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.

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.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

scholarly journals Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 601
Author(s):  
Orest Artemovych ◽  
Alexander Balinsky ◽  
Denis Blackmore ◽  
Anatolij Prykarpatski
Keyword(s):  
Dynamical Systems ◽  
Lie Algebras ◽  
Algebraic Structures ◽  
Loop Algebras ◽  
Type Symmetry ◽  
Novikov Algebras ◽  
Algebra Theory ◽  
Hamiltonian Operators ◽  
Adjoint Space ◽  
Algebraic Scheme

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

Summary of Elements of Algebraic Theory

1995 ◽  
Author(s):  
F. Iachello ◽  
R. D. Levine
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Structure ◽  
Binary Operation ◽  
Algebraic Theory ◽  
Quantum Mechanical ◽  
Ordinary Quantum

Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the essential concepts of Lie algebras. The binary operation (“multiplication”) in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, B] = AB - BA. A set of operators {X} is a Lie algebra when it is closed under commutation.

Noetherian Tensor Products

1972 ◽  
Vol 15 (2) ◽  
pp. 235-238
Author(s):  
E. A. Magarian ◽  
J. L. Motto
Keyword(s):  
Tensor Product ◽  
Jacobson Radical ◽  
Tensor Products ◽  
Minimum Condition ◽  
Ideal Structure ◽  
The Ideal

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

scholarly journals Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1737
Author(s):  
Mariia Myronova ◽  
Jiří Patera ◽  
Marzena Szajewska
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Coxeter Groups ◽  
Reflection Groups ◽  
Geometrical Structures ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Branching Rules ◽  
Definition Of ◽  
Representation Orbit

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

scholarly journals UNDECIDABILITY OF THE FIRST ORDER THEORIES OF FREE NONCOMMUTATIVE LIE ALGEBRAS

2018 ◽  
Vol 83 (3) ◽  
pp. 1204-1216 ◽  
Author(s):  
OLGA KHARLAMPOVICH ◽  
ALEXEI MYASNIKOV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Model Theory ◽  
Characteristic Zero ◽  
Elementary Theory ◽  
Independence Property ◽  
First Order ◽  
Free Lie Algebras ◽  
Image Position ◽  
Do So

AbstractLet R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L, viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left( L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left( L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.

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