THE MULTIPLIER AND THE COVER OF DIRECT SUMS OF LIE ALGEBRAS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250026 ◽  
Author(s):  
Ali Reza Salemkar ◽  
Behrouz Edalatzadeh
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Direct Sums ◽  
Schur Multipliers ◽  
Direct Sum ◽  
Group Case

In this paper, we prove that the Schur multiplier of the direct sum of two arbitrary Lie algebras is isomorphic to the direct sum of the Schur multipliers of the direct factors and the usual tensor product of the Lie algebras, which is similar to the work of Miller (1952) in the group case. Also, a cover for the direct sum of two Lie algebras in terms of given covers of them will be constructed.

SOME PROPERTIES ON ISOCLINISM OF LIE ALGEBRAS AND COVERS

2008 ◽  
Vol 07 (04) ◽  
pp. 507-516 ◽  
Author(s):  
ALI REZA SALEMKAR ◽  
HADI BIGDELY ◽  
VAHID ALAMIAN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Direct Sums ◽  
Schur Multipliers ◽  
Finite Dimensional ◽  
Equivalent Conditions

In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.

INTEGRABLE COUPLING SYSTEMS OF HAMILTONIAN LATTICE EQUATIONS BY SEMI-DIRECT SUMS OF LIE ALGEBRAS

2010 ◽  
Vol 24 (07) ◽  
pp. 681-694
Author(s):  
LI-LI ZHU ◽  
JUN DU ◽  
XIAO-YAN MA ◽  
SHENG-JU SANG
Keyword(s):  
Eigenvalue Problem ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Liouville Integrability ◽  
Direct Sums ◽  
Direct Sum ◽  
Hamiltonian Lattice ◽  
Integrable Couplings ◽  
Type Lattice ◽  
Integrable Coupling

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

scholarly journals The identity of certain representation algebra decompositions

1970 ◽  
Vol 3 (1) ◽  
pp. 73-74
Author(s):  
S. B. Conlon ◽  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Linear Algebra ◽  
Residue Field ◽  
Complex Field ◽  
Direct Sums ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
A Finite Group ◽  
Representation Algebra

Let G be a finite group and F a complete local noetherian commutative ring with residue field of characteristic p # 0. Let A(G) denote the representation algebra of G with respect to F. This is a linear algebra over the complex field whose basis elements are the isomorphism-classes of indecomposable finitely generated FG-representation modules, with addition and multiplication induced by direct sum and tensor product respectively. The two authors have separately found decompositions of A(G) as direct sums of subalgebras. In this note we show that the decompositions in one case have a common refinement given in the other's paper.

scholarly journals A non-abelian tensor product of Lie algebras

1991 ◽  
Vol 33 (1) ◽  
pp. 101-120 ◽  
Author(s):  
Graham J. Ellis
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Algebraic Theory ◽  
Tensor Products ◽  
Algebraic Structures ◽  
Commutative Algebras ◽  
Group Case ◽  
Do So ◽  
Product Of Groups

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

ON SCHUR MULTIPLIERS OF LIE ALGEBRAS AND GROUPS OF MAXIMAL CLASS

2010 ◽  
Vol 20 (06) ◽  
pp. 807-821 ◽  
Author(s):  
LINDSEY R. BOSKO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Additional Property ◽  
Maximal Class ◽  
Schur Multiplier ◽  
Schur Multipliers

The Lie algebra analogue to the Schur multiplier has been investigated in a number of recent articles. We consider the multipliers of Lie algebras of maximal class, classifying these algebras with a certain additional property. The classification leads to a conjecture about a bound on the dimension of the multiplier for each of these algebras and also for p-groups of maximal class. The conjectures are then shown to hold.

Some inequalities for the multiplier of a pair of n-Lie algebras

2019 ◽  
Vol 12 (02) ◽  
pp. 1950028
Author(s):  
Azam K. Mousavi ◽  
Mohammad Reza R. Moghaddam ◽  
Mehdi Eshrati
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Sufficient Condition ◽  
Schur Multiplier ◽  
Necessary And Sufficient Condition ◽  
Schur Multipliers ◽  
Finite Dimensional ◽  
Necessary And Sufficient

Let [Formula: see text] be a pair of [Formula: see text]-Lie algebras. Then, we introduce the concept of Schur multiplier of the pair [Formula: see text], denoted by [Formula: see text], and some inequalities for the dimension of [Formula: see text] are given. We also determine a necessary and sufficient condition, for which the Schur multiplier of a pair of [Formula: see text]-Lie algebras can be embedded into the Schur multipliers of their [Formula: see text]-Lie algebra factors. Moreover, some inequalities for the Schur multiplier of a pair of finite-dimensional nilpotent [Formula: see text]-Lie algebras are acquired.

scholarly journals Fusion frames and g-frames in tensor product and direct sum of Hilbert spaces

2012 ◽  
Vol 6 (2) ◽  
pp. 287-303
Author(s):  
Amir Khosravi ◽  
Azandaryani Mirzaee
Keyword(s):  
Tensor Product ◽  
Finite Number ◽  
Hilbert Spaces ◽  
Product Space ◽  
Tensor Products ◽  
Riesz Bases ◽  
Direct Sums ◽  
Fusion Frames ◽  
Direct Sum ◽  
Tensor Product Space

In this paper we study fusion frames and g-frames for the tensor products and direct sums of Hilbert spaces. We show that the tensor product of a finite number of g-frames (resp. fusion frames, g-Riesz bases) is a g-frame (resp. fusion frame, g-Riesz basis) for the tensor product space and vice versa. Moreover we obtain some important results in tensor products and direct sums of g-frames, fusion frames, resolutions of the identity and duals.

scholarly journals The Schur multipliers of Lie algebras of maximal class

2019 ◽  
Vol 29 (05) ◽  
pp. 795-801
Author(s):  
Afsaneh Shamsaki ◽  
Peyman Niroomand
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Class ◽  
Schur Multiplier ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Schur Multipliers ◽  
Dimension Formula

Let [Formula: see text] be a non-abelian nilpotent Lie algebra of dimension [Formula: see text] and [Formula: see text] be its Schur multiplier. It was proved by the second author the dimension of the Schur multiplier is equal to [Formula: see text] for some [Formula: see text]. In this paper, we classify all nilpotent Lie algebras of maximal class for [Formula: see text]. The dimension of Schur multiplier of such Lie algebras is also bounded by [Formula: see text]. Here, we give the structure of all nilpotent Lie algebras of maximal class [Formula: see text] when [Formula: see text] and then we show that all of them are capable.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

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