Derivations Of Algebras Research Articles

A simple algebraic derivation of the equation for a line of best fit

Abstract We present a simple algebraic derivation of the equation for a line of best fit. Unlike conventional derivations, which require multi-variable calculus, our argument relies on elementary algebra alone, and avoids more advanced notation (such as the sigma summation symbol) by exploiting routine data averages. In this way, our approach demystifies the process of computing a linear regression, and is likely to be of interest to students and instructors alike. We illustrate the main results with a worked example taken from an introductory physics context.

Description of Local Derivations on Jordan Algebras of Dimension Five

In the present paper we investigate local derivations on finite dimensional Jordan algebras. The Gleason--Kahane--\.{Z}elazko theorem, which is a fundamental contribution in the theory of Banach algebras, asserts that every unital linear functional $F$ on a complex unital Banach algebra $A$, such that $F(a)$ belongs to the spectrum $\sigma(a)$ of $a$ for every $a\in A,$ is multiplicative. In modern terminology this is equivalent to the following condition: every unital linear local homomorphism from a unital complex Banach algebra $A$ into ${\Bbb C}$ is multiplicative. We recall that a linear map $T$ from a Banach algebra $A$ into a~Banach algebra $B$ is said to be a~local homomorphism if, for every $a$ in $A$, there exists a homomorphism $\Phi_a : A\to B$, depending on $a$, such that $T(a)=\Phi_a(a)$. A similar notion was introduced and studied to give a characterization of derivations on operator algebras. Namely, the concept of local derivations was introduced by R.~Kadison and D.~Larson, A.~Sourour independently in 1990. R.~Kadison gave the description of all continuous local derivations from a von Neumann algebra into its dual Banach bemodule. B. Jonson extends the result of R. Kadison by proving that every local derivation from a $C^*$-algebra into its Banach bimodule is a derivation. It is known that, every local derivation on a JB-algebra is a derivation. In particular, every local derivation on a finite dimensional semisimple Jordan algebra is a derivation. In the present paper we investigate derivations and local derivations on five-dimensional nilpotent non-associative Jordan algebras. The description of local derivations of nilpotent Jordan algebras is an open problem. In~the~present paper we give the description of local derivations on five-dimensional nilpotent non-associative Jordan algebras over an algebraically closed field of characteristic $\neq 2$, $3$. We also give a~criterion of a~linear operator on Jordan algebras of dimension five to be a local derivation.

Extended derivations of algebras

We study the concept of extended derivations of algebras which expands diverse definitions of generalized derivations given in the literature. We concentrate on the family of the anti-commutative algebras and classify such spaces of derivations by considering a suitable notion of equivalence when we restrict attention to sucssh algebras. Afterwards, we investigate a link between the well-known ( α , β , γ ) -derivations and some new extended derivations obtained in our classification. Finally, we present applications of extended derivations to study degenerations of algebras.

A $\delta$-first Whitehead Lemma for Jordan algebras

We compute $\delta$-derivations of simple Jordan algebras with values in irreducible bimodules. They turn out to be either ordinary derivations ($\delta = 1$), or scalar multiples of the identity map ($\delta = \frac 12$). This can be considered as a generalization of the "First Whitehead Lemma" for Jordan algebras which claims that all such ordinary derivations are inner. The proof amounts to simple calculations in matrix algebras, or, in the case of Jordan algebras of a symmetric bilinear form, to more elaborated calculations in Clifford algebras.

Derivations of non-commutative group algebras

In this paper, we study the derivations of group algebras of some important groups, namely, Dihedral ([Formula: see text]), Dicyclic ([Formula: see text]) and Semi-dihedral ([Formula: see text]). First, we explicitly classify all inner derivations of a group algebra [Formula: see text] of a finite group [Formula: see text] over an arbitrary field [Formula: see text]. Then we classify all [Formula: see text]-derivations of the group algebras [Formula: see text], [Formula: see text] and [Formula: see text] when [Formula: see text] is a field of characteristic [Formula: see text] or an odd rational prime [Formula: see text] by giving the dimension and an explicit basis of these derivation algebras. We explicitly describe all inner derivations of these group algebras over an arbitrary field. Finally, we classify all derivations of the above group algebras when [Formula: see text] is an algebraic extension of a prime field.

Additive derivations of incidence algebras

Additive derivations of incidence algebras

Exceptional Periodicity and Magic Star algebras: Generalized roots, gradings, Hermitian Vinberg T-algebras and their derivations

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

On the Structure of the Algebra of Derivations for Some Low-Dimensional Leibniz Algebras

On the Structure of the Algebra of Derivations for Some Low-Dimensional Leibniz Algebras

PENGEMBANGAN E-MODUL BERBANTUAN FLIPBOOK PADA MATERI APLIKASI TURUNAN FUNGSI ALJABAR KELAS XI SMAN 8 LUBUKLINGGAU

The aim of this research is to determine the validity, practicality and effectiveness of the Flipbook-Assisted E-Module on the Algebra Function Derivative Application Material for Class XI IPA 2 SMA Negeri 8 Lubuklinggau. The research used is RD using the ADDIE model. Data collection techniques are interviews, observation, questionnaires and tests. The research subjects were students of class XI Science 2 at SMA Negeri 8 Lubuklinggau.It is-assisted moduleflipbook what was developed was declared valid, this was based on a language validator of 0.86, a media validator of 0.94 and a material validator of 0.95. The results of the validity test recapitulation show a high score with an average of 0.91. The teacher's practicality result is 4.85, testone to one evaluation was 4.52, small group practicality was 4.31, it was concluded thatIt is-assisted moduleflipbook in the application material for algebraic function derivatives, it shows an average score of 4.56 and is said to be practical. Effectiveness based on scoreN-gain amounting to 0.71 in the high category. therefore it is concluded thatIt is-The flipbook-assisted module on the application of algebraic function derivatives developed is valid, practical and effective.

