151 research outputs found
A characterisation of Lie algebras via algebraic exponentiation
In this article we describe varieties of Lie algebras via algebraic
exponentiation, a concept introduced by Gray in his Ph.D. thesis. For
an infinite field of characteristic different from , we prove
that the variety of Lie algebras over is the only variety of
non-associative -algebras which is a non-abelian locally
algebraically cartesian closed (LACC) category. More generally, a variety of
-algebras is a non-abelian (LACC) category if and only if
and . In characteristic the
situation is similar, but here we have to treat the identities and
separately, since each of them gives rise to a variety of
non-associative -algebras which is a non-abelian (LACC) category.Comment: The ancillary files contain the code used in the proofs. Final
version to appear in Advances in Mathematic
Computing topological zeta functions of groups, algebras, and modules, II
Building on our previous work (arXiv:1405.5711), we develop the first
practical algorithm for computing topological zeta functions of nilpotent
groups, non-associative algebras, and modules. While we previously depended
upon non-degeneracy assumptions, the theory developed here allows us to
overcome these restrictions in various interesting cases.Comment: 33 pages; sequel to arXiv:1405.571
Square root of a multivector in 3D Clifford algebras
The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl3;0, Cl2;1, Cl1;2 and Cl0;3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved. 
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