We will give a generalized framework of derived bracket construction. It will be shown that a deformation differential provides a strong homotopy (sh) Leibniz algebra structure by derived bracket construction. A relationship between the three concepts, homotopy algebra theory, deformation theory and derived bracket construction, will be discussed. We will prove that the derived bracket construc...

A triangular matrix algebra over a field k is defined by a triplet (R, S, M) where R and S are k-algebras and RMS is an SR-bimodule. We show that if R, S and M are finite dimensional and the global dimensions of R and S are finite, then the triangular matrix algebra corresponding to (R, S, M) is derived equivalent to the one corresponding to (S, R, DM), where DM = Homk(M, k) is the dual of M , ...

In this paper, first we introduce the notion of a nonabelian embedding tensor, which is generalization an tensor. Then Leibniz-Lie algebra, underlying algebraic structure and can also be viewed as Leibniz algebra. Next using derived bracket, construct differential graded Lie whose Maurer-Cartan elements are exactly tensors. Consequently, obtain algebra that governs deformations Finally, define ...

The diagonalization of the Heun–Racah operator is studied with help modified algebraic Bethe ansatz. This most general bilinear expression in two generators Racah algebra. A presentation this algebra given terms dynamical operators and allows construction vectors for operator. associated equations are derived both homogeneous inhomogeneous cases.

We prove that the m-generated Grassmann algebra can be embedded into a 2 × 2 matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley–Hamilton and standard identities for n× n matrices over the m-generated Grassmann algebra are derived from this embedding. Other related embedding results are also presented.