The Connection between Der(Uq+(g)) and Der(Ur,s+(g))

To facilitate the parallel development of the structures and properties of two-parameter quantum groups with those of one-parameter quantum groups, this paper primarily elucidates the interrelations and distinctions between the derivation algebras of these two types of quantum groups. Additionally, we also proposed a method for deriving the derivation algebra of one-parameter quantum groups from known two-parameter quantum group derivations, and vice versa.

Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras U T 2 ( W λ ) UT_2(W_\lambda ) or E n d ( W μ ) End(W_\mu ) for some integral dominant weight λ , μ \lambda ,\mu with μ ≠ 0 \mu \neq 0 . In the special case L = s l 2 L=\mathfrak {sl}_2 we prove that this is a sufficient condition too.

Algebraic formulas and geometric derivation of source identities

Source identities are fundamental identities between multivariable special functions. We give a geometric derivation of rational and trigonometric source identities. We also give a systematic derivation and extension of various determinant representations for source functions which appeared in previous literature as well as introducing the elliptic version of the determinants, and obtain identities between determinants. We also show several symmetrization formulas for the rational version.

On Schur algebras and derivations of free Lie algebras

We investigate the action of Schur algebra on the Lie algebras of derivations of free Lie algebras and operad structures constructed from it. We also show that the Lie algebra of derivations is generated by quadratic derivations together with the action of the Schur operad. Applications to certain subgroups of the automorphism group of a finitely generated free group are given as well.

Realization of Lie algebras of derivations and moduli spaces of some rational homotopy types

Realization of Lie algebras of derivations and moduli spaces of some rational homotopy types

Inverse parameter identification and model updating for Cross-laminated Timber substructures

Finite Element (FE) model updating is crucial for identifying key parameters in structural design and improving predictive accuracy. Despite extensive research on advanced FE procedures approved for user applications, persistent disparities remain in real-world scenarios, especially for complex materials like wood. Capturing accurate mechanical characteristics with traditional models poses challenges in sustainability projects. This study introduces a derivative-free model updating procedure using a Single-Objective Optimisation (SOO) incorporating observed and predicted natural frequencies and vibration modes. The objective function optimises tuning parameters to minimise discrepancies between predicted and observed outcomes. The focus is on Cross-laminated Timber (CLT), a composite wooden structure gaining traction as a sustainable alternative to materials like reinforced concrete and steel. However, the mechanical properties of CLT can vary due to inherent variability in wood’s mechanical characteristics. This research identifies sensitive mechanical properties — longitudinal Young’s modulus, internal shear moduli, and rolling shear modulus of CLT — using a model updating procedure based on a comprehensive set of data from Experimental Modal Analysis (EMA). The study provides mathematical algebraic derivations of the updating procedure and a step-by-step implementation algorithm to facilitate practical application in structural engineering.

A purely algebraic derivation of associated Laguerre polynomials

We present a fully algebraic derivation of the Laguerre polynomials. The derivation is based on the knowledge of the energy eigenvectors of quantum mechanics solution of hydrogen-like atom Schrödinger equation, and a suitable translation operator. The method is purely algebraic since it does not require any analytical calculations.

Generalized derivation on semiprime and prime Banach algebras

Abstract In this study, the commutativity conditions with generalized derivations, which have not been examined before, are discussed. Under these conditions, the subject of generalized derivations in (semi)prime Banach algebras is studied. New fundamental results will be provided for researchers in this field and generalize some of the results found in the literature.

Generalized derivations of current Lie algebras

Let L be a Lie algebra and let A be an associative commutative algebra with unity, both over the same field F. We consider the following question. Is every generalized derivation (resp. quasiderivation) of L ⊗ A the sum of a derivation and a map from the centroid of L ⊗ A , if the same holds true for L?

The associative algebra of derivations of a group algebra

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group [Formula: see text] over a finite field [Formula: see text] are determined such that the set of derivations of the group algebra form an associative [Formula: see text]-algebra. The derivations of [Formula: see text] form a nontrivial associative [Formula: see text]-algebra if and only if [Formula: see text] has characteristic 2 and [Formula: see text] is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting [Formula: see text]-algebra is determined.

Non-Nilpotent Leibniz Algebras with One-Dimensional Derived Subalgebra

In this paper we study non-nilpotent non-Lie Leibniz F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}$$\\end{document}-algebras with one-dimensional derived subalgebra, where F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}$$\\end{document} is a field with char(F)≠2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {char}}(\\mathbb {F}) \ e 2$$\\end{document}. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document}, where n=dimFLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=\\dim _\\mathbb {F}L_n$$\\end{document}. This generalizes the result found in Demir et al. (Algebras and Representation Theory 19:405-417, 2016), which is only valid when F=C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}=\\mathbb {C}$$\\end{document}. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document}. Eventually, we solve the coquecigrue problem for Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document} by integrating it into a Lie rack.