In this paper, we review deformation, cohomology and homotopy theories of relative Rota–Baxter ( RB $\mathsf {RB}$ ) Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an L ∞ $L_\infty$ -algebra whose Maurer–Cartan elements are algebras. Then using the twisting method, that controls deformations a algebra. Meanwhile, cohomolo...

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley–Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices such that M1,M2 ∈ L =⇒ M1M2 − M2M1 ∈ L. Two matrix Lie algebras are conjugate if there is an invertible matrix M such that L1 = ML...

We prove that (1) for any complete lattice $L$, the set $\mathcal {D}(L)$ of all non-empty saturated compact subsets Scott space $L$ is a Heyting algebra (with reverse inclusion order); and (2) if latti

The breadth of a Lie algebra $L$ is defined to be the maximal dimension image $ad_x=[x,-]:L\to L$, for $x\in L$. Here, we initiate an investigation into three families algebras by posets and provide combinatorial formulas members each family.

Let G be a group, F a field of prime characteristic p and V a finite-dimensional FGmodule. Let L(V ) denote the free Lie algebra on V regarded as an FG-submodule of the free associative algebra (or tensor algebra) T (V ). For each positive integer r, let Lr(V ) and T r(V ) be the rth homogeneous components of L(V ) and T (V ), respectively. Here Lr(V ) is called the rth Lie power of V . Our mai...

Let A be a Banach algebra. We call a pair (G,A) a Gelfand theory for A if the following axioms are satisfied: (G 1) A is a C∗-algebra, and G : A → A is a homomorphism; (G 2) the assignment L 7→ G−1(L) is a bijection between the sets of maximal modular left ideals of A and A, respectively; (G 3) for each maximal modular left ideal L of A, the linear map GL : A/G−1(L) → A/L induced by G has dense...

We study a W -algebra of central charge 2(k − 1)/(k + 2), k = 2, 3, . . . contained in the commutant of a Heisenberg algebra in a simple affine vertex operator algebra L(k, 0) of type A (1) 1 with level k. We calculate the operator product expansions of the W -algebra. We also calculate some singular vectors in the case k ≤ 6 and determine the irreducible modules and Zhu’s algebra. Furthermore,...

and the norm of an element is defined by ||/|| = ƒ | f(t) \ dt. In [4] Rudin showed that every function in L(R) is the convolution of two other functions. In other words, every element of the convolution algebra L(R) can be factored in L 1 ^ )» although this algebra lacks a unit. Subsequently, Cohen [ l ] observed that the essential ingredient in Rudin's argument is that ^(R) has an approximate...

Let C denote the algebra of functions continuous on the unit circle. With norm ||/¡| =sup¡xi-i \f(X)\, C is a Banach algebra. Let A denote the set of all / in C which are boundary values of functions analytic in \z\ <1 and continuous in \z\ ál. A is then a closed subalgebra of C, and by known results A consists of those and only those/ in C for which /|X|-l/(X)XBdX = 0, «^0. In [l ] the questio...

in this paper, we introduce the notion of multiplier in -algebra and study relationships between multipliers and some special mappings, likeness closure operators, homomorphisms and ( -derivations in -algebras. we introduce the concept of idempotent multipliers in bl-algebra and weak congruence and obtain an interconnection between idempotent multipliers and weak congruences. also, we introduce...