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Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$. It is conjectured that for each Dynkin node $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal. In this…

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and…

We develop a theory of bicrystalline ideals, synthesizing Gr\"obner basis techniques and Kashiwara's crystal theory. This provides a unified algebraic, combinatorial, and computational approach that applies to ideals of interest, old and…

We prove that each real semisimple Lie algebra G has a Q-form, such that every real representation of G can be realized over the rational numbers Q. This was previously proved by M.S.Raghunathan (and rediscovered by P.Eberlein) in the…

For each $n\geqslant3$, we construct an uncountable family of models of the crystal of the basic $U_q(\hat{\mathfrak{sl}}_n)$-module. These models are all based on partitions, and include the usual $n$-regular and $n$-restricted models, as…

We compute the Balmer spectrum of the category of perfect complexes on an algebraic stack admitting a finite locally free cover by an affine scheme and identify it with the homogeneous spectrum of the cohomology ring.

The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general…

Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way…

Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…

We provide a detailed analysis of atomic *-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic…

An infinite crystal can be constructed by an infinite number of parallel two-dimensional (hkl) crystal planes coupled to each other. For crystals with negligible spin-orbit coupling, we report a rigorous proof of a criterion on the…

We study the existence of post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent,…

We define a class of algebras describing links of binary isolating formulas on a set of realizations for a family of 1-types of a complete theory. We prove that a set of labels for binary isolating formulas on a set of realizations for a…

We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated.…

A certain Diophantine problem and 2D crystallography are linked through the notion of standard realizations which was introduced originally in the study of random walks. In the discussion, a complex projective quadric defined over Q is…

We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.

We suggest that a certain one-to-one parametrization of completely positive maps on the matrix algebra might be useful in the study of quantum channels. This is illustrated in the case of binary quantum channels. While the algorithm is…

Explicit formulas of the universal $R$-matrix are given for all quantized nontwisted rank 3 affine Lie algebras $U_q(A_2^{(1)})\,,~U_q(C_2^{(1)})$ and $U_q(G_2^{(1)})$.

Colored planar rook algebra is a semigroup algebra in which the basis element has a diagrammatic description. The category of finite dimensional modules over this algebra is completely reducible and suitable functors are defined on this…

A class of periodic soliton cellular automata is introduced associated with crystals of non-exceptional quantum affine algebras. Based on the Bethe ansatz at q=0, we propose explicit formulas for the dynamical period and the size of certain…