible representations is independent of n for each $$\mu $$ —that is, the multiplicities $$c_{\mu ,n}$$ are eventually independent of n for each $$\mu $$ . Church–Ellenberg–Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of Church–Ellenberg–Farb). We also explore some combinatorial consequences of this stability.

Equilibrium states and entropy theory for Nica-Pimsner algebras

We study the equilibrium simplex of Nica-Pimsner algebras arising from product systems of finite rank on the free abelian semigroup. First we show that every equilibrium state has a convex decomposition into parts parametrized by ideals on the unit hypercube. Secondly we associate every gauge-invariant part to a sub-simplex of tracial states of the diagonal algebra. We show how this parametrization lifts to the full equilibrium simplices of non-infinite type.The finite rank entails an entropy theory for identifying the two critical inverse temperatures: (a) the least upper bound for existence of non finite-type equilibrium states, and (b) the least positive inverse temperature below which there are no equilibrium states at all. We show that the first one can be at most the strong entropy of the product system whereas the second is the infimum of the tracial entropies (modulo negative values). Thus phase transitions can happen only in-between these two critical points and possibly at zero temperature.

Braid Group Action on Affine Yangian

We study braid group actions on Yangians associated with symmetrizable Kac-Moody Lie algebras. As an application, we focus on the affine Yangian of type A and use the action to prove that the image of the evaluation map contains the diagonal Heisenberg algebra inside $\hat{\mathfrak{gl}}_N$.

A proof of the shuffle conjecture

We present a proof of the compositional shuffle conjecture by Haglund, Morse and Zabrocki [Canad. J. Math., 64 (2012), 822–844], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra by Haglund, Haiman, Loehr, Remmel, and Ulyanov [Duke Math. J., 126 (2005), 195–232]. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector spaceV∗V_*whose degree zero part is the ring of symmetric functionsSym⁡[X]\operatorname {Sym}[X]overQ(q,t)\mathbb {Q}(q,t). We then extend these operators to an action of an algebraA~\tilde {\mathbb A}acting on this space, and interpret the right generalization of the∇\nablausing an involution of the algebra which is antilinear with respect to the conjugation(q,t)↦(q−1,t−1)(q,t)\mapsto (q^{-1},t^{-1}).

Diagonal reduction algebra and the reflection equation

We describe the diagonal reduction algebra D(gl n ) of the Lie algebra gl n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl n ).

The Center of Small Quantum Groups I: The Principal Block in Type A

We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The exemplary case of $\mathfrak{sl}_3$ is computed explicitly, and further evidence of $\mathfrak{sl}_4$ is sketched. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum $\mathfrak{sl}_m$ at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group $S_{m}$.