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Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagovi\'c the fixed-point subalgebra $\mathfrak{k} =…

Let g be a simple Lie algebra, with fixed Borel subalgebra b and with Weyl group W. Expanding on previous work of Fan and Stembridge in the simply laced case, this note aims to study the fully commutative elements of W, and their…

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv…

This paper is a continuation of the paper entitled "Subshifts, $\lambda$-graph bisystems and $C^*$-algebras", arXiv:1904.06464. A $\lambda$-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying certain compatibility…

We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is…

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z(G)}$ be the center of $\mathcal{G}$. Suppose that ${\mathfrak q}\colon \mathcal{G}\times \mathcal{G}\longrightarrow \mathcal{G}$ is an…

We say that a hypercomplex nilpotent Lie algebra is $\mathbb{H}$-solvable if there exists a sequence of $\mathbb{H}$-invariant subalgebras $\mathfrak{g}_1^{ \mathbb{H}}\supset\mathfrak{g}_2^{…

Let $G$ be a linear connected non-compact real simple Lie group and let $K\subset G$ be a maximal compact subgroup of $G$. Suppose that the centre of $K$ isomorphic to $\mathbb{S}^1$ so that $G/K$ is a global Hermitian symmetric space. Let…

We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the…

A Lie algebra is said to be quadratic if it admits a symmetric invariant and non-degenerated bilinear form. Semisimple algebras with the Killing form are examples of these algebras, while orthogonal subspaces provide abelian quadatric…

We reexamine different examples of reduction chains $\mathfrak{g} \supset \mathfrak{g}'$ of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra $\mathfrak{g}'$ leads to polynomial…

This work was inspired by two natural questions. The first question is when Lie(G')=Lie(G)', where G is a connected algebraic supergroup defined over a field of characteristic zero. The second question is whether the unipotent radical of…

Let $\mathfrak q$ be a finite-dimensional Lie algebra, $\vartheta\in Aut(\mathfrak q)$ a finite order automorphism, and $\mathfrak q_0$ the subalgebra of fixed points of $\vartheta$. Using $\vartheta$ one can construct a pencil $\mathcal P$…

For any Lie algebra of classical type or type $G_2$ we define a $K$-theoretic analog of Dunkl's elements, the so-called truncated {\it Ruijsenaars-Schneider-Macdonald elements}, $RSM$-elements for short, in the corresponding {\it…

An arbitrary proper parabolic subalgebra ${\mathfrak p}$ of a simple complex Lie algebra ${\mathfrak g}$ induces an embedding ${\mathfrak g}\hookrightarrow \mathbb W_n$, and more generally an embedding ${\mathfrak g}\hookrightarrow \mathbb…

We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that `self-commutes', we refer to such Lie algebroids as inner Q-algebroids: these provide natural examples of suitably "superised" Q-algebroids in…

We introduce parabolic presentations of twisted Yangians of types AI and AII, interpolating between the R-matrix presentation and the Drinfeld presentation. Then we formulate and provide parabolic presentations for the shifted twisted…

Let $\Gamma=(\mathcal{V},\mathcal{E})$ be a graph, whose vertices $v\in \mathcal{V}$ are colored black and white and labeled with invertible elements $\lambda_v$ from a commutative and associative ring $R$ containing $\pm 1$. Then we…

On a (pseudo-)Riemannian manifold (MM,g), some fields of endomorphisms i.e. sections of End(TMM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra,…

Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite dimensional real graded Lie algebra, with a Euclidian metric \langle \cdot , \cdot \rangle adapted to the gradation. The metric \langle\cdot , \cdot \rangle is…